Center for Quantum Information

Non-Gaussian evolution of high-order spin correlations characterizes important properties of quantum many-body systems. In practice, decoherence, statistical fluctuation and miscalibration of experimental parameters all hinder the witness of non-Gaussian dynamics. Here we demonstrate the crossover between Gaussian and non-Gaussian dynamics on a two-dimensional XY model with long-range and spatially structured interaction using a trapped ion quantum simulator. We prepare different initial densities of magnon excitations and verify the dynamics of single-spin observables for the engineered Hamiltonian. Then we compare the high-order spin correlations with the mean-field solution and the Holstein-Primakoff approximation, and demonstrate the non-Gaussian behavior in a way independent of the calibration errors. Our work provides a verifiable path from classically simulatable dynamics to regimes where quantum advantage may emerge.

Large ion crystals in Penning traps provide a platform for quantum simulation and sensing with hundreds of spins, but their continuous rigid-body rotation has so far limited flexible local qubit control. Here we demonstrate programmable site-selective spin control across large rotating ${}^{9}\mathrm{Be}^{+}$ crystals in a Penning trap. A tightly focused off-resonant laser beam drives local $R_z$ phase rotations via differential AC Stark shifts. Beam steering synchronised with crystal rotation enables addressing of arbitrary ions throughout the crystal. Ramsey-based characterisation shows $R_z(π)$ gate fidelity of 94.6% and nearest-neighbour crosstalk of 1.2%. We use this capability to prepare spatially structured spin patterns, generating a biskyrmion spin texture in a single-layer crystal, then extending the method to bilayer crystals we perform layer-selective addressing operations. We further demonstrate dual-quadrature Ramsey sensing by imprinting a relative $π/2$ phase shift between spatial sub-ensembles, enabling simultaneous measurement of orthogonal spin components within a single experimental realisation. These results establish programmable local control in large rotating ion crystals, opening new routes for engineering spatially structured quantum states in multidimensional trapped-ion systems.

As trapped-ion quantum computing scales to larger qubit registers and more complex control protocols, classical control systems face a fundamental tradeoff: sub-microsecond board-level feedback requires tight hardware coupling, whereas maintainability and extensibility require clean, modular software abstractions. This paper presents QuCtrl-BELL (Bell), a compiler-driven software stack for trapped-ion quantum control. The design resolves this tradeoff by decoupling control flow -- including loops, branches, and synchronization -- from hardware state data. A Python-embedded domain-specific language (DSL) is lowered through a six-stage transpilation pipeline covering control flow graph (CFG) construction, static single-assignment (SSA) conversion, liveness analysis, and graph-coloring register allocation. The compiler generates deterministic distributed board-level programs and compact step-table data. A cross-board synchronization protocol supports feedback loops with latency below 700~ns without host intervention. Bell is deployed and evaluated on the QuCtrl-BELL platform (RISC-V + PXIe), demonstrating that a compiler-based infrastructure can provide programmability, deterministic timing, and modularity for scalable trapped-ion quantum control.

We report precision spectroscopy of ionic-core transitions in alkaline-earth Rydberg atoms. We demonstrate high-resolution measurements of isotope shifts and hyperfine splitting of dipole transitions in ionic cores which have not been explored so far. A key element of this work is the reduction of the linewidth by more than two orders of magnitude enabled by dynamical control of Rydberg electron's orbit which significantly enhances the spectral resolution. Furthermore, to unambiguously identify the frequency shift, we directly compare core ion's spectrum with a signal from a single trapped ion serving as an ultimate frequency reference. This work provides an important foundation for quantum control of inner-core transitions, which offer an useful tool in manipulating Rydberg atom as well as a sensitive probe for electron-core interactions in atomic and molecular systems.

Trapped-ion quantum computers based on the quantum charge-coupled device architecture require on the order of ten trap electrodes per qubit, making the number of vacuum feedthroughs a bottleneck at the system scale. Time-division multiplexed (TDM)-based voltage control for trap electrodes provides a natural route to alleviate this constraint. However, previous studies have been limited to architectural proposals for static trap-potential compensation and room-temperature demonstrations of dynamic-electrode control, leaving cryogenic operation of TDM-based voltage control for static and dynamic electrodes experimentally unexplored. In this study, we develop and cryogenically validate TDM-based voltage control schemes for two distinct electrode classes. For static electrodes used in trap-potential compensation, we implement a 32-channel demultiplexed system operating at approximately 27~K, achieving an effective voltage update rate of 37.5~kHz with an output range of $\pm10~\mathrm{V}$ per channel. For dynamic electrodes used in ion operations, such as shuttling, we implement a four-channel demultiplexed system operating at approximately 14~K, achieving an effective voltage update rate of 1~MHz with a comparable output range. These results establish TDM-based voltage control as a practical approach for both electrode classes, providing a path for mitigating the vacuum feedthrough bottleneck in scalable trapped-ion quantum processors.

Quantum simulation with controllable many-body platforms offers a powerful route to exploring complex phases and dynamics that are difficult to access in natural materials. Among these, topological spin textures such as skyrmions are central to modern condensed-matter physics and play a key role in chiral quantum many-body systems. Their controlled realisation in large, programmable quantum platforms, however, remains an outstanding challenge. Here, we report deterministic generation and site-resolved reconstruction of topological spin textures in a two-dimensional crystal of more than 150 trapped ions. Using globally applied spin-dependent forces, we generate skyrmion configurations and reconstruct the full vector spin field with single-ion resolution, obtaining a winding number of 0.99$\pm$0.02 and a mean local fidelity of 0.87$\pm$0.04. In addition, we implement single-ion-resolved control to deterministically prepare domain-wall states, extending our approach to a broader class of non-uniform spin textures. These results establish trapped-ion crystals as a platform for engineering complex spin textures and open the door to exploring topology-dependent nonequilibrium dynamics in long-range interacting quantum systems.

Ferroelectric materials are an ideal platform for high-speed reconfigurable photonic integrated circuits (PICs) for classical and quantum photonic computations, communications, and sensing. Most reconfigurable PIC devices achieve their functionalities via interference and are therefore highly sensitive to phase errors. Under static bias, carrier drift in ferroelectric waveguides induces continuous phase drift, creating a severe bottleneck for both PIC functionality and scalability. Here we propose achieving bias-stable and scalable ferroelectric PICs by exploiting the intrinsically low carrier drift of lithium tantalate (LT). Taking one of the PIC devices that is most sensitive to phase drift, the optical phased array (OPA), as an example, we designed and fabricated an integrated LT OPA that can keep the far-field main lobe 8 dB higher than side lobes for over 4 hours, representing at least a two-order-of-magnitude improvement over the state of the art. We demonstrated our device's capability in generating arbitrary spatiotemporal waveforms with a modulation frequency as low as 0.1 Hz, leading to practical applications in optical tweezers, trapped-ion quantum computers, adaptive optics for astronomy, AR, 3D printers, LiDAR, and free-space optical communications. Beyond OPA, our work establishes LT as a bias-stable, scalable, and high-speed PIC platform for large-scale classical and quantum photonic systems.

Quantum systems promise to revolutionize information processing science and technology [1-3]. The preservation of quantum coherence, the defining property of qubits, fundamentally constrains the performance of quantum information processing with quantum memories [4]. While trapped atomic ions theoretically support million-year coherence based on spontaneous emission [5-7], experimental demonstrations have reached far less, only about an hour [8-13]. Here we combine clock-state qubits with decoherence-free subspace (DFS) encoding to achieve coherence exceeding ten hours. Using correlation-based phase tracking in 171Yb+ ion pairs sympathetically cooled by 138Ba+ ion, we demonstrate this without magnetic shielding or enhanced microwave phase stabilization that previously limited coherence times. DFS encoding references the qubit phase to the inter-ion energy difference to reject microwave phase noise and common-mode magnetic fluctuations, while clock states provide environmental insensitivity. Throughout measurements extended to 1600 seconds, we observe minimal coherence decay, with exponential fits yielding a coherence time of (3.77 +/- 1.09) x 10^4 seconds. Our results establish DFS encoding as a form of passive error correction that eliminates technical noise constraints, unlocking the million-year coherence potential of atomic ions for scalable quantum information processing.

Non-Hermitian many-body systems can be spectrally unstable, so small perturbations may induce large eigenvalue shifts. The pseudospectrum quantifies this instability and provides a perturbation-robust diagnostic. For inverse-polynomially small $ε$, we show that deciding whether a point $z\in\mathbb{C}$ is $ε$-close to the spectrum is PSPACE-hard for $5$-local operators, whereas deciding whether $z$ lies in the $ε$-pseudospectrum is QMA-complete for $4$-local operators. This identifies pseudospectrum membership as a natural computational target. We then present a concrete end-to-end quantum framework for deciding pseudospectrum membership, which combines a singular-value estimation step with a dissipative state preparation algorithm. Our Quantum Singular-value Gaussian-filtered Search (QSIGS) combines quantum singular value transformation (QSVT) with classical post-processing to achieve Heisenberg-limited query scaling for singular-value estimation. To prepare suitable input states, we introduce an algorithmic Lindbladian protocol for approximate ground right singular vectors and prove its effectiveness for the Hatano--Nelson model. Finally, we demonstrate the full pipeline on a trapped-ion quantum computer and distinguish points inside and outside the target pseudospectrum near the exceptional point of a minimal non-Hermitian qubit model.

Efficient entanglement distillation is a central task in quantum information science and future quantum networks. At the core of distillation protocols are the quantum error correction and detection schemes which enhance the fidelity of entangled pairs. Conventional protocols focus on digital systems, which typically require complicated compiled circuits, high-fidelity multi-qubit operations and delicate pulse-level control that impose high demands on near-term hardware. Crucially, the leading physical platforms for quantum networks, trapped ions and neutral atoms, are governed by native many-body Hamiltonians inherently suited for analog, continuous-time evolution. Adopting these natural dynamics is simpler than engineering digital logic via delicate pulse-level control. Motivated by this experimental reality, we seek to leverage the intrinsic analog capabilities for efficient entanglement distillation. In this work, we introduce the Hamiltonian entanglement distillation protocol, which exploits the intrinsic information scrambling generated by random time evolution under native Hamiltonians. We establish a quantitative connection between output fidelity and Out-of-Time-Order Correlators, showing that efficient scrambling directly implies good distillation performance. Since generic Hamiltonians are naturally efficient scramblers, the capability for distillation is ubiquitous: almost all Hamiltonians in the Hilbert space suffice for high-fidelity distillation. Numerical simulations of representative Rydberg-atom and trapped-ion systems further confirm that robust performance could be achieved using only short-range interactions and evolution times feasible in current experiments. By avoiding the complexity of digital circuit control, our approach substantially relaxes experimental requirements, providing a scalable route to entanglement engineering on current analog quantum platforms.

The maintenance of quantum entanglement lays the elementary building block of quantum information processing, requiring an integration of long coherence time, sufficient storage capacity, and high-fidelity entangling gates. Here we encode two-qubit entangled states into the decoherence-free subspaces (DFS) of four ions in a cryogenic trap. By crosstalk-free sympathetic cooling under dual-type encoding and multi-state detection which discards the collision-induced leakage error, we achieve a storage lifetime of about one hour for the entangled logical states. We further study the second-order DFS and show its advantage in suppressing the spatially nonuniform noise over the first-order DFS. Our work paves the way for applications of DFS quantum memories in quantum computing, quantum network and precision measurement.

Large, 3D trapped ion crystals offer improved sensitivity in quantum sensing protocols, and are expected to be implemented as platforms in near-future experiments. However, numerical techniques used to study the laser cooling of such crystals are inefficient as the number of ions, $N$, in the crystal increases. Here we develop a powerful numerical framework to simulate laser cooling of up to $10^5$ ions stored in a Penning trap. We apply this framework to characterize and optimize the cooling of ellipsoidal 3D crystals. We document new pathways to enhanced cooling based on the addition of an axial component to the potential energy-dominated $\boldsymbol{E}\times\boldsymbol{B}$ modes. Furthermore, we observe greatly enhanced cooling of the perpendicular kinetic energy to below 1 mK in prolate ion crystals, enabling a simplified cooling beam setup for such crystals. We propose specific values of trap and laser beam parameters which lead to optimal cooling in a variety of examples. This work illustrates the feasibility of preparing large 3D crystals for high-sensitivity quantum science protocols, motivating their use in future experiments.

Quantum networks, integrating quantum communication, quantum metrology, and distributed quantum computing, could provide secure and efficient information transfer, high-resolution sensing, and an exponential speed-up in information processing. Deterministic entanglement distribution over long distances is a prerequisite for scalable quantum networks, enabling the utilization of device-independent quantum key distribution (DI-QKD) and quantum teleportation to achieve secure and efficient information transfer. However, the exponential photon loss in optical fibres prohibits efficient and deterministic entanglement distribution. Quantum repeaters, incorporating entanglement swapping and entanglement purification with quantum memories, offer the most promising means to overcome this limitation in fibre-based quantum networks. Despite numerous pioneering efforts toward realizing quantum repeaters, a critical bottleneck remains, as remote memory-memory entanglement suffers from decoherence more rapidly than it can be established and purified over long distances. We overcome this by developing long-lived trapped-ion memories, an efficient telecom interface, and a high-visibility single-photon entanglement protocol. This allows us to establish and maintain memory-memory entanglement over a 10 km fibre within the average entanglement establishment time for the same distance. As a direct application, we demonstrate metropolitan-scale DI-QKD, distilling 1,917 secret keys out of 4.05*10^5 Bell pairs over 10 km. We further report a positive key rate over 101 km in the asymptotic limit, extending the achievable distance by more than two orders of magnitude. Our work provides a critical building block for quantum repeaters and marks an important step toward scalable quantum networks.

Quantum many-body systems driven far from equilibrium can exhibit chaos, entanglement, and non-classical correlations, yet directly observing these phenomena in large, closed quantum systems remains challenging. Here we realize the Dicke model -- a fundamental description of light-matter interactions -- in a two-dimensional crystal of approximately 100 trapped ions. The ions' internal state is optically coupled to the center of mass vibrational mode via an optical spin-dependent force, enabling unitary many-body dynamics beyond the mean-field and few-body limits. In the integrable regime, where the phonons can be adiabatically eliminated, we observe a dynamical phase transition between ferromagnetic to paramagnetic spin phases. In contrast, when the spins and phonons are strongly coupled, we observe clear signatures of non-integrable chaotic dynamics, including erratic phase-space trajectories and the exponential growth of excitations and entanglement quantified by the one-body Rényi entropy. By quenching from an unstable fixed point in the near-integrable regime, quantum noise can generate correlated spin-phonon excitations. Our numerical calculations, in clear agreement with experiment, reveal the generation of two-mode spin-phonon squeezing, 2.6 dB below the standard quantum limit (4.6 dB relative to the initial thermal state), followed by generalized vacuum Rabi collapses and revivals. Our results establish large ion crystals as scalable analog quantum simulators of non-equilibrium light-matter dynamics and provide a controlled platform for experimental studies of information scrambling and entanglement in closed many-body systems.

Transport phenomena play a key role in a variety of application domains, and efficient simulation of these dynamics remains an outstanding challenge. While quantum computers offer potential for significant speedups, existing algorithms either lack rigorous theoretical guarantees or demand substantial quantum resources, preventing scalable and efficient validation on realistic quantum hardware. To address this gap, we develop a comprehensive framework for simulating classes of transport equations, offering both rigorous theoretical guarantees -- including exponential speedups in specific cases -- and a systematic, hardware-efficient implementation. Central to our approach is the Hamiltonian embedding technique, a white-box approach for end-to-end simulation of sparse Hamiltonians that avoids abstract query models and retains near-optimal asymptotic complexity. Empirical resource estimates indicate that our approach can yield an order-of-magnitude (e.g., $42\times$) reduction in circuit depth given favorable problem structures. We then apply our framework to solve linear and nonlinear transport PDEs, including the first experimental demonstration of a 2D advection equation on a trapped-ion quantum computer.

We develop a unified many-body theory of two-photon dark-state laser cooling, the workhorse for preparing trapped ions close to their motional quantum ground state. For ions with a $Λ$ level structure, driven by Raman lasers, we identify an ion-number-dependent crossover between weak and strong coupling where both the cooling rate and final temperature are simultaneously optimized. We obtain simple analytic results in both extremes: In the weak coupling limit, we show a Lorentzian spin-absorption spectrum determines the cooling rate and final occupation of the motional state, which are both independent of the number of ions. We also highlight the benefit of including an additional spin dependent force in this case. In the strong coupling regime, our theory reveals the role of collective dynamics arising from phonon exchange between dark and bright states, allowing us to explain the enhancement of the cooling rate with increasing ion number. Our analytic results agree closely with exact numerical simulations and provide experimentally accessible guidelines for optimizing cooling in large ion crystals, a key step toward scalable, high-fidelity trapped-ion quantum technologies.

Fault-tolerant quantum computing requires extremely precise knowledge and control of qubit dynamics during the application of a gate. We develop a data-driven learning protocol for characterizing quantum gates that builds off previous work on learning the Nakajima-Mori-Zwanzig (NMZ) formulation of open system dynamics from time series data, which allows detailed reconstruction of quantum evolution, including non-Markovian dynamics. We demonstrate this learning technique on three different systems: a simulation of a qubit whose dynamics are purely Markovian, a simulation of a driven qubit coupled to stochastic noise produced by an Ornstein-Uhlenbeck process, and trapped-ion experimental data of a driven qubit whose noise environment is not characterized ahead of time. Our technique is able to learn the generators of time evolution, or the NMZ operators, in all three cases and can learn the timescale in which the qubit dynamics can no longer be accurately described by a purely Markovian model. Our technique complements existing quantum gate characterization methods such as gate set tomography by explicitly capturing non-Markovianity in the gate generator, thus allowing for more thorough diagnosis of noise sources.

The dual-type qubit scheme is an emerging method to suppress crosstalk errors in scalable trapped-ion quantum computation and quantum network. Here we report a hardware-economic way to control dual-type $^{171}\mathrm{Yb}^+$ qubits using a single $355\,$nm mode-locked pulsed laser. Utilizing its broad frequency comb structure, we drive the Raman transitions of both qubit types encoded in the $S_{1/2}$ and the $F_{7/2}$ hyperfine levels, and probe their carrier transitions and the motional sidebands. We further demonstrate a direct entangling gate between the two qubit types. Our work can simplify the manipulation of the $^{171}\mathrm{Yb}^+$ qubits both at the hardware and the software level.

While quantum simulators promise to explore quantum many-body physics beyond classical computation, their capabilities are limited by the available native interactions in the hardware. On many platforms, accessible Hamiltonians are largely restricted to one- and two-body interactions, limiting access to multi-body Hamiltonians and to systems governed by simultaneous, non-commuting interaction terms that are central to condensed matter, quantum chemistry, and high-energy physics. We introduce and experimentally demonstrate a hybrid digital-analog protocol that overcomes these limitations by embedding analog evolution between shallow entangling-gate layers. This method produces effective Hamiltonians with simultaneous non-commuting three- and four-body interactions that are generated non-perturbatively and without Trotter error -- capabilities not practically attainable on near-term hardware using purely digital or purely analog schemes. We implement our scheme on a trapped-ion quantum processor and use it to realize a topological spin chain exhibiting prethermal strong zero modes persisting at high temperature, as well as models featuring three- and four-body interactions. Our hardware-agnostic and scalable method opens new routes to realizing complex many-body physics across quantum platforms.

Quantum computers are believed to solve a class of computational problems that are based on modular arithmetic faster than classical computers. Among the arithmetic building blocks, comparison of integer pairs is a primitive. Here we report its demonstration in the Reimei quantum computer at RIKEN, whose trapped-ion architecture provides all-to-all qubit connectivity together with high gate fidelities. We observe high success probabilities for bit widths n = 3, 5, 7, and 9: Under a conventional output-only success criterion we obtain 95% at n=9; under a stricter criterion additionally requiring the ancilla to be correct, the success is 69% at n=9. These results demonstrate reliable quantum comparison at scales far beyond those previously achieved experimentally, not only for comparators but also in the broader context of quantum arithmetic circuits.

Quantum supremacy has been explored extensively in gate-model settings. Here, we introduce a quantum-supremacy framework for a hybrid digital-analog-digital quantum computing (DADQC) model. We consider a device that applies an initial layer of single-qubit gates, a single transverse-field Ising analog block, and a final single-qubit layer before $Z$-basis readout. The analog block approximates $Z$-diagonal Ising evolution, and we prove that the resulting output distribution is within constant total-variation (TV) distance of an Instantaneous Quantum Polynomial-time (IQP) circuit. Our bounds and constructions are established for fully connected as well as bounded-degree hardware graphs, matching a variety of architectures, including trapped-ion, neutral atom, and superconducting platforms. Assuming anticoncentration (which we prove for all-to-all hardware graphs and conjecture for bounded-degree hardware graphs) and an average-case hardness conjecture for the associated complex-temperature Ising partition functions, standard reductions imply that any efficient classical sampler achieving constant TV error collapses the polynomial hierarchy. Our results imply that quantum-supremacy tests are possible on today's quantum annealers, as well as other devices capable of hybrid digital-analog quantum evolution.

We propose the use of motional states of two interacting atoms trapped in a potential stroboscopically engineered by an optical tweezer as a means to implement a qubit-oscillator system, in analogy to those implemented in circuit quantum electrodynamics and trapped ions. In our setting, the center of mass degree of freedom of the atoms plays the role of a photon or phonon mode, while the interacting, relative mode acts as a qubit. No internal state is involved in our system, which makes this motional qubit robust to spin-dependent noise. We show that a universal set of bosonic operations, including displacement, rotation, squeezing, and the corresponding set of gates controlled by the qubit, can be implemented through precise temporal modulation of the optical tweezers. We numerically check that these gates can be generated with high fidelity, and discuss possible schemes for initial state preparation and final state readout. While we restrict the discussion to a single qubit-oscillator module, scalability can be achieved by coupling arrays of atoms via dipolar or Rydberg-dressed interactions.

Neutral atom quantum computers and to a lesser extent trapped ions may suffer from atom loss. In this work, we investigate the impact of atom loss in long chains of trapped ions. Even though this is a relatively rare event, ion loss in long chains must be addressed because it destabilizes the entire chain resulting in the loss of all the qubits of the chain. We propose a solution to the chain loss problem based on (1) a quantum error correction code distributed over multiple long chains, (2) beacon qubits within each long chain to detect the loss of a chain, and (3) a decoder adapted to correct a combination of circuit faults and erasures after beacon qubits convert chain losses into erasures. We verify the chain loss correction capability of our scheme through circuit level simulations with a distributed $[[72,12,6]]$ BB code with beacon qubits.

Displacement sensing is a fundamental task in metrology. However, the development of quantum-enhanced sensors that fully utilize the available degrees of freedom in many-body quantum systems remains an outstanding challenge. We propose novel many-body displacement sensing schemes that use spin-dependent squeezed (SDS) states -- hybrid spin-boson states whose bosonic squeezed quadrature is conditioned on an auxiliary spin. We prove that SDS states are \emph{optimal}, i.e. their quantum Cramér-Rao bound saturates the Heisenberg limit. We propose explicit measurement sequences that can be readily implemented in systems such as trapped ions. We also introduce a scalable state-preparation protocol and numerically demonstrate the preparation of $8.7$~dB of spin-dependent squeezing $15$ times faster than the standard approach using second-order sidebands in trapped ions. The potential applications of our sensing protocols range from measuring single-photon scattering to searches for dark matter.

Quantum networks and quantum repeaters represent the promising avenues for building large-scale quantum information systems, serving as foundational infrastructure for distributed quantum computing, long-distance quantum communication, and networked quantum sensing. A critical step in realizing a functional quantum network is the efficient and high-fidelity establishment of heralded entanglement between remote quantum nodes. Multiplexing offers a powerful strategy to accelerate remote entanglement distribution, particularly over long optical fibers. Here, we demonstrate the first multiplexing-enhanced heralded entanglement between two trapped-ion quantum network nodes. By multiplexing $10$ temporal photonic modes, we achieve a 4.59-fold speedup in ion-ion entanglement generation and attain an entanglement fidelity of $95.9\pm1.5\%$ over $1.2$ km of fiber. Employing a dual-type architecture, our system is readily scalable to multiple nodes, thereby establishing a key building block for future large-scale quantum networks.

Probing quantum dynamics in the strong-field regime is critical for advancing our understanding of controlled quantum systems and developing robust quantum technologies. In this work, we experimentally investigate the dynamics of a trapped ion where the Rabi frequency (Omega) approaches the vibrational mode frequency (nu), pushing the system beyond the weak-field regime, where non-trivial quantum correlations emerge. We begin by setting the detuning (delta) - the frequency offset between the qubit transition and the driving field - to zero and varying Omega from low to high values, eventually reaching the vibrational frequency. Using quantum state tomography, we reconstruct the density matrix and track its evolution to assess non-Markovianity, revealing significant memory effects governed by the interplay between internal and motional degrees of freedom. Furthermore, by exploring the dynamics across various parameter pairs (Omega, delta), we find that non-Markovianity does not always increase monotonically with Omega for a fixed delta. Strikingly, when the condition delta squared plus Omega squared equals nu squared is met, the non-Markovianity exhibits a circular pattern of maxima. At this parameter combination, the system's Hamiltonian takes a form similar to the Jaynes-Cummings model, enabling the possibility of analytical insights into the observed dynamics. These results go beyond the conventional carrier and sideband regimes, uncovering novel features of strong-field quantum dynamics. Our findings establish a pathway for using trapped-ion platforms to investigate non-Markovianity, coherent control, and the fundamental behavior of open quantum systems in extreme regimes.

We propose a measurement-based FTQC (MB-FTQC) architecture for high-connectivity platforms such as trapped ions and neutral atoms. The key idea is to use verified logical ancillas combined with Knill's error-correcting teleportation, eliminating repeated syndrome measurements and simplifying decoding to logical Pauli corrections, thus keeping classical overhead low. To align with near-term device scales, we present two implementations benchmarked under circuit-level depolarizing noise: (i) a Steane-code version that uses analog $R_Z(θ)$ rotations, akin to the STAR architecture [Akahoshi et al., PRX Quantum 5, 010337], aiming for the megaquop regime ($\sim 10^6$ $T$ gates) on devices with thousands of qubits; and (ii) a Golay-code version with higher-order zero-level magic-state distillation, targeting the gigaquop regime ($\sim 10^9$ $T$ gates) on devices with tens of thousands of qubits. At a physical error rate $p=10^{-4}$, the Steane path supports $5\times 10^{4}$ logical $R_Z(θ)$ rotations, corresponding to $\sim 2.4\times 10^{6}$ $T$ gates and enabling megaquop-scale computation. With about $2{,}240$ physical qubits, it achieves $\log_{2}\mathrm{QV}=64$. The Golay path supports more than $2\times 10^{9}$ $T$ gates, enabling gigaquop-scale computation. These results suggest that our architecture can deliver practical large-scale quantum computation on near-term high-connectivity hardware without relying on resource-intensive surface codes or complex code concatenation.

Autonomous quantum machines (AQMs) execute tasks without requiring time-dependent external control. Motivations for AQMs include the restrictions imposed by classical control on quantum machines' coherence times and geometries. Most AQM work is theoretical and abstract; yet an experiment recently demonstrated AQMs' usefulness in qubit reset, crucial to quantum computing. To further reduce quantum computing's classical control, we propose realizations of (fully and partially) quantum-autonomous gates on three platforms: Rydberg atoms, trapped ions, and superconducting qubits. First, we show that a Rydberg-blockade interaction or an ultrafast transition can quantum-autonomously effect entangling gates on Rydberg atoms. One can perform $Z$ or entangling gates on trapped ions mostly quantum-autonomously, by sculpting a linear Paul trap or leveraging a ring trap. Passive lasers control these gates, as well as the Rydberg-atom gates, quantum-autonomously. Finally, circuit quantum electrodynamics can enable quantum-autonomous $Z$ and $XY$ gates on superconducting qubits. The gates can serve as building blocks for (fully or partially) quantum-autonomous circuits, which may reduce classical-control burdens.

Simulating out-of-equilibrium dynamics of quantum field theories in nature is challenging with classical methods, but is a promising application for quantum computers. Unfortunately, simulating interacting bosonic fields involves a high boson-to-qubit encoding overhead. Furthermore, when mapping to qubits, the infinite-dimensional Hilbert space of bosons is necessarily truncated, with truncation errors that grow with energy and time. A qubit-based quantum computer, augmented with an active bosonic register, and with qubit, bosonic, and mixed qubit-boson quantum gates, offers a more powerful platform for simulating bosonic theories. We demonstrate this capability experimentally in a hybrid analog-digital trapped-ion quantum computer, where qubits are encoded in the internal states of the ions, and the bosons in the ions' motional states. Specifically, we simulate nonequilibrium dynamics of a (1+1)-dimensional Yukawa model, a simplified model of interacting nucleons and pions, and measure fermion- and boson-occupation-state probabilities. These dynamics populate high bosonic-field excitations starting from an empty state, and the experimental results capture well such high-occupation states. This simulation approaches the regime where classical methods become challenging, bypasses the need for a large qubit overhead, and removes truncation errors. Our results, therefore, open the way to achieving demonstrable quantum advantage in qubit-boson quantum computing.

Recent advances in quantum technologies have enabled quantum simulation of gauge theories -- some of the most fundamental frameworks of nature -- in regimes far from equilibrium, where classical computation is severely limited. These simulators, primarily based on neutral atoms, trapped ions, and superconducting circuits, hold the potential to address long-standing questions in nuclear, high-energy, and condensed-matter physics, and may ultimately allow first-principles studies of matter evolution in settings ranging from the early universe to high-energy collisions. Research in this rapidly growing field is also driving the convergence of concepts across disciplines and uncovering new phenomena. In this Review, we highlight recent experimental and theoretical developments, focusing on phenomena accessible in current and near-term quantum simulators, including particle production and string breaking, collision dynamics, thermalization, ergodicity breaking, and dynamical quantum phase transitions. We conclude by outlining promising directions for future research and opportunities enabled by available quantum hardware.

Optical crosstalk due to imperfect addressing in trapped-ion entangling gates generates unwanted non-local entanglement between target ions and their neighbors that is difficult to mitigate using standard quantum error correction. We present a method to design entangling operations that are inherently insensitive to crosstalk by engineering the effective qubit coupling matrix. By controlling the geometric phases generated in the motional modes of the ion string, we construct a coupling matrix that selectively excludes crosstalk-affected neighbor ions from the entangling operation. This approach requires no knowledge of the amount of crosstalk present and avoids the need for additional gate operations or modifications to the optical setup. We numerically demonstrate the construction of crosstalk-insensitive entangling pulses for target ion pairs within an equispaced 20-ion string and provide experimental validation of crosstalk-insensitive entanglement in a three-ion string.

The low-energy isomeric transition in Thorium-229 offers a unique interface between nuclear and atomic physics, presenting a resource for quantum technologies that is notably resilient to environmental decoherence. While early experiments focused on nuclei in solid-state crystals, the recent advent of a continuous-wave vacuum ultraviolet laser at 148.4~nm now enables direct coherent control of individual trapped Th-229 ions. Building on this breakthrough, we present a theoretical framework for utilizing trapped Th-229^{3+} ions as high-fidelity nuclear-level qubits, wherein quantum state preparation, single-qubit control, and entangling operations based on nuclear energy levels can all be efficiently realized. We analyze a scheme to generate entanglement between the nuclear isomeric states of two ions through phonon-mediated coupling, driven by optimized red- and blue-detuned laser sideband pulses. Our analysis, grounded in realistic experimental parameters, also demonstrates that high-fidelity entanglement is achievable, leveraging the nucleus's intrinsically long coherence times. These results provide a practical roadmap for developing nuclear-based quantum information processors and suggest that entangled nuclear-level qubits could potentially unlock new frontiers in precision metrology.

Independent control of numerous electrodes in quantum charge-coupled device architectures presents a significant challenge for wiring and hardware scalability. To address this issue, we demonstrate a voltage control method based on time-division multiplexing (TDM). This approach utilizes a single high-update-rate digital-to-analog converter (DAC) to sequentially generate control signals for multiple electrodes, thereby reducing both the number of required DACs and associated wiring. We experimentally validate this concept by developing a 10-channel system that operates with only two DACs. The developed TDM-based voltage control system is applied to a surface-electrode trap, where we successfully trap a single $^{40}\mathrm{Ca}^+$ ion and demonstrate a simple ion transport primitive. This approach offers a resource-efficient and scalable solution for advanced quantum computing systems based on trapped ions.

Three-dimensional (3D) crystals offer a route to scale up trapped ion systems for quantum sensing and quantum simulation applications. However, engineering coherent spin-motion couplings and effective spin-spin interactions in large crystals poses technical challenges associated with decoherence and prolonged timescales to generate appreciable entanglement. Here, we explore the possibility to speed up these interactions in 3D crystals via parametric amplification. We derive a general Hamiltonian for the parametric amplification of spin-motion coupling that is applicable to crystals of any dimension in both rf Paul traps and Penning traps. Unlike in lower dimensional crystals, we find that the ability to faithfully (uniformly) amplify the spin-spin interactions in 3D crystals depends on the physical implementation of the spin-motion coupling. We consider the light-shift (LS) gate, and the so-called phase-insensitive and phase-sensitive Mølmer-Sørensen (MS) gates, and find that only the latter gate can be faithfully amplified in general 3D crystals. We discuss a situation where non-uniform amplification can be advantageous. We also reconsider the impact of counter-rotating terms on parametric amplification and find that they are not as detrimental as previous studies suggest.

A quantum memory is an essential element for quantum computation, quantum network and quantum metrology. Previously, a single-qubit quantum memory with a coherence time of about an hour has been realized in a dual-species setup where a coolant ion provides sympathetic cooling for a memory ion of different species. However, the frequent random position hopping between the ions in the room-temperature trap limits the technique there only applicable to single-qubit storage. Here we report a multi-ion quantum memory in a cryogenic trap based on the dual-type scheme, and demonstrate a coherence time above two hours for a logical qubit encoded in the decoherence-free subspace, i.e. two-ion entangled states, after correcting the dominant leakage error. Our scheme alleviates the necessity of an ultra-stable frequency reference for the stored qubit, and has a preferable scalability owing to the same mass of the metastable-state memory ions and the ground-state coolant ion.

Two-dimensional planar ion crystals in a Penning trap are a platform for quantum information science experiments. However, the low-frequency planar modes of these crystals are not efficiently cooled by laser cooling, which can limit the utility of the drumhead modes for quantum information processing. Recently, it has been shown that nonlinear mode coupling can enhance the cooling of the low-frequency planar modes. Here, we demonstrate in numerical simulations that this coupling can be dynamically tuned by adiabatically changing the rotation frequency of the ion crystal during experiments. Furthermore, we show that this technique can, in addition, produce lower temperatures for the low-frequency planar modes via an adiabatic cooling process. This result allows cooling of the planar modes to sub-millikelvin temperatures, resulting in improved spectral resolution of the drumhead modes at experimentally relevant rotation frequencies, which is crucial for quantum information processing applications.

Trapped ions constitute a promising platform for implementation of a quantum network. Recently, a dual-type qubit scheme has been realized in a quantum network node where the communication qubits and the memory qubits are encoded in different energy levels of the same ion species, such that the generation of ion-photon entanglement on the communication qubits has negligible crosstalk error on the preloaded quantum information in the memory qubits. However, to achieve the versatile applications of a quantum network, a crucial component of the dual-type node, namely the entangling gate between the communication and the memory qubits, is still missing. Here we report a dual-type quantum network node equipped with ion-photon entanglement generation, crosstalk-free quantum memory and entangling gates between the dual-type qubits simultaneously. We demonstrate its practical applications including the quantum state teleportation and the preparation of multipartite entangled state. Our work achieves the necessary components of a dual-type quantum network node and paves the way toward its applications in a large-scale quantum internet.

Fock-state lattice (FSL) offers a powerful quantum simulator for topological phenomena due to the unbounded scalability and ease of implementation. Nevertheless, the unique topological properties induced by its site-dependent coupling have remained elusive, mainly due to the challenge of handling an infinite state space without translational symmetry. Here, we rigorously analyze the topological features of a semi-infinite FSL-based Su-Schrieffer-Heeger (SSH) model, in both Hermitian and non-Hermitian realms, by mapping it to the solvable Jaynes-Cummings (JC) model via a unitary displacement transformation. We find a more stable topological zero mode than the conventional SSH model, originating from the bound state at the inherent domain wall under anisotropic conditions. With gain and loss introduced, we predict a non-Hermitian bound effect (NHBE), i. e., any state overlapping with the bound state will quickly stabilize to the domain wall, with the minimal stabilization time occurring in the vicinity of exceptional point (EP). The paritytime (PT ) phase transition can be observed by the oscillating-to-steady crossover of dynamics in the subspace orthogonal to the bound state. Furthermore, a concrete experimental proposal based on the trapped-ion setup is provided.

Analog quantum simulators have provided key insights into quantum many-body dynamics. However, in such systems, both coherent and incoherent errors limit their scalability, hindering simulations in regimes that challenge classical simulations. In this work, we introduce an error mitigation technique that addresses and effectively suppresses a key source of error in leading simulator platforms: shot-to-shot fluctuations in the parameters for the Hamiltonian governing the system dynamics. We rigorously prove that amplifying this shot-to-shot noise and extrapolating to the zero-noise limit recovers noiseless results for realistic noise distributions. Experimentally, we demonstrate this technique on a 27-ion trapped-ion quantum simulator, extending the two-qubit exchange oscillation lifetime threefold. Numerically, we predict a significant enhancement in the effective many-body coherence time for Rydberg atom arrays under realistic conditions. Our scheme provides a possible route towards extending the effective coherence time in analog quantum experiments, enabling deeper explorations of quantum many-body dynamics.

Electromagnetically Induced Transparency (EIT) cooling is a well-established method for preparing trapped ion systems in their motional ground state. However, isolating a three-level system, as required for EIT cooling, is often challenging or impractical. Nonetheless, multilevel systems can inherently host dark states. In this work, we extend the EIT cooling framework to such multilevel systems. We develop a formalism to accurately determine the cooling rate in the weak sideband coupling regime and provide an approximate estimate for cooling rates beyond this regime, without the need for explicit simulation of the motional degree of freedom. We clarify the connection between the cooling rate and the absorption spectrum, offering a pathway for efficient near-ground-state cooling of ions with complex electronic structures.

The computation of excited states in strongly interacting quantum many-body systems is of fundamental importance. Yet, it is notoriously challenging due to the exponential scaling of the Hilbert space dimension with the system size. Here, we introduce a neural network-based algorithm that can simultaneously output multiple low-lying excited states of a quantum many-body spin system in an accurate and efficient fashion. This algorithm, dubbed the neural quantum excited-state (NQES) algorithm, requires no explicit orthogonalization of the states and is generally applicable to higher dimensions. We demonstrate, through concrete examples including the Haldane-Shastry model with all-to-all interactions, that the NQES algorithm is capable of efficiently computing multiple excited states and their related observable expectations. In addition, we apply the NQES algorithm to two classes of long-range interacting trapped-ion systems in a two-dimensional Wigner crystal. For non-decaying all-to-all interactions with alternating signs, our computed low-lying excited states bear spatial correlation patterns similar to those of the ground states, which closely match recent experimental observations that the quasi-adiabatically prepared state accurately reproduces analytical ground-state correlations. For a system of up to 300 ions with power-law decaying antiferromagnetic interactions, we successfully uncover its gap scaling and correlation features. Our results establish a scalable and efficient algorithm for computing excited states of interacting quantum many-body systems, which holds potential applications ranging from benchmarking quantum devices to photoisomerization.

We introduce a fault-tolerant protocol for code concatenation of a generalized Shor code using a butterfly network architecture with high noise thresholds and low ancilla overhead to allow implementation on current devices. We develop a probability passing decoder using tensor networks that applies Bayesian updates to the marginal error probabilities after each layer of checks, achieving a state preparation threshold of $e_c \approx 0.089$ for erasure errors, and $\approx 0.015$ for unheralded noise. We implement our state preparation protocol on ion-trap hardware with added noise to demonstrate the threshold behavior in a real quantum device. We further theoretically test the performance of our scheme as a quantum memory and for universal quantum computation through the preparation of low-noise magic states for state distillation and $T$-gate injection.

We present a digital quantum computation of two-hadron scattering in a $Z_2$ lattice gauge theory in 1+1 dimensions. We prepare well-separated single-particle wave packets with desired momentum-space wavefunctions, and simulate their collision through digitized time evolution. Multiple hadronic wave packets can be produced using the efficient, systematically improvable algorithm of this work, achieving high fidelity with the target initial state. Specifically, employing a trapped-ion quantum computer (IonQ Forte), we prepare up to three meson wave packets using 11 and 27 system qubits, and simulate collision dynamics of two meson wave packets for the smaller system. Results for local observables are consistent with numerical simulations at early times, but decoherence effects limit evolution into long times. We demonstrate the critical role of high-fidelity initial states for precision measurements of state-sensitive observables, such as $S$-matrix elements. Our work establishes the potential of quantum computers in simulating hadron-scattering processes in strongly interacting gauge theories.

Vibrational modes of trapped ions have traditionally served as quantum buses to mediate internal qubits. However, with recent advances in quantum control, it has become possible to use these vibrational modes directly as quantum computational resources, such as bosonic qubits. Here, we propose a dual-rail encoding scheme in which a dual-rail qubit is encoded by two vibrational modes that share a single phonon. We present the preparation, measurement, and implementation of single- and two-qubit gates, enabling universal quantum computation. The dual-rail qubit system offers scalability and all-to-all connectivity. Moreover, we extend the dual-rail qubit system to a logical internal qubit--dual-rail qubit hybrid system by incorporating internal qubits into the dual-rail qubit system as another type of logical qubit. The hybrid system nearly doubles the number of available logical qubits compared to conventional trapped-ion quantum computers while maintaining all-to-all connectivity. Additionally, we propose a method for implementing multi-qubit controlled gates and discuss potential applications that can leverage the advantages of the hybrid system. Our scheme provides a practical framework for an internal qubit-boson qubit hybrid system.

The liquid-vapor transition is a classic example of a discontinuous (first-order) phase transition. Such transitions underlie many phenomena in cosmology, nuclear and particle physics, and condensed-matter physics. They give rise to long-lived metastable states, whose decay can be driven by either thermal or quantum fluctuations. Yet, direct experimental observations of how these states collapse into a stable phase remain elusive in the quantum regime. Here, we use a trapped-ion quantum simulator to observe the real-time dynamics of ``bubble nucleation'' induced by quantum fluctuations. Bubbles are localized domains of the stable phase which spontaneously form, or nucleate, and expand as the system is driven across a discontinuous quantum phase transition. Implementing a mixed-field Ising spin model with tunable and time-dependent interactions, we track the microscopic evolution of the metastable state as the Hamiltonian parameters are varied in time with various speeds, bringing the system out of equilibrium. Site-resolved measurements reveal the emergence and evolution of finite-size quantum bubbles, providing direct insight into the mechanism by which the metastable phase decays. We also identify nonequilibrium scaling behavior near the transition, consistent with a generalized Kibble-Zurek mechanism. Our results demonstrate the power of quantum simulators to probe out-of-equilibrium many-body physics, including quantum bubble nucleation, a key feature of discontinuous quantum phase transitions, with application to studies of matter formation in the early universe.

Recent advances of quantum technologies rely on precise control and integration of quantum objects, and technological breakthrough is anticipated for further scaling up to realize practical applications. Trapped-ion quantum technology is a promising candidates to realize them, while its scalability depends on the development of intra-node scaling up, reproducibility of quantum nodes and photonic interconnection among them. Utilization of integrated photonics instead of free-space optics is a crucial step toward mass production of trapped-ion quantum nodes and manifests itself as useful for laser delivery for various quantum operations and photon detection. However, whole architecture of the scalable photonic circuits for them is left unexplored. In this work, we discuss photonic architectures for trapped-ion quantum devices, in which lasers of multiple wavelengths are delivered to multiple trapping zones within a single chip. We analyze two methods of configuring nanophotonic waveguides and compare them in terms of loss of total laser power. This work opens up a new landscape of photonic architecture for various applications not only to quantum ones such as trapped-ion devices but also to classical photonics, e.g. optoelectronic devices and biochemical sensing.

Practical knowledge about troubleshooting and error handling in trapped-ion systems remains largely undocumented and held within individual labs, creating a barrier to cross-disciplinary collaboration towards engineering scalable systems. This paper presents a structured troubleshooting framework for trapping ions, developed through hands-on experience in the lab. The framework categorizes standard failure modes across subsystems -- vacuum, electronics, optics, and imaging -- and organizes them into a modular decision-tree structure. Each troubleshooting action is annotated with estimates of cost and operational risk inspired by the principles of Failure Mode and Effects Analysis (FMEA). By categorizing failure modes and their associated costs, this work bridges the gap between physicists and engineers, enabling collaborative system design and setting the stage for turning experimental setups into robust, reproducible products. This framework provides a structured basis for future error-handled ion-trap systems, designed with diagnostics and maintainability in mind -- for use in distributed quantum computing and networks.

Estimating partition functions of Ising spin glasses is a cornerstone of statistical physics and computational science, yet it remains classically challenging due to its $\#$P-hard complexity. While Jarzynski's equality offers a theoretical pathway, its practical application is crippled at low temperatures by rare, divergent statistical fluctuations. Here, we introduce a quantum protocol that overcomes this fundamental limitation by synergizing reverse quantum annealing with optimized nonequilibrium initial distributions. Our method dramatically suppresses the estimator variance, achieving saturation in the low-temperature regime where existing methods fail. Numerical benchmarks on the Sherrington-Kirkpatrick spin glass and the 3-SAT problem demonstrate that our protocol reduces computational scaling exponents by over an order of magnitude (e.g., from $\sim 8.5$ to $\sim 0.5$), despite retaining exponential system-size dependence. Crucially, our protocol circumvents stringent adiabatic constraints, making it feasible for near-term quantum devices like superconducting qubits, trapped ions, and Rydberg atom arrays. This work provides a methodological framework for quantum-enhanced estimation in spin glass thermodynamics and beyond by harnessing non-adiabatic quantum dynamics to address a classically difficult problem.

Trapped-ion qubits are among the most promising candidates for quantum computing, quantum information processing, and quantum simulation. In general, trapped ions are considered to have sufficiently long coherence times, which are mainly characterized under laser-free conditions. However, in reality, essential laser fields for quantum manipulation introduce residual light shift, which seriously degrades the coherence due to power fluctuations. Here, we present a comprehensive study of AC Stark shifts in the hyperfine energy levels of the $^{171}\mathrm{Yb}^+$ ion, revealing an asymmetric light shift between two circular polarizations in the clock qubit and pronounced vector light shifts in the Zeeman qubits. By precisely tuning these polarizations, a remarkable enhancement in coherence time is observed, reaching over a hundredfold for the clock qubit and more than tenfold for the Zeeman qubits, when comparing conditions of maximum and minimum shifts. These findings advance the practical realization of scalable trapped-ion quantum processors, enabling deep quantum circuit execution and long duration adiabatic operations.

Continuously parameterized two-qubit gates are a key feature of state-of-the-art trapped-ion quantum processors as they have favorable error scalings and show distinct improvements in circuit performance over more restricted maximally entangling gatesets. In this work, we provide a comprehensive and pedagogical discussion on how to practically implement these continuously parameterized Mølmer-Sørensen gates on the Quantum Scientific Computing Open User Testbed (QSCOUT), a low-level trapped-ion processor. To generate the arbitrary entangling angles, $θ$, we simply scale the amplitude of light used to generate the entanglement. However, doing so requires careful consideration of amplifier saturation as well as the variable light shifts that result. As such, we describe a method to calibrate and cancel the dominant fourth-order effects, followed by a dynamic virtual phase advance during the gate to cancel any residual light shifts, and find a linear scaling between $θ$ and the residual light shift. Once, we have considered and calibrated these effects, we demonstrate performance improvement with decreasing $θ$. Finally, we describe nuances of hardware control to transform the XX-type interaction of the arbitrary-angle Mølmer-Sørensen gate into a phase-agnostic and crosstalk-mitigating ZZ interaction.

Quantum network and quantum repeater are promising ways to scale up a quantum information system to enable various applications with unprecedented performance. As a current bottleneck of building a long-distance quantum network, the distribution rate of heralded entanglement between remote network nodes is typically much lower than the decoherence rate of each local node, which obstructs the implementation of a metropolitan-scale quantum network with more than two remote nodes. A promising scheme to accelerate the remote entanglement distribution is through multiplexing enhancement based on a multimode quantum network node. In this work, we experimentally realize a functional $5$-ion quantum network node with two different types of qubits inside. We employ a hybrid multiplexing scheme combining the methods of multiple excitation and ion shuttling, in which maximally $44$ time-bin modes are generated and sent through a long fiber to boost the entangling rate. Via this scheme, we can generate heralded ion-photon entanglement with a high fidelity of $96.8\%$/$94.6\%$/$89.8\%$ with a success rate of $263\,\text{s}^{-1}$/$40\,\text{s}^{-1}$/$4.28\,\text{s}^{-1}$, over a fiber of $3\,$m/$1\,$km/$12\,$km, respectively. In addition, the memory qubit can protect the stored quantum information from the destructive ion-photon entangling attempts via dual-type encoding and a memory coherence time of $366\,$ms is achieved. This coherence time has exceeded the expected entanglement generation time $234\,$ms over a $12\,$km fiber, which is realized for the first time in a metropolitan-scale quantum network node.

We study the ground-state phase diagram of a one-dimensional $\mathbb{Z}_2$ lattice gauge theory coupled to soft-core bosonic matter at unit filling, inspired by the Higgs sector of the standard model. Through a combination of analytical perturbative approaches, exact diagonalization, and density-matrix-renormalization-group simulations, we uncover a rich phase diagram driven by gauge-field-mediated resonant pair hopping and the confinement of single particles. The pair hopping results in a bunching state with superextensive energy and macroscopic particle number fluctuations at strong electric field strengths and weak on-site interactions. The bunching state crosses over into a pair superfluid phase as the on-site interaction increases, characterized by a finite superfluid density and powerlaw-decaying pair correlations. At large on-site interaction strengths and driven by effective interactions induced by the gauge constraint, the superfluid transitions into an incompressible pair Mott insulator phase. At weak field strengths and on-site interactions, we find a plasma-like region, where single bosons exhibit large short-range correlations and the ground state is composed almost equally of states with even and odd local boson occupation. The presence of a bunching state with large number fluctuations, which is difficult to study using classical numerics, motivates experimental realizations in hybrid boson-qubit quantum simulation platforms such as circuit QED, neutral atoms, and trapped ions. Our findings highlight the rich interplay between gauge fields and soft-core bosonic matter.

We report the determination of the Landé $g_J$ factor and Zeeman coefficients for the ground-state of $^{171}$Yb$^+$, relevant to microwave quantum frequency standards (QFSs). The $g_J$ factor is obtained by using two independent methods: multiconfiguration Dirac-Hartree-Fock and multireference configuration interaction, yielding a consistent value of 2.002615(70). The first- and second-order Zeeman coefficients are determined as 14,010.78(49) Hz/$μ$T and 31.0869(22) mHz/$μ$T$^2$, respectively, based on the calculated $g_J$ factor. These coefficients enable reduced magnetic-field-induced uncertainties, improving the accuracy of the $^{171}$Yb$^+$ microwave QFSs. The results reported in this work also offer potential for improved constraints on variations in fundamental constants through frequency comparisons, and advancing trapped-ion quantum computers based on the ground-state hyperfine splitting of $^{171}$Yb$^+$.

The ability to execute a large number of quantum gates in parallel is a fundamental requirement for quantum error correction, allowing an error threshold to exist under the finite coherence time of physical qubits. Recently, two-dimensional ion crystals have been demonstrated as a plausible approach to scale up the qubit number in a trapped ion quantum computer. However, although the long-range Coulomb interaction between the ions enables their strong connectivity, it also complicates the design of parallel gates and leads to intrinsic crosstalk errors. Here we examine the effects of crosstalk errors on a rotated surface code. We show that, instead of the distance-3 code considered in previous works, a distance-5 code is necessary to correct the two-qubit crosstalk error. We numerically calculate the logical error rates and coherence times under various crosstalk errors, gate infidelities and coherence times of the physical qubits, and we optimize the parallelism level according to the competition between different error sources. We show that a break-even point can be reached under realistic parameters. We further analyze the spatial dependence of the crosstalk, and discuss the scaling of the logical error rate versus the code distance for the long-term goal of a logical error rate below $10^{-10}$.

The calibration of quantum measurements is limited by the ability to accurately prepare quantum states under unknown device errors. We develop an accurate calibration protocol for the measurement apparatus of a quantum computer that is `blind' to the state preparation. Blind calibration quantifies and corrects measurement errors from simple tomographic data on a noisy quantum state. Importantly, it calibrates multiple error mechanisms in a single experiment, eliminating the need for bespoke, separate calibration experiments. Using a trapped-ion quantum computer, we systematically demonstrate the accuracy of the method. We use blind calibration to estimate the native calibration parameters of the experimental system. The recovered calibrations are consistent with directly measured values and perform similarly in predicting the state properties.

To achieve the entangling gate fidelity above the quantum error correction threshold, it is critical to suppress errors due to experimental imperfection. We consider the Mølmer-Sørensen gates in trapped-ion systems, and develop a general approach to suppress a family of noise sources that appeared as either symmetric or asymmetric errors. Using the time-average displacement minimization technique, both symmetric error and displacement-dependent part of the asymmetric errors are eliminated. Then, by analyzing the tangent space of displacement-independent errors, we obtain the analytic form of the generators of the correction operator to the remaining error terms. We then develop a compensation pulse to fully suppress the remaining displacement-independent errors. The effectiveness of our scheme is further verified by numerical analysis, through which we observe a significant reduction of entangling gate infidelity. Our findings enhance gate fidelity and robustness to noise for ion trap quantum computing.

The quantum charge-coupled device (QCCD) is one of the notable architectures to achieve large-scale trapped-ion quantum computers. To realize QCCD architecture, ions must be transported quickly while minimizing motional excitation. High-voltage sources are necessary to achieve such high-quality ion transport through a high secular frequency. In this study, we report the development of a field programmable gate array (FPGA)-based digital-to-analog converter (DAC) system with an output voltage range of +/-50 V and demonstrate its effectiveness in ion transport operations. The device provides 16-channel analog output, maximum update rate of 16 mega updates per second (MUPS), slew rate of 20 V/us, and bandwidth of > 200 kHz. By optimizing the voltage sets with quadratic programming, we experimentally confirmed that this DAC system can achieve more than twice the secular frequency attainable when its output range is restricted to +/-10 V, which is consistent with the fact that scaling all electrode voltages by a factor of 5 will scale the secular frequency by the square root of 5. Since the output range of many commercially available DACs is commonly limited to +/-10 V, this increase is effective for ion shuttling operations, such as transport, split and merge. The developed DAC system has potential to increase the speed of ion transport thereby reducing processing times in QCCD-based quantum computers.

The requirement for Hermiticity in quantum mechanics ensures the reality of energies, while the parity-time symmetry offers an alternative route to achieve this goal. Interestingly, in a three-level system, the parity-time symmetry-breaking can lead to a third-order exceptional point with distinctive topological properties and enhanced sensitivity. To experimentally implement this in open quantum systems, it is essential to introduce two well-controlled loss channels. However, the requirement for these two loss channels presents a challenge in experimental implementation due to the lack of methods to realize the dynamics governed by an effective non-Hermitian Hamiltonian. Here we address the challenge by employing two approaches to eliminate the effects of quantum jump terms so that the dynamics is governed by an effective non-Hermitian Hamiltonian in a dissipative trapped ion with two loss channels. Based on this, we experimentally observe the parity-time symmetry-breaking-induced third-order exceptional point through non-Hermitian absorption spectroscopy. In particular, we perform quantum state tomography to directly demonstrate the coalescence of three eigenstates into a single eigenstate at the exceptional point. Finally, we identify an intrinsic third order Liouvillian exceptional point associated with a parity-time symmetry breaking via quench dynamics. Our experiments can be extended to observe other non-Hermitian phenomena involving more than two levels and potentially find applications in quantum information technology.

We report the achievement of single-qubit gates with sub-part-per-million error rates, in a trapped-ion $^{43}$Ca$^{+}$ hyperfine clock qubit. We explore the speed/fidelity trade-off for gate times $4.4\leq t_{g}\leq35~μ$s, and benchmark a minimum error per Clifford gate of $1.5(4) \times 10^{-7}$. Calibration errors are suppressed to $< 10^{-8}$, leaving qubit decoherence ($T_{2}\approx 70$ s), leakage, and measurement as the dominant error contributions. The ion is held above a microfabricated surface-electrode trap which incorporates a chip-integrated microwave resonator for electronic qubit control; the trap is operated at room temperature without magnetic shielding.

We systematically investigate local phonon hopping in the radial direction of a linear trapped-ion string. We measure the decay of hopping as a function of key trap parameters and analyze the results in terms of the decay time and the number of oscillations. We attribute the loss of coherence to nonlinear coupling between different modes. Despite quantitative differences, the overall trends in our numerical simulations are similar to those of the experimental results. This work establishes a method for evaluating phonon hopping coherence and provides insight into the underlying decoherence mechanisms.

State-of-the-art noisy-intermediate-scale quantum (NISQ) processors are currently implemented across a variety of hardware platforms, each with their own distinct gatesets. As such, circuit compilation should not only be aware of, but also deeply connect to, the native gateset and noise properties of each. Trapped-ion processors are one such platform that provides a gateset that can be continuously parameterized across both one- and two-qubit gates. Here we use the Quantum Scientific Computing Open User Testbed (QSCOUT) to study noise-aware compilations focused on continuously parameterized two-qubit $\mathcal{ZZ}$ gates (based on the Mølmer-Sørensen interaction) using $\textbf{Superstaq}$, a quantum software platform for hardware-aware circuit compiler optimizations. We discuss the realization of $\mathcal{ZZ}$ gates with arbitrary angle on the all-to-all connected trapped-ion system. Then we discuss a variety of different compiler optimizations that innately target these $\mathcal{ZZ}$ gates and their noise properties. These optimizations include moving from a restricted maximally entangling gateset to a continuously parameterized one, swap mirroring to further reduce total entangling angle of the operations, focusing the heaviest $\mathcal{ZZ}$ angle participation on the best performing gate pairs, and circuit approximation to remove the least impactful $\mathcal{ZZ}$ gates. We demonstrate these compilation approaches on the hardware with randomized quantum volume circuits, observing the potential to realize a larger quantum volume as a result of these optimizations. Using differing yet complementary analysis techniques, we observe the distinct improvements in system performance provided by these noise-aware compilations and study the role of stochastic and coherent error channels for each compilation choice.

Topological defects are discontinuities of a system protected by global properties, with wide applications in mathematics and physics. While previous experimental studies mostly focused on their classical properties, it has been predicted that topological defects can exhibit quantum superposition. Despite the fundamental interest and potential applications in understanding symmetry-breaking dynamics of quantum phase transitions, its experimental realization still remains a challenge. Here, we report the observation of quantum superposition of topological defects in a trapped-ion quantum simulator. By engineering long-range spin-spin interactions, we observe a spin kink splitting into a superposition of kinks at different positions, creating a ``Schrodinger kink'' that manifests non-locality and quantum interference. Furthermore, by preparing superposition states of neighboring kinks with different phases, we observe the propagation of the wave packet in different directions, thus unambiguously verifying the quantum coherence in the superposition states. Our work provides useful tools for non-equilibrium dynamics in quantum Kibble-Zurek physics.

Spontaneous particle-pair formation is a fundamental phenomenon in nature. It can, for example, appear when the potential energy between two particles increases with separation, as if they were connected by a tense string. Beyond a critical separation, new particle pairs can form, causing the string to break. String-breaking dynamics in quantum chromodynamics play a vital role in high-energy particle collisions and early universe evolution. Simulating string evolution and hadron formation is, therefore, a grand challenge in modern physics. Quantum simulators, well-suited for studying dynamics, are expected to outperform classical computing methods. However, the required experimental capabilities to simulate string-breaking dynamics have not yet been demonstrated, even for simpler models of the strong force. We experimentally probe, for the first time, the spatiotemporal dynamics of string-breaking in a (1+1)-dimensional $\mathbb{Z}_2$ lattice gauge theory using a fully programmable trapped-ion quantum simulator. We emulate external static charges and strings via site-dependent magnetic-field control enabled by a dual array of tightly focused laser beams targeting individual ions. First, we study how confinement affects isolated charges, finding that they freely spread without string tension but exhibit localized oscillations when tension is increased. Then, we observe and characterize string-breaking dynamics of a string stretched between two static charges after an abrupt increase in string tension. Charge pairs appear near the string edges and spread into the bulk, revealing a route to dynamical string-breaking distinct from the conventional Schwinger mechanism. Our work demonstrates that analog quantum simulators have achieved the necessary control to explore string-breaking dynamics, which may ultimately be relevant to nuclear and high-energy physics.

Dual-type qubits have become a promising way to suppress the crosstalk error of auxiliary operations in large-scale ion trap quantum computation. Here we demonstrate a direct entangling gate between dual-type qubits encoded in the $S_{1/2}$ and $D_{5/2}$ hyperfine manifolds of $^{137}\mathrm{Ba}^{+}$ ions. Our scheme is economic in the hardware, requiring only a single $532\,$nm laser system to entangle both qubit types by driving their Raman transitions. We achieve a Bell state fidelity of $96.3(4)\%$ for the dual-type Molmer-Sorensen gate between an $S$-$D$ ion pair, comparable to that for the same-type $S$-$S$ or $D$-$D$ gates. This technique can reduce the overhead for back-and-forth conversions between dual-type qubits in the quantum circuit with wide applications in quantum error correction and ion-photon quantum networks.

Quantum computing holds unparalleled potentials to enhance machine learning. However, a demonstration of quantum learning advantage has not been achieved so far. We make a step forward by rigorously establishing a noise-robust, unconditional quantum learning advantage in expressivity, inference speed, and training efficiency, compared to commonly-used classical models. Our proof is information-theoretic and pinpoints the origin of this advantage: entanglement can be used to reduce the communication required by non-local tasks. In particular, we design a task that can be solved with certainty by quantum models with a constant number of parameters using entanglement, whereas commonly-used classical models must scale linearly to achieve a larger-than-exponentially-small accuracy. We show that the quantum model is trainable with constant resources and robust against constant noise. Through numerical and trapped-ion experiments on IonQ Aria, we demonstrate the desired advantage. Our results provide valuable guidance for demonstrating quantum learning advantages with current noisy intermediate-scale devices.

We develop a hybrid oscillator-qubit processor framework for quantum simulation of strongly correlated fermions and bosons that avoids the boson-to-qubit mapping overhead encountered in qubit hardware. This framework gives exact decompositions of particle interactions such as density-density terms and gauge-invariant hopping, as well as approximate methods based on the Baker-Campbell Hausdorff formulas including the magnetic field term for the $U(1)$ quantum link model in $(2+1)$D. We use this framework to show how to simulate dynamics using Trotterisation, perform ancilla-free partial error detection using Gauss's law, measure non-local observables, estimate ground state energies using a oscillator-qubit variational quantum eigensolver as well as quantum signal processing, and we numerically study the influence of hardware errors in circuit QED experiments. To show the advantages over all-qubit hardware, we perform an end-to-end comparison of the gate complexity for the gauge-invariant hopping term and find an improvement of the asymptotic scaling with the boson number cutoff $S$ from $\mathcal{O}(\log(S)^2)$ to $\mathcal{O}(1)$ in our framework as well as, for bosonic matter, a constant factor improvement of better than $10^4$. We also find an improvement from $\mathcal{O}(\log(S))$ to $\mathcal{O}(1)$ for the $U(1)$ magnetic field term. While our work focusses on an implementation in superconducting hardware, our framework can also be used in trapped ion, and neutral atom hardware. This work establishes digital quantum simulation with hybrid oscillator-qubit hardware as a viable and advantageous method for the study of qubit-boson models in materials science, chemistry, and high-energy physics.

Throughout its history, the theory of quantum error correction has heavily benefited from translating classical concepts into the quantum setting. In particular, classical notions of weight enumerators, which relate to the performance of an error-correcting code, and MacWilliams' identity, which helps to compute enumerators, have been generalized to the quantum case. In this work, we establish a distinct relationship between the theoretical machinery of quantum weight enumerators and a seemingly unrelated physics experiment: we prove that Rains' quantum shadow enumerators - a powerful mathematical tool - arise as probabilities of observing fixed numbers of triplets in a Bell sampling experiment. This insight allows us to develop here a rigorous framework for the direct measurement of quantum weight enumerators, thus enabling experimental and theoretical studies of the entanglement structure of any quantum error-correcting code or state under investigation. On top of that, we derive concrete sample complexity bounds and physically-motivated robustness guarantees against unavoidable experimental imperfections. Finally, we experimentally demonstrate the possibility of directly measuring weight enumerators on a trapped-ion quantum computer. Our experimental findings are in good agreement with theoretical predictions and illuminate how entanglement theory and quantum error correction can cross-fertilize each other once Bell sampling experiments are combined with the theoretical machinery of quantum weight enumerators.

Information engines produce mechanical work through measurement and adaptive control. For information engines, the principal challenge lies in how to store the generated work for subsequent utilization. Here, we report an experimental demonstration where quantized mechanical motion serves as a quantum battery and gets charged in repeated cycles by a single trapped-ion information engine. This is enabled by a key technological advancement in rapid state discrimination, allowing us to suppress measurement-induced disturbances. Consequently, we were able to obtain a charging efficiency over 50\% of the theoretical limit at the optimal temperature. The experimental results substantiate that this approach can render trapped ions a promising platform for microscopic information engines with potential applications in the future upon scaling up.

We report the electromagnetically-induced-transparency (EIT) cooling of $^{137}\mathrm{Ba}^{+}$ ions with a nuclear spin of $I=3/2$, which are a good candidate of qubits for future large-scale trapped ion quantum computing. EIT cooling of atoms or ions with a complex ground-state level structure is challenging due to the lack of an isolated $Λ$ system, as the population can escape from the $Λ$ system to reduce the cooling efficiency. We overcome this issue by leveraging an EIT pumping laser to repopulate the cooling subspace, ensuring continuous and effective EIT cooling. We cool the two radial modes of a single $^{137}\mathrm{Ba}^{+}$ ion to average motional occupations of 0.08(5) and 0.15(7) respectively. Using the same laser parameters, we also cool all the ten radial modes of a five-ion chain to near their ground states. Our approach can be adapted to atomic species possessing similar level structures. It allows engineering of the EIT Fano-like spectrum, which can be useful for simultaneous cooling of modes across a wide frequency range, aiding in large-scale trapped-ion quantum information processing.

Single-shot quantum error correction has the potential to speed up quantum computations by removing the need for multiple rounds of syndrome extraction in order to be fault-tolerant. Using Quantinuum's H2 trapped-ion quantum computer, we implement the [[33,1,4]] 4D surface code and perform the first experimental demonstration of single-shot quantum error correction with bare ancilla qubits. We conduct memory experiments comparing the 2D and 4D surface codes and find that despite differences in qubit use and syndrome extraction circuit depth, the 4D surface code matches or outperforms the 2D surface code in both the fault-tolerant and single-shot regimes.

Quantum simulators with hundreds of qubits and engineerable Hamiltonians have the potential to explore quantum many-body models that are intractable for classical computers. However, learning the simulated Hamiltonian, a prerequisite for any applications of a quantum simulator, remains an outstanding challenge due to the fast increasing time cost with the qubit number and the lack of high-fidelity universal gate operations in the noisy intermediate-scale quantum era. Here we demonstrate the Hamiltonian learning of a two-dimensional ion trap quantum simulator with 300 qubits. We employ global manipulations and single-qubit-resolved state detection to efficiently learn the all-to-all-coupled Ising model Hamiltonian, with the required quantum resources scaling at most linearly with the qubit number. Our work paves the way for wide applications of large-scale ion trap quantum simulators.

Simulating non-equilibrium phenomena in strongly-interacting quantum many-body systems, including thermalization, is a promising application of near-term and future quantum computation. By performing experiments on a digital quantum computer consisting of fully-connected optically-controlled trapped ions, we study the role of entanglement in the thermalization dynamics of a $Z_2$ lattice gauge theory in 2+1 spacetime dimensions. Using randomized-measurement protocols, we efficiently learn a classical approximation of non-equilibrium states that yields the gap-ratio distribution and the spectral form factor of the entanglement Hamiltonian. These observables exhibit universal early-time signals for quantum chaos, a prerequisite for thermalization. Our work, therefore, establishes quantum computers as robust tools for studying universal features of thermalization in complex many-body systems, including in gauge theories.

We study a hardware-efficient variational quantum algorithm ansatz tailored for the trapped-ion quantum simulator, HEA-TI. We leverage programmable single-qubit rotations and global spin-spin interactions among all ions, reducing the dependence on resource-intensive two-qubit gates in conventional gate-based methods. We apply HEA-TI to state engineering of cluster states and analyze the scaling of required quantum resources. We also apply HEA-TI to solve the ground state problem of chemical molecules $\mathrm{H_{2}}$, $\mathrm{LiH}$ and $\mathrm{F_{2}}$. We numerically analyze the quantum computing resources required to achieve chemical accuracy and examine the performance under realistic experimental noise and statistical fluctuation. The efficiency of this ansatz is shown to be comparable to other commonly used variational ansatzes like UCCSD, with the advantage of substantially easier implementation in the trapped-ion quantum simulator. This approach showcases the hardware-efficient ansatz as a powerful tool for the application of the near-term quantum computer.

Two-dimensional (2D) ion crystals have become a promising way to scale up qubit numbers for ion trap quantum information processing. However, to realize universal quantum computing in this system, individually addressed high-fidelity two-qubit entangling gates still remain challenging due to the inevitable micromotion of ions in a 2D crystal as well as the technical difficulty in 2D addressing. Here we demonstrate two-qubit entangling gates between any ion pairs in a 2D crystal of four ions. We use symmetrically placed crossed acousto-optic deflectors (AODs) to drive Raman transitions and achieve an addressing crosstalk error below 0.1%. We design and demonstrate a gate sequence by alternatingly addressing two target ions, making it compatible with any single-ion addressing techniques without crosstalk from multiple addressing beams. We further examine the gate performance versus the micromotion amplitude of the ions and show that its effect can be compensated by a recalibration of the laser intensity without degrading the gate fidelity. Our work paves the way for ion trap quantum computing with hundreds to thousands of qubits on a 2D ion crystal.

The Lieb-Schultz-Mattis (LSM) theorem asserts that microscopic details of the system can impose non-trivial constraints on the system's low-energy properties. While traditionally applied to short-range interaction systems, where locality ensures a vanishing spectral gap in large system size limit, the impact of long-range interactions on the LSM theorem remains an open question. Long-range interactions are prevalent in experimental platforms such as Rydberg atoms, dipolar quantum gases, polar molecules, optical cavities, and trapped ions, where the interaction decay exponent can be experimentally tuned. We extend the LSM theorem in one dimension to long-range interacting systems and find that the LSM theorem holds for exponentially or power-law two-body interactions with a decay exponent $α> 2$. However, for power-law interactions with $α< 2$, the constraints of the LSM theorem on the ground state do not apply. Numerical simulations of long-range versions of the Heisenberg and Majumdar-Ghosh models, both satisfying the LSM symmetry requirements, are also provided. Our results suggest promising directions for experimental validation of the LSM theorem in systems with tunable long-range interactions.

Trapped atomic ions constitute one of the leading physical platforms for building the quantum repeater nodes to realize large-scale quantum networks. In a long-distance trapped-ion quantum network, it is essential to have crosstalk-free dual-type qubits: one type, called the communication qubit, to establish an entangling interface with telecom photons; and the other type, called the memory qubit, to store quantum information immune from photon scattering under entangling attempts. Here, we report the first experimental implementation of a telecom-compatible and crosstalk-free quantum network node based on two trapped $^{40}$Ca$^{+}$ ions. The memory qubit is encoded on a long-lived metastable level to avoid crosstalk with the communication qubit encoded in another subspace of the same ion species, and a quantum wavelength conversion module is employed to generate ion-photon entanglement over a $12\,$km fiber in a heralded style. Our work therefore constitutes an important step towards the realization of quantum repeaters and long-distance quantum networks.

Photo stimulated luminescent materials are one of the most attractive alternatives for next generation optical information storage technologies. However, there are still some challenges in regulating appropriate energy levels in luminescent materials for optical information storage. Herein, a green emission nanophosphor Zn2SiO4: Cr3+, Mn2+ with the trap depth of 1.05 eV, fulfilling the requirements for optical information storage, was fabricated for the first time through the solution combustion method and subsequent heat treatment at 1000 degree centigrade for 2h. The crystal structure, micromorphology, photoluminescence (PL), photoluminescence excitation (PLE), and afterglow properties of Zn2SiO4: xCr3+, yMn2+ were studied systematically. By applying the strategy of trap depth engineering, high trap density with proper trap depth was observed when Cr3+ ions were introduced into Zn2SiO4: Mn2+. Thermoluminescence (TL) glow curve analysis through the initial rise (IR) method was conducted to gain some insight into the information of traps. As proof of application, information storage was experimentally achieved by choosing 275 nm illumination for information writing and 980 nm NIR excitation for information reading. The results indicate that Zn2SiO4: Cr3+, Mn2+ phosphor holds promise for potential applications in the field of optical information storage.

Studying high-energy collisions of composite particles, such as hadrons and nuclei, is an outstanding goal for quantum simulators. However, preparation of hadronic wave packets has posed a significant challenge, due to the complexity of hadrons and the precise structure of wave packets. This has limited demonstrations of hadron scattering on quantum simulators to date. Observations of confinement and composite excitations in quantum spin systems have opened up the possibility to explore scattering dynamics in spin models. In this article, we develop two methods to create entangled spin states corresponding to wave packets of composite particles in analog quantum simulators of Ising spin Hamiltonians. One wave-packet preparation method uses the blockade effect enabled by beyond-nearest-neighbor Ising spin interactions. The other method utilizes a quantum-bus-mediated exchange, such as the native spin-phonon coupling in trapped-ion arrays. With a focus on trapped-ion simulators, we numerically benchmark both methods and show that high-fidelity wave packets can be achieved in near-term experiments. We numerically study scattering of wave packets for experimentally realizable parameters in the Ising model and find inelastic-scattering regimes, corresponding to particle production in the scattering event, with prominent and distinct experimental signals. Our proposal, therefore, demonstrates the potential of observing inelastic scattering in near-term quantum simulators.

In trapped ion system, accurate thermometry of ion is crucial for evaluating the system state and precisely performing quantum operations. However, when the motional state of a single ion is far away from the ground state, the spatial dimension of the phonon state sharply increases, making it difficult to realize accurate and mode-resolved thermometry with existing methods. In this work, we apply deep learning for the first time to the thermometry of trapped ion, providing an efficient and mode-resolved method for accurately estimating large mean phonon numbers. Our trained neural network model can be directly applied to other experimental setups without retraining or post-processing, as long as the related parameters are covered by the model's effective range, and it can also be conveniently extended to other parameter ranges. We have conducted experimental verification based on our surface trap, of which the result has shown the accuracy and efficiency of the method for thermometry of single ion under large mean phonon number, and its mode resolution characteristic can make it better applied to the characterization of system parameters, such as evaluating cooling effectiveness, analyzing surface trap noise.

The spin-boson model is a prototypical model for open quantum dynamics. Here we simulate the spin-boson model using a chain of trapped ions where a spin is coupled to a structured reservoir of bosonic modes. We engineer the spectral density of the reservoir by adjusting the ion number, the target ion location, the laser detuning to the phonon sidebands, and the number of frequency components in the laser, and we observe their effects on the collapse and revival of the initially encoded information. Our work demonstrates the ion trap as a powerful platform for simulating open quantum dynamics with complicated reservoir structures.

Quantum simulation holds promise of enabling a complete description of high-energy scattering processes rooted in gauge theories of the Standard Model. A first step in such simulations is preparation of interacting hadronic wave packets. To create the wave packets, one typically resorts to adiabatic evolution to bridge between wave packets in the free theory and those in the interacting theory, rendering the simulation resource intensive. In this work, we construct a wave-packet creation operator directly in the interacting theory to circumvent adiabatic evolution, taking advantage of resource-efficient schemes for ground-state preparation, such as variational quantum eigensolvers. By means of an ansatz for bound mesonic excitations in confining gauge theories, which is subsequently optimized using classical or quantum methods, we show that interacting mesonic wave packets can be created efficiently and accurately using digital quantum algorithms that we develop. Specifically, we obtain high-fidelity mesonic wave packets in the $Z_2$ and $U(1)$ lattice gauge theories coupled to fermionic matter in 1+1 dimensions. Our method is applicable to both perturbative and non-perturbative regimes of couplings. The wave-packet creation circuit for the case of the $Z_2$ lattice gauge theory is built and implemented on the Quantinuum H1-1 trapped-ion quantum computer using 13 qubits and up to 308 entangling gates. The fidelities agree well with classical benchmark calculations after employing a simple symmetry-based noise-mitigation technique. This work serves as a step toward quantum computing scattering processes in quantum chromodynamics.

Many promising quantum applications depend on the efficient quantum simulation of an exponentially large sparse Hamiltonian, a task known as sparse Hamiltonian simulation, which is fundamentally important in quantum computation. Although several theoretically appealing quantum algorithms have been proposed for this task, they typically require a black-box query model of the sparse Hamiltonian, rendering them impractical for near-term implementation on quantum devices. In this paper, we propose a technique named Hamiltonian embedding. This technique simulates a desired sparse Hamiltonian by embedding it into the evolution of a larger and more structured quantum system, allowing for more efficient simulation through hardware-efficient operations. We conduct a systematic study of this new technique and demonstrate significant savings in computational resources for implementing prominent quantum applications. As a result, we can now experimentally realize quantum walks on complicated graphs (e.g., binary trees, glued-tree graphs), quantum spatial search, and the simulation of real-space Schrödinger equations on current trapped-ion and neutral-atom platforms. Given the fundamental role of Hamiltonian evolution in the design of quantum algorithms, our technique markedly expands the horizon of implementable quantum advantages in the NISQ era.

Hybridizing different degrees of freedom or physical platforms potentially offers various advantages in building scalable quantum architectures. We here introduce a fault-tolerant hybrid quantum computation by taking the advantages of both discrete variable (DV) and continuous variable (CV) systems. Particularly, we define a CV-DV hybrid qubit with bosonic cat-code and single photon, which is implementable in current photonic platforms. By the cat-code encoded in the CV part, the dominant loss errors are readily correctable without multi-qubit encoding, while the logical basis is inherently orthogonal due to the DV part. We design fault-tolerant architectures by concatenating hybrid qubits and an outer DV quantum error correction code such as topological codes, exploring their potential merits in developing scalable quantum computation. We demonstrate by numerical simulations that our scheme is at least an order of magnitude more resource-efficient over all previous proposals in photonic platforms, allowing to achieve a record-high loss threshold among existing CV and hybrid approaches. We discuss its realization not only in all-photonic platforms but also in other hybrid platforms including superconduting and trapped-ion systems, which allows us to find various efficient routes towards fault-tolerant quantum computing.

A large qubit capacity and an individual readout capability are two crucial requirements for large-scale quantum computing and simulation. As one of the leading physical platforms for quantum information processing, the ion trap has achieved quantum simulation of tens of ions with site-resolved readout in 1D Paul trap, and that of hundreds of ions with global observables in 2D Penning trap. However, integrating these two features into a single system is still very challenging. Here we report the stable trapping of 512 ions in a 2D Wigner crystal and the sideband cooling of their transverse motion. We demonstrate the quantum simulation of long-range quantum Ising models with tunable coupling strengths and patterns, with or without frustration, using 300 ions. Enabled by the site resolution in the single-shot measurement, we observe rich spatial correlation patterns in the quasi-adiabatically prepared ground states, which allows us to verify quantum simulation results by comparing with the calculated collective phonon modes and with classical simulated annealing. We further probe the quench dynamics of the Ising model in a transverse field to demonstrate quantum sampling tasks. Our work paves the way for simulating classically intractable quantum dynamics and for running NISQ algorithms using 2D ion trap quantum simulators.

A promising paradigm of quantum computing for achieving practical quantum advantages is quantum annealing or quantum approximate optimization algorithm, where the classical problems are encoded in Ising interactions. However, it is challenging to build a quantum system that can efficiently map any structured problems. Here, we present a programmable trapped-ion quantum simulator of an Ising model with all-to-all connectivity with up to four spins. We implement the spin-spin interactions by using the coupling of trapped ions to multiple collective motional modes and realize the programmability through phase modulation of the Raman laser beams that are individually addressed on ions. As an example, we realize several Ising lattices with different interaction connectivities, where the interactions can be ferromagnetic or anti-ferromagnetic. We confirm the programmed interaction geometry by observing the ground states of the corresponding models through quantum state tomography. Our experimental demonstrations serve as an important basis for realizing practical quantum advantages with trapped ions.

One of the most striking many-body phenomena in nature is the sudden change of macroscopic properties as the temperature or energy reaches a critical value. Such equilibrium transitions have been predicted and observed in two and three spatial dimensions, but have long been thought not to exist in one-dimensional (1D) systems. Fifty years ago, Dyson and Thouless pointed out that a phase transition in 1D can occur in the presence of long-range interactions, but an experimental realization has so far not been achieved due to the requirement to both prepare equilibrium states and realize sufficiently long-range interactions. Here we report on the first experimental demonstration of a finite-energy phase transition in 1D. We use the simple observation that finite-energy states can be prepared by time-evolving product initial states and letting them thermalize under the dynamics of a many-body Hamiltonian. By preparing initial states with different energies in a 1D trapped-ion quantum simulator, we study the finite-energy phase diagram of a long-range interacting quantum system. We observe a ferromagnetic equilibrium phase transition as well as a crossover from a low-energy polarized paramagnet to a high-energy unpolarized paramagnet in a system of up to $23$ spins, in excellent agreement with numerical simulations. Our work demonstrates the ability of quantum simulators to realize and study previously inaccessible phases at finite energy density.

Calculating the equilibrium properties of condensed matter systems is one of the promising applications of near-term quantum computing. Recently, hybrid quantum-classical time-series algorithms have been proposed to efficiently extract these properties from a measurement of the Loschmidt amplitude $\langle ψ| e^{-i \hat H t}|ψ\rangle$ from initial states $|ψ\rangle$ and a time evolution under the Hamiltonian $\hat H$ up to short times $t$. In this work, we study the operation of this algorithm on a present-day quantum computer. Specifically, we measure the Loschmidt amplitude for the Fermi-Hubbard model on a $16$-site ladder geometry (32 orbitals) on the Quantinuum H2-1 trapped-ion device. We assess the effect of noise on the Loschmidt amplitude and implement algorithm-specific error mitigation techniques. By using a thus-motivated error model, we numerically analyze the influence of noise on the full operation of the quantum-classical algorithm by measuring expectation values of local observables at finite energies. Finally, we estimate the resources needed for scaling up the algorithm.

Universality and scaling laws are hallmarks of equilibrium phase transitions and critical phenomena. However, extending these concepts to non-equilibrium systems is an outstanding challenge. Despite recent progress in the study of dynamical phases, the universality classes and scaling laws for non-equilibrium phenomena are far less understood than those in equilibrium. In this work, using a trapped-ion quantum simulator with single-spin resolution, we investigate the non-equilibrium nature of critical fluctuations following a quantum quench to the critical point. We probe the scaling of spin fluctuations after a series of quenches to the critical Hamiltonian of a long-range Ising model. With systems of up to 50 spins, we show that the amplitude and timescale of the post-quench fluctuations scale with system size with distinct universal critical exponents, depending on the quench protocol. While a generic quench can lead to thermal critical behavior, we find that a second quench from one critical state to another (i.e.~a double quench) results in a new universal non-equilibrium behavior, identified by a set of critical exponents distinct from their equilibrium counterparts. Our results demonstrate the ability of quantum simulators to explore universal scaling beyond equilibrium.

Emulation of energy transfer processes in natural systems on quantum platforms can further our understanding of complex dynamics in nature. One notable example is the demonstration of vibrationally assisted energy transfer (VAET) on a trapped-ion quantum emulator, which offers insights for the energetics of light harvesting. In this work, we expand the study of VAET simulation with trapped ions to a non-Hermitian quantum system comprising a $\mathscr{PT}$-symmetric chromophore dimer weakly coupled to a vibrational mode. We first characterize exceptional points (EPs) and non- Hermitian features of the excitation energy transfer processes in the absence of the vibration, finding a degenerate pair of second-order EPs. Exploring the non-Hermitian dynamics of the whole system including vibrations, we find that energy transfer accompanied by absorption of phonons from a vibrational mode can be significantly enhanced near such a degenerate EP. Our calculations reveal a unique spectral feature accompanying the coalescing of eigenstates and eigenenergies that provides a novel approach to probe the degenerate EP by fluorescence-detected vibrational spectroscopy. Enhancement of the VAET process near the EP is found to be due to maximal favorability of phonon absorption at the degenerate EP, enabling multiple simultaneous excitations. Our work on improving VAET processes in non-Hermitian quantum systems paves the way for leveraging non-Hermiticity in quantum dynamics related to excitation energy transfer.

Quantum computers are now on the brink of outperforming their classical counterparts. One way to demonstrate the advantage of quantum computation is through quantum random sampling performed on quantum computing devices. However, existing tools for verifying that a quantum device indeed performed the classically intractable sampling task are either impractical or not scalable to the quantum advantage regime. The verification problem thus remains an outstanding challenge. Here, we experimentally demonstrate efficiently verifiable quantum random sampling in the measurement-based model of quantum computation on a trapped-ion quantum processor. We create and sample from random cluster states, which are at the heart of measurement-based computing, up to a size of 4 x 4 qubits. By exploiting the structure of these states, we are able to recycle qubits during the computation to sample from entangled cluster states that are larger than the qubit register. We then efficiently estimate the fidelity to verify the prepared states -- in single instances and on average -- and compare our results to cross-entropy benchmarking. Finally, we study the effect of experimental noise on the certificates. Our results and techniques provide a feasible path toward a verified demonstration of a quantum advantage.

Simulating quantum systems is one of the most promising avenues to harness the computational power of quantum computers. However, hardware errors in noisy near-term devices remain a major obstacle for applications. Ideas based on the randomization of Suzuki-Trotter product formulas have been shown to be a powerful approach to reducing the errors of quantum simulation and lowering the gate count. In this paper, we study the performance of non-unitary simulation channels and consider the error structure of channels constructed from a weighted average of unitary circuits. We show that averaging over just a few simulation circuits can significantly reduce the Trotterization error for both single-step short-time and multi-step long-time simulations. We focus our analysis on two approaches for constructing circuit ensembles for averaging: (i) permuting the order of the terms in the Hamiltonian and (ii) applying a set of global symmetry transformations. We compare our analytical error bounds to empirical performance and show that empirical error reduction surpasses our analytical estimates in most cases. Finally, we test our method on an IonQ trapped-ion quantum computer accessed via the Amazon Braket cloud platform, and benchmark the performance of the averaging approach.

We reported the realization of a $^{174}\mathrm{Yb}^+$-$^{113}\mathrm{Cd}^+$ bi-species Coulomb crystal comprising $^{174}\mathrm{Yb}^+$ ions as coolant and verified its potential for application as a $^{113}\mathrm{Cd}^+$ microwave frequency standard employing sympathetic cooling.The two species of massive ions stably trapped in a Paul trap make up this large two-component crystal. The $^{113}\mathrm{Cd}^+$ ions are trapped in the center, which reduces considerably RF heating and excess micromotion to which the $^{113}\mathrm{Cd}^+$ ions are subjected. Under this scheme, the uncertainty due to the second-order Doppler effect is reduced to $5\times10^{-16}$, which represents an order of magnitude improvement over sympathetic cooled $^{40}\mathrm{Ca}^+$-$^{113}\mathrm{Cd}^+$ crystal. The uncertainty from the second-order Zeeman effect, which contributes the largest uncertainty to the microwave-ion frequency standard, is reduced to $4\times10^{-16}$. The relevant AC Stark shift uncertainty is estimated to be $4\times10^{-19}$. These results indicate using $^{174}\mathrm{Yb}^+$ as coolant ions for $^{113}\mathrm{Cd}^+$ is far superior and confirm the feasibility of a sympathetic-cooled cadmium-ion microwave clock system employing a $^{174}\mathrm{Yb}^+$-$^{113}\mathrm{Cd}^+$ two-component crystal.

Generating ion-photon entanglement is a crucial step for scalable trapped-ion quantum networks. To avoid the crosstalk on memory qubits carrying quantum information, it is common to use a different ion species for ion-photon entanglement generation such that the scattered photons are far off-resonant for the memory qubits. However, such a dual-species scheme requires elaborate control of the portion and the location of different ion species, and can be subject to inefficient sympathetic cooling. Here we demonstrate a trapped-ion quantum network node in the dual-type qubit scheme where two types of qubits are encoded in the $S$ and $F$ hyperfine structure levels of ${}^{171}\mathrm{Yb}^+$ ions. We generate ion photon entanglement for the $S$-qubit in a typical timescale of hundreds of milliseconds, and verify its small crosstalk on a nearby $F$-qubit with coherence time above seconds. Our work demonstrates an enabling function of the dual-type qubit scheme for scalable quantum networks.

We report the first hybrid matter-photon implementation of verifiable blind quantum computing. We use a trapped-ion quantum server and a client-side photonic detection system networked via a fibre-optic quantum link. The availability of memory qubits and deterministic entangling gates enables interactive protocols without post-selection - key requirements for any scalable blind server, which previous realisations could not provide. We quantify the privacy at <~0.03 leaked classical bits per qubit. This experiment demonstrates a path to fully verified quantum computing in the cloud.

Non-Hermitian systems generically have complex energies, which may host topological structures, such as links or knots. While there has been great progress in experimentally engineering non-Hermitian models in quantum simulators, it remains a significant challenge to experimentally probe complex energies in these systems, thereby making it difficult to directly diagnose complex-energy topology. Here, we experimentally realize a two-band non-Hermitian model with a single trapped ion whose complex eigenenergies exhibit the unlink, unknot or Hopf link topological structures. Based on non-Hermitian absorption spectroscopy, we couple one system level to an auxiliary level through a laser beam and then experimentally measure the population of the ion on the auxiliary level after a long period of time. Complex eigenenergies are then extracted, illustrating the unlink, unknot or Hopf link topological structure. Our work demonstrates that complex energies can be experimentally measured in quantum simulators via non-Hermitian absorption spectroscopy, thereby opening the door for exploring various complex-energy properties in non-Hermitian quantum systems, such as trapped ions, cold atoms, superconducting circuits or solid-state spin systems.

Based on the Rb$^+$-Rb hybrid trap, we investigate the effect of ion-atom elastic collisions on the number and temperature of the remaining atoms. We measured the remaining atomic number and temperature as a function of the wavelength and intensity of the ionization laser, and whether the ion trap was turned on. Fittings with a single exponential decay function plus an offset to the number and radius of the remaining atoms are found to be in good agreement. We found a difference in the exponential factor of different wavelengths of ionization laser with the ion trap on or off. We suppose that the presence of electrons affects ion-atom collisions through disorder-induced heating. Our research contributes to a better understanding of how ultracold neutral plasma evolves, particularly the subsequent kinetics of atomic processes, which also serves as a useful reference for high-energy-density plasma.

Polyatomic cations $^{87}$Rb$_M^+$ ($M$ = 2, 3,$\ldots$) have been produced by cascade two-body ion-atom reactive collisions in the two-step CW-laser photoionization of laser-cooled $^{87}$Rb atoms and accumulated in the ion trap. Using resonant-excitation mass spectrometry and resonant excitation-assisted time-of-flight mass spectrometry, we directly observed and distinguished the charged reaction products. We experimentally verified the cascade generation and cascade dissociation of $^{87}$Rb$_M^+$. The populations of $^{87}$Rb$_M^+$ are quantitatively investigated by solving the rate equations. The $^{87}$Rb$^+$-$^{87}$Rb reaction rate coefficient was derived as 9.10$\times10^{-11}$ cm$^3$/s accordingly. The methods developed here for assembling and detecting homonuclear polyatomic cations can be applied to any experiment in ion-atom hybrid traps. The present study lays the foundation for exploring atomically precise metal clusters and physics from few- to many-body perspective.

Quantum error mitigation has been extensively explored to increase the accuracy of the quantum circuits in noisy-intermediate-scale-quantum (NISQ) computation, where quantum error correction requiring additional quantum resources is not adopted. Among various error-mitigation schemes, probabilistic error cancellation (PEC) has been proposed as a general and systematic protocol that can be applied to numerous hardware platforms and quantum algorithms. However, PEC has only been tested in two-qubit systems and a superconducting multi-qubit system by learning a sparse error model. Here, we benchmark PEC using up to four trapped-ion qubits. For the benchmark, we simulate the dynamics of interacting fermions with or without spins by applying multiple Trotter steps. By tomographically reconstructing the error model and incorporating other mitigation methods such as positive probability and symmetry constraints, we are able to increase the fidelity of simulation and faithfully observe the dynamics of the Fermi-Hubbard model, including the different behavior of charge and spin of fermions. Our demonstrations can be an essential step for further extending systematic error-mitigation schemes toward practical quantum advantages.

The trapped-ion system has been a leading platform for practical quantum computation and quantum simulation since the first scheme of a quantum gate was proposed by Cirac and Zoller in 1995. Quantum gates with trapped ions have shown the highest fidelity among all physical platforms. Recently, sophisticated schemes of quantum gates such as amplitude, phase, frequency modulation, or multi-frequency application, have been developed to make the gates fast, robust to many types of imperfections, and applicable to multiple qubits. Here, we review the basic principle and recent development of quantum gates with trapped ions.

We demonstrate the site-dependent control of polaritons in the Jaynes Cummings Hubbard (JCH) model with trapped ions. In a linear ion crystal under illumination by optical beams nearly resonant to the red-sideband (RSB) transition for the radial vibrational direction, quasiparticles called polaritonic excitations or polaritons, each being a superposition of one internal excitation and one vibrational quantum (phonon), can exist as conserved particles. Polaritons can freely hop between ion sites in a homogeneous configuration, while their motion can be externally controlled by modifying the parameters for the optical beams site-dependently. We demonstrate the blockade of polariton hopping in a system of two ions by the individual control of the frequency of the optical beams illuminating each ion. A JCH system consisting of polaritons in a large number of ion sites can be considered an artificial many-body system of interacting particles and the technique introduced here can be used to exert fine local control over such a system, enabling detailed studies of both its quasi-static and dynamic properties.

One-dimensional systems exhibiting a continuous symmetry can host quantum phases of matter with true long-range order only in the presence of sufficiently long-range interactions. In most physical systems, however, the interactions are short-ranged, hindering the emergence of such phases in one dimension. Here we use a one-dimensional trapped-ion quantum simulator to prepare states with long-range spin order that extends over the system size of up to $23$ spins and is characteristic of the continuous symmetry-breaking phase of matter. Our preparation relies on simultaneous control over an array of tightly focused individual-addressing laser beams, generating long-range spin-spin interactions. We also observe a disordered phase with frustrated correlations. We further study the phases at different ranges of interaction and the out-of-equilibrium response to symmetry-breaking perturbations. This work opens an avenue to study new quantum phases and out-of-equilibrium dynamics in low-dimensional systems.

Imaginary time evolution is a powerful tool applied in quantum physics, while existing classical algorithms for simulating imaginary time evolution suffer high computational complexity as the quantum systems become larger and more complex. In this work, we propose a probabilistic algorithm for implementing imaginary time evolution based on non-unitary quantum circuit. We demonstrate the feasibility of this method by solving the ground state energy of several quantum many-body systems, including H2, LiH molecules and the quantum Ising chain. Moreover, we perform experiments on superconducting and trapped ion cloud platforms respectively to find the ground state energy of H2 and its most stable molecular structure. We also analyze the successful probability of the algorithm, which is a polynomial of the output error and introduce an approach to increase the success probability by rearranging the terms of Hamiltonian.

Strongly-coupled gauge theories far from equilibrium may exhibit unique features that could illuminate the physics of the early universe and of hadron and ion colliders. Studying real-time phenomena has proven challenging with classical-simulation methods, but is a natural application of quantum simulation. To demonstrate this prospect, we quantum compute non-equal time correlation functions and perform entanglement tomography of non-equilibrium states of a simple lattice gauge theory, the Schwinger model, using a trapped-ion quantum computer by IonQ Inc. As an ideal target for near-term devices, a recently-predicted [Zache et al., Phys. Rev. Lett. 122, 050403 (2019)] dynamical quantum phase transition in this model is studied by preparing, quenching, and tracking the subsequent non-equilibrium dynamics in three ways: i) overlap echos signaling dynamical transitions, ii) non-equal time correlation functions with an underlying topological nature, and iii) the entanglement structure of non-equilibrium states, including entanglement Hamiltonians. These results constitute the first observation of a dynamical quantum phase transition in a lattice gauge theory on a quantum computer, and are a first step toward investigating topological phenomena in nuclear and high-energy physics using quantum technologies.

In this work, we design and implement frequency-robust Molmer-Sorensen gates on a linear chain of trapped ions, using Gaussian amplitude modulation and a constant laser frequency. We select this frequency to balance the entanglement accumulation of all motional modes during the gate to produce a strong robustness to frequency error, even for long ion chains. We demonstrate this technique on a three-ion chain, achieving $<\,1\%$ reduction from peak fidelity over a $20\,$kHz range of frequency offset, and we analyze the performance of this gate design through numerical simulations on chains of two to 33 ions.

Storage lifetime and capacity are two important factors to characterize the performance of a quantum memory. Here we report the stable trapping of above 200 ions in a cryogenic setup, and demonstrate the combination of the multi-qubit capacity and long storage lifetime by measuring the coherence time of randomly chosen ions to be on the order of hundreds of milliseconds. We apply composite microwave pulses to manipulate qubit states globally for efficient characterization of different storage units simultaneously, and we compare the performance of the quantum memory with and without the sympathetic cooling laser, thus unambiguously show the necessity of sympathetic cooling for the long-time storage of multiple ionic qubits.

A test of quantumness is a protocol where a classical user issues challenges to a quantum device to determine if it exhibits non-classical behavior, under certain cryptographic assumptions. Recent attempts to implement such tests on current quantum computers rely on either interactive challenges with efficient verification, or non-interactive challenges with inefficient (exponential time) verification. In this paper, we execute an efficient non-interactive test of quantumness on an ion-trap quantum computer. Our results significantly exceed the bound for a classical device's success.

An external drive can improve the coherence of a quantum many-body system by averaging out noise sources. It can also be used to realize models that are inaccessible in the static limit, through Floquet Hamiltonian engineering. The full possibilities for combining these tools remain unexplored. We develop the requirements needed for a pulse sequence to decouple a quantum many-body system from an external field without altering the intended dynamics. Demonstrating this technique experimentally in an ion-trap platform, we show that it can provide a large improvement to coherence in real-world applications. Finally, we engineer an approximate quantum simulation of the Haldane-Shastry model, an exactly solvable paradigm for long-range interacting spins. Our results expand and unify the quantum simulation toolbox.

The idea of exploiting maximally-entangled states as a resource lies at the core of several modalities of quantum information processing, including secure quantum communication, quantum computation, and quantum sensing. However, due to imperfections during or after the entangling gates used to prepare such states, the amount of entanglement decreases and their quality as a resource gets degraded. We introduce a low-overhead protocol to reverse this degradation by partially filtering out a specific type of noise relevant to many quantum technologies. We present two trapped-ion schemes for the implementation of a non-unitary probabilistic filter against amplitude damping noise, which can protect any maximally-entangled pair from spontaneous photon scattering during or after the two-qubit trapped-ion entangling gates. This filter can be understood as a protocol for single-copy quasi-distillation, as it uses only local operations to realise a reversal operation that can be understood in terms of weak measurements.

The trapped ion quantum simulator has demonstrated qualitative properties of different physical models for up to tens of ions. In particular, a linear ion chain naturally hosts long-range Ising interactions under the laser driving, which has been used for various phenomena such as quantum phase transition, localization, thermalization and information propagation. For near-term practical usage, a central task is to find more quantitative applications of the noisy quantum simulators that are robust to small errors in the parameters. Here we report the quantum simulation of a long-range transverse-field Ising model using up to 61 ions and probe the critical behavior of its quantum phase transition through the Kibble-Zurek mechanism. By calibrating and verifying the coupling coefficients, we realize the same model for increasing ion numbers, so as to extract a critical exponent free of the finite size effect. For ferromagnetic interaction, our experimental result agrees well with the previous numerical predictions. As for the anti-ferromagnetic case, signals are too weak to fit a critical exponent due to the frustration in the interaction, but still consistent with the theory.

At Sandia National Laboratories, QSCOUT (the Quantum Scientific Computing Open User Testbed) is an ion-trap based quantum computer built for the purpose of allowing users low-level access to quantum hardware. Commands are executed on the hardware using Jaqal (Just Another Quantum Assembly Language), a programming language designed in-house to support the unique capabilities of QSCOUT. In this work, we describe a batching implementation of our custom software that speeds the experimental run-time through the reduction of communication and upload times. Reducing the code upload time during experimental runs improves system performance by mitigating the effects of drift. We demonstrate this implementation through a set of quantum chemistry experiments using a variational quantum eigensolver (VQE). While developed specifically for this testbed, this idea finds application across many similar experimental platforms that seek greater hardware control or reduced overhead.

Controllable bosonic systems can provide post-classical computational power with sub-universal quantum computational capability. A network that consists of a number of bosons evolving through beam-splitters and phase-shifters between different modes, has been proposed and applied to demonstrate quantum advantages. While the network has been implemented mostly in optical systems with photons, recently alternative realizations have been explored, where major limitations in photonic systems such as photon loss, and probabilistic manipulation can be addressed. Phonons, the quantized excitations of vibrational modes, of trapped ions can be a promising candidate to realize the bosonic network. Here, we experimentally demonstrate a minimal-loss phononic network that can be programmed and in which any phononic states are deterministically prepared and detected. We realize the network with up to four collective-vibrational modes, which can be straightforwardly extended to reveal quantum advantage. We benchmark the performance of the network with an exemplary algorithm of tomography for arbitrary multi-mode states with a fixed total phonon number. We obtain reconstruction fidelities of 94.5 $\pm$ 1.95 % and 93.4 $\pm$ 3.15 % for single-phonon and two-phonon states, respectively. Our experiment demonstrates a clear and novel pathway to scale up a phononic network for various quantum information processing beyond the limitations of classical and other quantum systems.

Measurement-induced state disturbance is a major challenge in obtaining quantum statistics at multiple time points. We propose a method to extract dynamic information from a quantum system at intermediate time points, namely snapshotting quantum dynamics. To this end, we apply classical post-processing after performing the ancilla-assisted measurements to cancel out the impact of the measurements at each time point. Based on this, we reconstruct a multi-time quasi-probability distribution (QPD) that correctly recovers the probability distributions at the respective time points. Our approach can also be applied to simultaneously extract exponentially many correlation functions with various time-orderings. We provide a proof-of-principle experimental demonstration of the proposed protocol using a dual-species trapped-ion system by employing $^{171}\rm{Yb}^+$ and $^{138}\rm{Ba}^+$ ions as the system and the ancilla, respectively. Multi-time measurements are performed by repeated initialization and detection of the ancilla state without directly measuring the system state. The two- and three-time QPDs and correlation functions are reconstructed reliably from the experiment, negativity and complex values in the QPDs clearly indicate a contribution of the quantum coherence throughout dynamics.

The accurate evaluation of the second-order Doppler frequency shift (SODFS) of trapped ions in a linear Paul trap has been studied with experiments and molecular dynamics (MD) simulations. The motion of trapped ions in the trap has three contributions, and we focus on the ion excess micromotion, which is rarely discussed when evaluating the SODFS. Based on the hypothesis that the ion density is uniformly distributed in the radial direction, we propose a new model to accurately evaluate the total SODFS for ion microwave clocks. The effectiveness of the model has been verified both in simulation and experiment, especially for ion ensemble with temperature less than 100 mK. We believe that our new model offers advantages in accurately evaluating the SODFS for the ion trap, especially those of laser-cooled ion microwave clocks based on large ion clouds.

Realizing the potential of near-term quantum computers to solve industry-relevant constrained-optimization problems is a promising path to quantum advantage. In this work, we consider the extractive summarization constrained-optimization problem and demonstrate the largest-to-date execution of a quantum optimization algorithm that natively preserves constraints on quantum hardware. We report results with the Quantum Alternating Operator Ansatz algorithm with a Hamming-weight-preserving XY mixer (XY-QAOA) on trapped-ion quantum computer. We successfully execute XY-QAOA circuits that restrict the quantum evolution to the in-constraint subspace, using up to 20 qubits and a two-qubit gate depth of up to 159. We demonstrate the necessity of directly encoding the constraints into the quantum circuit by showing the trade-off between the in-constraint probability and the quality of the solution that is implicit if unconstrained quantum optimization methods are used. We show that this trade-off makes choosing good parameters difficult in general. We compare XY-QAOA to the Layer Variational Quantum Eigensolver algorithm, which has a highly expressive constant-depth circuit, and the Quantum Approximate Optimization Algorithm. We discuss the respective trade-offs of the algorithms and implications for their execution on near-term quantum hardware.

Jaynes-Cummings-Hubbard (JCH) model is a fundamental many-body model for light-matter interaction. As a leading platform for quantum simulation, the trapped ion system has realized the JCH model for two to three ions. Here we report the quantum simulation of the JCH model using up to 32 ions. We verify the simulation results even for large ion numbers by engineering low excitations and thus low effective dimensions; then we extend to 32 excitations for an effective dimension of $2^{77}$, which is difficult for classical computers. By regarding the phonon modes as baths, we explore Markovian or non-Markovian spin dynamics in different parameter regimes of the JCH model, similar to quantum emitters in a structured photonic environment. We further examine the dependence of the non-Markovian dynamics on the effective Hilbert space dimension. Our work demonstrates the trapped ion system as a powerful quantum simulator for many-body physics and open quantum systems.

Supersymmetry (SUSY) helps solve the hierarchy problem in high-energy physics and provides a natural groundwork for unifying gravity with other fundamental interactions. While being one of the most promising frameworks for theories beyond the Standard Model, its direct experimental evidence in nature still remains to be discovered. Here we report experimental realization of a supersymmetric quantum mechanics (SUSY QM) model, a reduction of the SUSY quantum field theory for studying its fundamental properties, using a trapped ion quantum simulator. We demonstrate the energy degeneracy caused by SUSY in this model and the spontaneous SUSY breaking. By a partial quantum state tomography of the spin-phonon coupled system, we explicitly measure the supercharge of the degenerate ground states, which are superpositions of the bosonic and the fermionic states. Our work demonstrates the trapped-ion quantum simulator as an economic yet powerful platform to study versatile physics in a single well-controlled system.

Quantum computing testbeds exhibit high-fidelity quantum control over small collections of qubits, enabling performance of precise, repeatable operations followed by measurements. Currently, these noisy intermediate-scale devices can support a sufficient number of sequential operations prior to decoherence such that near term algorithms can be performed with proximate accuracy (like chemical accuracy for quantum chemistry). While the results of these algorithms are imperfect, these imperfections can help bootstrap quantum computer testbed development. Demonstrations of these algorithms over the past few years, coupled with the idea that imperfect algorithm performance can be caused by several dominant noise sources in the quantum processor, which can be measured and calibrated during algorithm execution or in post-processing, has led to the use of noise mitigation to improve computational results. Conversely, benchmark algorithms coupled with noise mitigation can help diagnose the nature of noise, whether systematic or purely random. Here, we outline the use of coherent noise mitigation techniques as a characterization tool in trapped-ion testbeds. We perform model-fitting of the noisy data to determine the noise source based on realistic noise models and demonstrate that systematic noise amplification coupled with error mitigation schemes provides useful data for noise model deduction. Further, in order to connect lower level noise model details with application specific performance of near term algorithms, we experimentally construct the loss landscape of a variational algorithm under various injected noise sources coupled with error mitigation techniques. This type of connection enables application-aware hardware codesign, in which the most important noise sources in specific applications, like quantum chemistry, become foci of improvement in subsequent hardware generations.

Ising model is a widely studied class of models in quantum computation. In this paper we investigate the computational characteristics of the random field Ising model (RFIM) with long-range interactions that decays as an inverse polynomial of distance, which can be achieved in current ion trap system. We prove that for an RFIM with long-range interaction embedded on a 2-dimensional plane, solving its ground state is NP-complete for all diminishing exponent, and prove that the 1-dimensional RFIM with long-range interaction can be efficiently approximated when the interaction decays fast enough.

Quantum simulation of 1D relativistic quantum mechanics has been achieved in well-controlled systems like trapped ions, but properties like spin dynamics and response to external magnetic fields that appear only in higher dimensions remain unexplored. Here we simulate the dynamics of a 2D Weyl particle. We show the linear dispersion relation of the free particle and the discrete Landau levels in a magnetic field, and we explicitly measure the spatial and spin dynamics from which the conservation of helicity and properties of antiparticles can be verified. Our work extends the application of an ion trap quantum simulator in particle physics with the additional spatial and spin degrees of freedom.

Most near-term quantum information processing devices will not be capable of implementing quantum error correction and the associated logical quantum gate set. Instead, quantum circuits will be implemented directly using the physical native gate set of the device. These native gates often have a parameterization (e.g., rotation angles) which provide the ability to perform a continuous range of operations. Verification of the correct operation of these gates across the allowable range of parameters is important for gaining confidence in the reliability of these devices. In this work, we demonstrate a procedure for sample-efficient verification of continuously-parameterized quantum gates for small quantum processors of up to approximately 10 qubits. This procedure involves generating random sequences of randomly-parameterized layers of gates chosen from the native gate set of the device, and then stochastically compiling an approximate inverse to this sequence such that executing the full sequence on the device should leave the system near its initial state. We show that fidelity estimates made via this technique have a lower variance than fidelity estimates made via cross-entropy benchmarking. This provides an experimentally-relevant advantage in sample efficiency when estimating the fidelity loss to some desired precision. We describe the experimental realization of this technique using continuously-parameterized quantum gate sets on a trapped-ion quantum processor from Sandia QSCOUT and a superconducting quantum processor from IBM Q, and we demonstrate the sample efficiency advantage of this technique both numerically and experimentally.

We introduce the notion of "generalized bosons" whose exchange statistics resemble those of bosons, but the local bosonic commutator $[a_i,a_i^\dagger]=1$ is replaced by an arbitrary single-mode operator that is diagonal in the generalized Fock basis. Examples of generalized bosons include boson pairs and spins. We consider the analogue of the boson sampling task for these particles and observe that its output probabilities are still given by permanents, so that the results regarding hardness of sampling directly carry over. Finally, we propose implementations of generalized boson sampling in circuit-QED and ion-trap platforms.

Two-dimensional (2D) quantum magnetism is a paradigm in strongly correlated many-body physics. The understanding of 2D quantum magnetism can be expedited by employing a controllable quantum simulator that faithfully maps 2D-spin Hamiltonians. The 2D quantum simulators can exhibit exotic phenomena such as frustrated quantum magnetism and topological order and can be used to show quantum computational advantages. Many experimental platforms are being developed, including Rydberg atoms and superconducting annealers. However, with trapped-ion systems, which showed the most advanced controllability and quantum coherence, quantum magnetism was explored in one-dimensional chains. Here, we report simulations of frustrated quantum magnetism with 2D ion crystals. We create a variety of spin-spin interactions for quantum magnets, including those that exhibit frustration by driving different vibrational modes and adiabatically prepare the corresponding ground states. The experimentally measured ground states are consistent with the theoretical predictions and are highly degenerate for geometrically frustrated spin models in two dimensions. Quantum coherence of the ground states is probed by reversing the time evolution of the B-field to the initial value and then measuring the extent to which the remaining state coincides with the initial state. Our results open the door for quantum simulations with 2D ion crystals.

The ion microwave frequency standard is a candidate for the next generation of microwave frequency standard with the potential for very wide applications. The Dick effect and second-order Doppler frequency shift (SODFS) limit the performance of ion microwave frequency standards. The introduction of sympathetic cooling technology can suppress the Dick effect and SODFS and improve the stability and accuracy of the frequency standard. However, the sympathetically-cooled ion microwave frequency standard has seldom been studied before. This paper reports the first sympathetically-cooled ion microwave frequency standard in a Paul trap. Using laser-cooled ${}^{40}\mathrm{Ca}^{+}$ as coolant ions, ${}^{113}\mathrm{Cd}^{+}$ ion crystal is cooled to below 100 mK and has a coherence lifetime of over 40 s. The short-term frequency stability reached $3.48 \times 10^{-13}/τ^{1/2}$, which is comparable to that of the mercury ion frequency standard. Its uncertainty is $1.5\times 10^{-14}$, which is better than that of directly laser-cooled cadmium ion frequency standard.

Mitigating errors in quantum information processing devices is especially important in the absence of fault tolerance. An effective method in suppressing state-preparation errors is using multiple copies to distill the ideal component from a noisy quantum state. Here, we use classical shadows and randomized measurements to circumvent the need for coherent access to multiple copies at an exponential cost. We study the scaling of resources using numerical simulations and find that the overhead is still favorable compared to full state tomography. We optimize measurement resources under realistic experimental constraints and apply our method to an experiment preparing Greenberger-Horne-Zeilinger (GHZ) state with trapped ions. In addition to improving stabilizer measurements, the analysis of the improved results reveals the nature of errors affecting the experiment. Hence, our results provide a directly applicable method for mitigating errors in near-term quantum computers.

Quantum simulators have recently enabled experimental observations of quantum many-body systems' internal thermalization. Often, the global energy and particle number are conserved, and the system is prepared with a well-defined particle number - in a microcanonical subspace. However, quantum evolution can also conserve quantities, or charges, that fail to commute with each other. Noncommuting charges have recently emerged as a subfield at the intersection of quantum thermodynamics and quantum information. Until now, this subfield has remained theoretical. We initiate the experimental testing of its predictions, with a trapped-ion simulator. We prepare 6-21 spins in an approximate microcanonical subspace, a generalization of the microcanonical subspace for accommodating noncommuting charges, which cannot necessarily have well-defined nontrivial values simultaneously. We simulate a Heisenberg evolution using laser-induced entangling interactions and collective spin rotations. The noncommuting charges are the three spin components. We find that small subsystems equilibrate to near a recently predicted non-Abelian thermal state. This work bridges quantum many-body simulators to the quantum thermodynamics of noncommuting charges, whose predictions can now be tested.

Dissipation is often considered as a detrimental effect in quantum systems for unitary quantum operations. However, it has been shown that suitable dissipation can be useful resources both in quantum information and quantum simulation. Here, we propose and experimentally simulate a dissipative phase transition (DPT) model using a single trapped ion with an engineered reservoir. We show that the ion's spatial oscillation mode reaches a steady state after the alternating application of unitary evolution under a quantum Rabi model Hamiltonian and sideband cooling of the oscillator. The average phonon number of the oscillation mode is used as the order parameter to provide evidence for the DPT. Our work highlights the suitability of trapped ions for simulating open quantum systems and shall facilitate further investigations of DPT with various dissipation terms.

Regarding trapped-ion microwave-frequency standards, we report on the determination of hyperfine splittings and Landé $g_J$ factors of $^{111,113}$Cd$^+$. The hyperfine splittings of the $5p~^2P_{3/2}$ state of $^{111,113}$Cd$^+$ ions were measured using laser-induced fluorescence spectroscopy. The Cd$^+$ ions were confined in a linear Paul trap and sympathetically cooled by Ca$^+$ ions. Furthermore, the hyperfine splittings and Landé $g_J$ factors of the $5s~^2S_{1/2}$ and $5p~^2P_{1/2,3/2}$ levels of $^{111,113}$Cd$^+$ were calculated with greater accuracy using the multiconfiguration Dirac--Hartree--Fock scheme. The measured hyperfine splittings and the Dirac--Hartree--Fock calculation values were cross-checked, thereby further guaranteeing the reliability of our results. The results provided in this work can improve the signal-to-noise ratio of the clock transition and the accuracy of the second-order Zeeman shift correction, and subsequently the stability and accuracy of the microwave frequency standard based on trapped Cd$^+$ ions.

We propose and study ways speeding up of the entangling operations in the trapped ions system with high fidelity. First, we find a scheme to increase the speed of a two-qubit gate without the limitation of trap frequency, which was considered as the fundamental limit. Second, we study the fast gate scheme for entangling more than two qubits simultaneously. We apply the method of applying multiple frequency components on laser beams for the gate operations. In particular, in order to avoid infinite terms from the coupling to carrier transition, we focus on the phase-insensitive gate scheme here. We carefully study the effect of large excitation of motional mode beyond the limit of Lamb-Dicke approximation by including up to second order terms of the Lamb-Dicke parameter. We study the speed limit of multi-qubit global entangling gates without individual addressing requirements. Furthermore, our gates can be made insensitive to the fluctuation of initial motional phases which are difficult to stabilise in the phase-insensitive gate scheme.

We have sympathetically cooled and crystallized ${}^{113}\mathrm{Cd}^{+}$ ions with laser-cooled ${}^{40}\mathrm{Ca}^{+}$ ions, and directly observed the complete large bicrystal structure in a linear Paul trap. The large two-component crystal contains up to $3.5\times10^3$ ${}^{40}\mathrm{Ca}^{+}$ ions and $6.8\times10^3$ ${}^{113}\mathrm{Cd}^{+}$ ions. The temperature of the crystallized ${}^{113}\mathrm{Cd}^{+}$ ions was measured to be as low as 41 mK with a large mass ratio. Moreover, we have studied several properties and structures of the ultracold sample. The factors affecting the sympathetic cooling effect were studied, including the electrical parameters and the number ratio between laser-cooled ions and sympathetically-cooled ions. The results of this paper enrich the experimental research of large two-component ion crystals, and the ultracold sample of ${}^{113}\mathrm{Cd}^{+}$ ions makes it possible to further improve the accuracy of the cadmium-ion microwave frequency standard.

We report on progress towards a microwave frequency standard based on a laser-cooled 171Yb+ ion trap system. The electronics, lasers, and magnetic shields are integrated into a single physical package. With over 1E5 ions are stably trapped, the system offers a high signal-to-noise ratio Ramsey line-shape. In comparison with previous work, the frequency instability of a 171Yb+ microwave clock was further improved to $8.5 \times {10^{ - 13}}/\sqrt τ$ for averaging times between 10 and 1000 s.

In this paper, we present a dynamical decoupling (DD) technique to coherently control the dynamics of a single local phonon in trapped ions. A 2$π$ rotation at a motional sideband transition flips the sign of the relevant local phonon state, resulting in cancellation of the phonon dynamics. In this work, we implement DD using single and multiple blue-sideband pulses to control a single local phonon in two $^{40}{\rm Ca}^{+}$ ions in a linear Paul trap. Our proposed DD technique can be used to engineer coupling between local phonon modes.

Quantum measurements cannot be thought of as revealing preexisting results, even when they do not disturb any other measurement in the same trial. This feature is called contextuality and is crucial for the quantum advantage in computing. Here, we report the first observation of quantum contextuality simultaneously free of the detection, sharpness and compatibility loopholes. The detection and sharpness loopholes are closed by adopting a hybrid two-ion system and highly efficient fluorescence measurements offering a detection efficiency of $100\%$ and a measurement repeatability $>98\%$. The compatibility loophole is closed by targeting correlations between observables for two different ions in a Paul trap, a $^{171}\mathrm{Yb}^{+}$ ion and a $^{138}\mathrm{Ba}^{+}$ ion, chosen so measurements on each ion use different operation laser wavelengths, fluorescence wavelengths, and detectors. The experimental results show a violation of the bound for the most adversarial noncontextual models and open a new way to certify quantum systems.

We discuss how to simplify the Molmer-Sorensen (MS) gate which is used for the trapped ion quantum computer. The original MS gate is implemented by illuminating two ions with bichromatic coherent light fields separately at the same time. In this paper, we propose a method for transforming a separable state of two ions into one of the Bell states by illuminating the two ions with monochromatic coherent light fields individually and this point is the advantage of our scheme over the original MS gate. The length of the execution time of our proposed gate is comparable to that of the original MS gate, however, numerical calculations show that our proposed gate is weakly sensitive to thermal fluctuations of the phonons. By giving another example of a simple two-ion gate that can generate entanglement but is strongly vulnerable to thermal fluctuations, we show that our simplified MS gate is more marked than usual.

Achieving quantum computational advantage requires solving a classically intractable problem on a quantum device. Natural proposals rely upon the intrinsic hardness of classically simulating quantum mechanics; however, verifying the output is itself classically intractable. On the other hand, certain quantum algorithms (e.g. prime factorization via Shor's algorithm) are efficiently verifiable, but require more resources than what is available on near-term devices. One way to bridge the gap between verifiability and implementation is to use "interactions" between a prover and a verifier. By leveraging cryptographic functions, such protocols enable the classical verifier to enforce consistency in a quantum prover's responses across multiple rounds of interaction. In this work, we demonstrate the first implementation of an interactive quantum advantage protocol, using an ion trap quantum computer. We execute two complementary protocols -- one based upon the learning with errors problem and another where the cryptographic construction implements a computational Bell test. To perform multiple rounds of interaction, we implement mid-circuit measurements on a subset of trapped ion qubits, with subsequent coherent evolution. For both protocols, the performance exceeds the asymptotic bound for classical behavior; maintaining this fidelity at scale would conclusively demonstrate verifiable quantum advantage.

We theoretically propose and experimentally demonstrate a parallel-electromagnetically-induced transparency (parallel-EIT) cooling technique for ion crystals in the Paul trap. It has less stringent requirements on the cooling resonance condition than the standard electromagnetically-induced transparency (EIT) cooling, thus allowing, in principle, to simultaneously cool the motional mode spectrum with an arbitrary range. A proof-of-principle validation for this cooling scheme is experimentally demonstrated with up to 4 trapped 40Ca+ ions. We observe simultaneous near-ground-state cooling for all motional modes with best average phonon number about 0.2. By tuning the trap frequency in a large range to imitate a broadband motional mode spectrum, we can still reach almost the same cooling limit for all the modes while standard EIT cooling shows limited cooling range. Our method has a simple experimental configuration, requiring only appropriate modulation of the probe beam of standard EIT cooling, and can be applied to various types of ions (e.g., 171Yb+, 40Ca+). This cooling scheme provides a powerful tool for the initialization of the trapped-ion quantum computers and simulators.

Complementarity is an essential feature of quantum mechanics. The preparation of an eigenstate of one observable implies complete randomness in its complementary observable. In quantum cryptography, complementarity allows us to formulate security analyses in terms of phase-error correction. However, in the device-independent regime that offers security without device characterization, the concept becomes much subtler. Security proofs of device-independent quantum cryptography tasks are often complex and quite different from those of their more standard device-dependent cousins. The existing proofs pose huge challenges to experiments, among which large data-size requirement is a crux. Here, we show the complementarity security origin of the device-independent tasks. By linking complementarity with quantum nonlocality, we recast the device-independent scheme into a quantum error correction protocol. Going beyond the identical-and-independent-distribution case, we consider the most general attack. We generalize the sample entropy in classical Shannon theory for the finite-size analysis. Our method exhibits good finite-size performance and brings the device-independent scheme to a more practical regime. Applying it to the data in a recent ion-trap-based device-independent quantum key distribution experiment, one could reduce the requirement on data size to less than a third. Furthermore, the complementarity approach can be naturally extended to advantage key distillation to ease experiments by tolerating higher loss and lower transmittance.

The rotation of subspaces by a chosen angle is a fundamental quantum computing operation, with applications in error correction and quantum algorithms such as the Quantum Approximate Optimization Algorithm, the Variational Quantum Eigensolver and the quantum singular value transformation. Such rotations are usually implemented at the hardware level via multiple-controlled-phase gates, which lead to large circuit depth when decomposed into one- and two-qubit gates. Here, we propose a fast, high-fidelity way to implement such operations via topological quantum walks, where a sequence of single-qubit $z$ rotations of an ancilla qubit are interleaved with the evolution of a system Hamiltonian in which a matrix $A$ is embedded. The subspace spanned by the left or right singular vectors of $A$ with non-zero singular values is rotated, depending on the state of the ancilla. This procedure can be implemented in superconducting qubits, ion-traps and Rydberg atoms with star-type connectivity, significantly reducing the total gate time required compared to previous proposals.

We measured the ground-state hyperfine splitting of trapped ${}^{113}$Cd${}^{+}$ ions to be 15199862855.02799(27) Hz with a fractional uncertainty of $1.8\times10^{-14}$. The ions were trapped and laser-cooled in a linear quadrupole Paul trap. The fractional frequency stability was measured to be ${4.2} \times 10^{-13}/\sqrtτ $, obtained from Ramsey fringes of high signal-to-noise ratio and taken over a measurement time of nearly 5 hours, which is close to the short-term stability limit estimated from the Dick effect. Our result is consistent with previous reported values, but the measurement precision is four times better than the best result obtained to date.

We propose a set of Bell-type nonlocal games that can be used to prove an unconditional quantum advantage in an objective and hardware-agnostic manner. In these games, the circuit depth needed to prepare a cyclic cluster state and measure a subset of its Pauli stabilizers on a quantum computer is compared to that of classical Boolean circuits with the same, nearest-neighboring gate connectivity. Using a circuit-based trapped-ion quantum computer, we prepare and measure a six-qubit cyclic cluster state with an overall fidelity of 60.6% and 66.4%, before and after correcting for measurement-readout errors, respectively. Our experimental results indicate that while this fidelity readily passes conventional (or depth-0) Bell bounds for local hidden-variable models, it is on the cusp of demonstrating a higher probability of success than what is possible by depth-1 classical circuits. Our games offer a practical and scalable set of quantitative benchmarks for quantum computers in the pre-fault-tolerant regime as the number of qubits available increases.

Quantum simulation provides important tools in studying strongly correlated many-body systems with controllable parameters. As a hybrid of two fundamental models in quantum optics and in condensed matter physics, the Rabi-Hubbard model demonstrates rich physics through the competition between local spin-boson interactions and long-range boson hopping. Here we report an experimental realization of the Rabi-Hubbard model using up to $16$ trapped ions and present a controlled study of its equilibrium properties and quantum dynamics. We observe the ground-state quantum phase transition by slowly quenching the coupling strength, and measure the quantum dynamical evolution in various parameter regimes. With the magnetization and the spin-spin correlation as probes, we verify the prediction of the model Hamiltonian by comparing theoretical results in small system sizes with experimental observations. For larger-size systems of $16$ ions and $16$ phonon modes, the effective Hilbert space dimension exceeds $2^{57}$, whose dynamics is intractable for classical supercomputers.

Noisy intermediate-scale quantum (NISQ) hardware is typically limited to low-depth quantum circuits to limit the number of opportunities for introduction of error by unreliable quantum gates. A less-explored alternative approach is to repeatedly apply a quantum channel with a desired quantum state as a stable fixed point. Increased circuit depth can in this case be beneficial rather than harmful due to dissipative self-correction. The quantum channels constructed from MERA circuits can be interpreted in terms of the renormalization group(RG), and their fixed points are RG fixed points, i.e. scale-invariant systems such as conformal field theories. Here, building upon the theoretical proposal of Kim and Swingle, we numerically and experimentally study the robust preparation of the ground state of the critical Ising model using circuits adapted from the work of Evenbly and White. The experimental implementation exhibits self-correction through renormalization seen in the convergence and stability of local observables, and makes essential use of the ability to measure and reset individual qubits afforded by the "quantum CCD" architecture of the Honeywell ion-trap. We also numerically test error mitigation by zero-noise extrapolation schemes specially adapted for renormalization circuits, which are able to outperform typical extrapolation schemes using lower gate overhead.

Parallel operations in conventional computing have proven to be an essential tool for efficient and practical computation, and the story is not different for quantum computing. Indeed, there exists a large body of works that study advantages of parallel implementations of quantum gates for efficient quantum circuit implementations. Here, we focus on the recently invented efficient, arbitrary, simultaneously entangling (EASE) gates, available on a trapped-ion quantum computer. Leveraging its flexibility in selecting arbitrary pairs of qubits to be coupled with any degrees of entanglement, all in parallel, we show an $n$-qubit Clifford circuit can be implemented using $6\log(n)$ EASE gates, an $n$-qubit multiply-controlled NOT gate can be implemented using $3n/2$ EASE gates, and an $n$-qubit permutation can be implemented using six EASE gates. We discuss their implications to near-term quantum chemistry simulations and the state of the art pattern matching algorithm. Given Clifford + multiply-controlled NOT gates form a universal gate set for quantum computing, our results imply efficient quantum computation by EASE gates, in general.

Trapped ions constitute one of the most promising systems for implementing quantum computing and networking. For large-scale ion-trap-based quantum computers and networks, it is critical to have two types of qubits, one for computation and storage, while the other for auxiliary operations like runtime qubit detection, sympathetic cooling, and repetitive entanglement generation through photon links. Dual-type qubits have previously been realized in hybrid systems using two ion species, which, however, introduces significant experimental challenges for laser setup, gate operations as well as the control of the fraction and positioning of each qubit type within an ion crystal. Here we solve these problems by implementing two coherently-convertible qubit types using the same ion species. We encode the qubits into two pairs of clock states of the 171Yb+ ions, and achieve fast and high-fidelity conversion between the two types using narrow-band lasers. We further demonstrate that operations on one qubit type, including sympathetic laser cooling, gates and qubit detection, have crosstalk errors less than 0.03% on the other type, well below the error threshold for fault-tolerant quantum computing. Our work showcases the feasibility and advantages of using coherently convertible dual-type qubits with the same ion species for future large-scale quantum computing and networking.

Cooling down a trapped ion into its motional ground state is a central step for trapped ions based quantum information processing. State of the art cooling schemes often work under a set of optimal cooling conditions derived analytically using a perturbative approach, in which the sideband coupling is assumed to be the weakest of all the relevant transitions. As a result the cooling rate is severely limited. Here we propose to use quantum control technique powered with automatic differentiation to speed up the classical cooling schemes. We demonstrate the efficacy of our approach by applying it to find the optimal cooling conditions for classical sideband cooling and electromagnetically induced transparency cooling schemes, which are in general beyond the weak sideband coupling regime. Based on those numerically found optimal cooling conditions, we show that faster cooling can be achieved while at the same time a low average phonon occupation can be retained.

Many-body open quantum systems balance internal dynamics against decoherence from interactions with an environment. Here, we explore this balance via random quantum circuits implemented on a trapped ion quantum computer, where the system evolution is represented by unitary gates with interspersed projective measurements. As the measurement rate is varied, a purification phase transition is predicted to emerge at a critical point akin to a fault-tolerent threshold. We probe the "pure" phase, where the system is rapidly projected to a deterministic state conditioned on the measurement outcomes, and the "mixed" or "coding" phase, where the initial state becomes partially encoded into a quantum error correcting codespace. We find convincing evidence of the two phases and show numerically that, with modest system scaling, critical properties of the transition clearly emerge.

We propose and demonstrate a scalable scheme for the simultaneous determination of internal and motional states in trapped ions with single-site resolution. The scheme is applied to the study of polaritonic excitations in the Jaynes- Cummings Hubbard model with trapped ions, in which the internal and motional states of the ions are strongly correlated. We observe quantum phase transitions of polaritonic excitations in two ions by directly evaluating their variances per ion site. Our work establishes an essential technological method for large-scale quantum simulations of polaritonic systems.

We study variants of Shor's code that are adept at handling single-axis correlated idling errors, which are commonly observed in many quantum systems. By using the repetition code structure of the Shor's code basis states, we calculate the logical channel applied to the encoded information when subjected to coherent and correlated single qubit idling errors, followed by stabilizer measurement. Changing the signs of the stabilizer generators allows us to change how the coherent errors interfere, leading to a quantum error correcting code which performs as well as a classical repetition code of equivalent distance against these errors. We demonstrate a factor of 4 improvement of the logical memory in a distance-3 logical qubit implemented on a trapped-ion quantum computer. Even-distance versions of our Shor code variants are decoherence-free subspaces and fully robust to identical and independent coherent idling noise.

Electromagnetically induced transparency (EIT) cooling has established itself as one of the most widely used cooling schemes for trapped ions during the past twenty years. Compared to its alternatives, EIT cooling possesses important advantages such as a tunable effective linewidth, a very low steady state phonon occupation, and applicability for multiple ions. However, existing analytic expression for the steady state phonon occupation of EIT cooling is limited to the zeroth order of the Lamb-Dicke parameter. Here we extend such calculations and present the explicit expression to the second order of the Lamb-Dicke parameter. We discuss several implications of our refined formula and are able to resolve certain difficulties in existing results.

Electrons and ions trapped with electromagnetic fields have long served as important high-precision metrological instruments, and more recently have also been proposed as a platform for quantum information processing. Here we point out that these systems can also be used as highly sensitive detectors of passing charged particles, due to the combination of their extreme charge-to-mass ratio and low-noise quantum readout and control. In particular, these systems can be used to detect energy depositions many orders of magnitude below typical ionization scales. As illustrations, we suggest some applications in particle physics. We outline a non-destructive time-of-flight measurement capable of sub-eV energy resolution for slowly moving, collimated particles. We also show that current devices can be used to provide competitive sensitivity to models where ambient dark matter particles carry small electric millicharges $\ll e$. Our calculations may also be useful in the characterization of noise in quantum computers coming from backgrounds of charged particles.

The Quantum Scientific Computing Open User Testbed (QSCOUT) at Sandia National Laboratories is a trapped-ion qubit system designed to evaluate the potential of near-term quantum hardware in scientific computing applications for the US Department of Energy (DOE) and its Advanced Scientific Computing Research (ASCR) program. Similar to commercially available platforms, most of which are based on superconducting qubits, it offers quantum hardware that researchers can use to perform quantum algorithms, investigate noise properties unique to quantum systems, and test novel ideas that will be useful for larger and more powerful systems in the future. However, unlike most other quantum computing testbeds, QSCOUT uses trapped $^{171}$Yb$^{+}$ ions as the qubits, provides full connectivity between qubits, and allows both quantum circuit and low-level pulse control access to study new modes of programming and optimization. The purpose of this manuscript is to provide users and the general community with details of the QSCOUT hardware and its interface, enabling them to take maximum advantage of its capabilities.

Vibrational degrees of freedom in trapped-ion systems have recently been gaining attention as a quantum resource, beyond the role as a mediator for entangling quantum operations on internal degrees of freedom, because of the large available Hilbert space. The vibrational modes can be represented as quantum harmonic oscillators and thus offer a Hilbert space with infinite dimension. Here we review recent theoretical and experimental progress in the coherent manipulation of the vibrational modes, including bosonic encoding schemes in quantum information, reliable and efficient measurement techniques, and quantum operations that allow various quantum simulations and quantum computation algorithms. We describe experiments using the vibrational modes, including the preparation of non-classical states, molecular vibronic sampling, and applications in quantum thermodynamics. We finally discuss the potential prospects and challenges of trapped-ion vibrational-mode quantum information processing.

Noncommuting conserved quantities have recently launched a subfield of quantum thermodynamics. In conventional thermodynamics, a system of interest and an environment exchange quantities -- energy, particles, electric charge, etc. -- that are globally conserved and are represented by Hermitian operators. These operators were implicitly assumed to commute with each other, until a few years ago. Freeing the operators to fail to commute has enabled many theoretical discoveries -- about reference frames, entropy production, resource-theory models, etc. Little work has bridged these results from abstract theory to experimental reality. This paper provides a methodology for building this bridge systematically: We present a prescription for constructing Hamiltonians that conserve noncommuting quantities globally while transporting the quantities locally. The Hamiltonians can couple arbitrarily many subsystems together and can be integrable or nonintegrable. Our Hamiltonians may be realized physically with superconducting qudits, with ultracold atoms, and with trapped ions.

We have experimentally observed the dynamics of a single polariton and two polaritons in a two-ion chain. By driving two trapped ions at a motional blue-sideband transition, we realize the anti-Jaynes-Cummings-Hubbard model. When a single polariton exists in a trapped-ion chain, the polariton hops between the ion sites. On the other hand, when there are single polaritons at each ion site, the hopping of the polaritons is suppressed because of the polariton-polariton interaction induced by the nonlinearity of the anti-Jaynes-Cummings interaction, thereby realizing the blockade of polariton hopping in the anti-Jaynes-Cummings-Hubbard model with trapped ions. Our work is a step towards the development of a trapped-ion based quantum simulator for strongly interacting polaritonic systems.

Thermalization is a ubiquitous process of statistical physics, in which details of few-body observables are washed out in favor of a featureless steady state. Even in isolated quantum many-body systems, limited to reversible dynamics, thermalization typically prevails. However, in these systems, there is another possibility: many-body localization (MBL) can result in preservation of a non-thermal state. While disorder has long been considered an essential ingredient for this phenomenon, recent theoretical work has suggested that a quantum many-body system with a uniformly increasing field -- but no disorder -- can also exhibit MBL, resulting in `Stark MBL.' Here we realize Stark MBL in a trapped-ion quantum simulator and demonstrate its key properties: halting of thermalization and slow propagation of correlations. Tailoring the interactions between ionic spins in an effective field gradient, we directly observe their microscopic equilibration for a variety of initial states, and we apply single-site control to measure correlations between separate regions of the spin chain. Further, by engineering a varying gradient, we create a disorder-free system with coexisting long-lived thermalized and nonthermal regions. The results demonstrate the unexpected generality of MBL, with implications about the fundamental requirements for thermalization and with potential uses in engineering long-lived non-equilibrium quantum matter.

Quantum phase transitions (QPTs) are usually associated with many-body systems with large degrees of freedom approaching the thermodynamic limit. In such systems, the many-body ground state shows abrupt changes at zero temperature when the control parameter of the Hamiltonian is scanned across a quantum critical point. Recently it has been realized that a QPT can also occur in a simple system composed of only a two-level atom and a single-mode bosonic field, described by the quantum Rabi model (QRM). Here we report the first experimental demonstration of a QPT in the QRM using a single trapped ion. We measure the average spin-up state population of the ion and the average phonon number in its spatial oscillation mode as two order parameters and observe the clear evidences of the phase transition via slow quench of the coupling between the ion and its spatial motion. An experimental probe of the phase transitions in a fundamental quantum optics model without imposing the thermodynamic limit opens up a new window for the controlled study of QPTs and quantum critical phenomena.

The conventional framework for defining and understanding phases of matter requires thermodynamic equilibrium. Extensions to non-equilibrium systems have led to surprising insights into the nature of many-body thermalization and the discovery of novel phases of matter, often catalyzed by driving the system periodically. The inherent heating from such Floquet drives can be tempered by including strong disorder in the system, but this can also mask the generality of non-equilibrium phases. In this work, we utilize a trapped-ion quantum simulator to observe signatures of a non-equilibrium driven phase without disorder: the prethermal discrete time crystal (PDTC). Here, many-body heating is suppressed not by disorder-induced many-body localization, but instead via high-frequency driving, leading to an expansive time window where non-equilibrium phases can emerge. We observe a number of key features that distinguish the PDTC from its many-body-localized disordered counterpart, such as the drive-frequency control of its lifetime and the dependence of time-crystalline order on the energy density of the initial state. Floquet prethermalization is thus presented as a general strategy for creating, stabilizing and studying intrinsically out-of-equilibrium phases of matter.

We demonstrate a microfabricated surface-electrode ion trap that is applicable as a nanofriction emulator and studies of many-body dynamics of interacting systems. The trap enables both single-well and double-well trapping potentials in the radial direction, where the distance between the two potential wells can be adjusted by the applied RF voltage. In the double-well configuration, parallel ion strings can be formed, which is a suitable system for the emulation of the Frenkel-Kontorova (FK) model. We derive the condition under which the trap functions as a FK model emulator. The trap is designed so that the Coulomb interaction between two ion strings becomes significant. We report on the microfabrication process for such downsized trap electrodes and experimental results of single-well and double-well operation with calcium ions. With the trap demonstrated in this work we can create atomically accessible, self-assembled Coulomb systems with a wide tuning range of the corrugation parameter in the FK model. This makes it a promising system for quantum simulations, but also for the study of nanofriction in one and higher dimensional systems.

The generation and verification of large-scale entanglement are essential to the development of quantum technologies. In this paper, we present an efficient scheme to generate genuine multipartite entanglement of a large number of qubits by using the Heisenberg interaction. This method can be conveniently implemented in various physical platforms, including superconducting, trapped-ion, and cold-atom systems. In order to characterize the entanglement of the output quantum state, we generalize the stabilizer formalism and develop an entanglement witness method. In particular, we design a generic searching algorithm to optimize entanglement witness with a minimal number of measurement settings under a given noise level. From the perspective of practical applications, we numerically study the trade-off between the experiment efficiency and the detection robustness.

Trapped ion in the Lamb-Dicke regime with the Lamb-Dicke parameter $η\ll1$ can be cooled down to its motional ground state using sideband cooling. Standard sideband cooling works in the weak sideband coupling limit, where the sideband coupling strength is small compared to the natural linewidth $γ$ of the internal excited state, with a cooling rate much less than $γ$. Here we consider cooling schemes in the strong sideband coupling regime, where the sideband coupling strength is comparable or even greater than $γ$. We derive analytic expressions for the cooling rate and the average occupation of the motional steady state in this regime, based on which we show that one can reach a cooling rate which is proportional to $γ$, while at the same time the steady state occupation increases by a correction term proportional to $η^{2}$ compared to the weak sideband coupling limit. We demonstrate with numerical simulations that our analytic expressions faithfully recover the exact dynamics in the strong sideband coupling regime.

Ion trap is one of the most promising candidates for quantum computing. Current schemes mainly focus on a linear chain of up to about one hundred ions in a Paul trap. To further scale up the qubit number, one possible direction is to use 2D or 3D ion crystals (Wigner crystals). In these systems, ions are generally subjected to large micromotion due to the strong fast-oscillating electric field, which can significantly influence the performance of entangling gates. In this work, we develop an efficient numerical method to design high-fidelity entangling gates in a general 3D ion crystal. We present numerical algorithms to solve the equilibrium configuration of the ions and their collective normal modes. We then give a mathematical description of the micromotion and use it to generalize the gate scheme for linear ion chains into a general 3D crystal. The involved time integral of highly oscillatory functions is expanded into a fast-converging series for accurate and efficient evaluation and optimization. As a numerical example, we show a high-fidelity entangling gate design between two ions in a 100-ion crystal, with a theoretical fidelity of 99.9\%.

Quantum error correction protects fragile quantum information by encoding it into a larger quantum system. These extra degrees of freedom enable the detection and correction of errors, but also increase the operational complexity of the encoded logical qubit. Fault-tolerant circuits contain the spread of errors while operating the logical qubit, and are essential for realizing error suppression in practice. While fault-tolerant design works in principle, it has not previously been demonstrated in an error-corrected physical system with native noise characteristics. In this work, we experimentally demonstrate fault-tolerant preparation, measurement, rotation, and stabilizer measurement of a Bacon-Shor logical qubit using 13 trapped ion qubits. When we compare these fault-tolerant protocols to non-fault tolerant protocols, we see significant reductions in the error rates of the logical primitives in the presence of noise. The result of fault-tolerant design is an average state preparation and measurement error of 0.6% and a Clifford gate error of 0.3% after error correction. Additionally, we prepare magic states with fidelities exceeding the distillation threshold, demonstrating all of the key single-qubit ingredients required for universal fault-tolerant operation. These results demonstrate that fault-tolerant circuits enable highly accurate logical primitives in current quantum systems. With improved two-qubit gates and the use of intermediate measurements, a stabilized logical qubit can be achieved.

Interacting quantum systems illustrate complex phenomena including phase transitions to novel ordered phases. The universal nature of critical phenomena reduces their description to determining only the transition temperature and the critical exponents. Numerically calculating these results for systems in new universality classes is complicated due to critical slowing down, requiring increasing resources near the critical point. An alternative approach analytically continues the calculation onto the complex plane and determines the partition function via its zeros. Here we show how to robustly perform this analysis on noisy intermediate scale trapped ion quantum computers in a scalable manner, using the XXZ model as a prototype. We illustrate the transition from XY-like behavior to Ising-like behavior as a function of the anisotropy. While quantum computers cannot yet scale to the thermodynamic limit, our work provides a pathway to do so as hardware improves, allowing the determination of critical phenomena for systems that cannot be solved otherwise.

Individual trapped atomic qubits represent one of the most promising technologies to scale quantum computers, owing to their negligible idle errors and the ability to implement a full set of reconfigurable gate operations via focused optical fields. However, the fidelity of quantum gate operations can be limited by weak confinement of the atoms transverse to the laser. We present measurements of this effect by performing individually-addressed entangling gates in chains of up to 25 trapped atomic ions that are weakly confined along the chain axis. We present a model that accurately describes the observed decoherence from the residual heating of the ions caused by noisy electric fields. We propose to suppress these effects through the use of ancilla ions interspersed in the chain to sympathetically cool the qubit ions throughout a quantum circuit.

Parity-time($\mathcal{PT}$)-symmetric systems, featuring real eigenvalues despite its non-Hermitian nature, have been widely utilized to achieve exotic functionalities in the classical realm, such as loss-induced transparency or lasing revival. By approaching the exceptional point (EP) or the coalescences of both eigenvalues and eigenstates, unconventional effects are also expected to emerge in pure quantum $\mathcal{PT}$ devices. Here, we report experimental evidences of spontaneous $\mathcal{PT}$ symmetry breaking in a single cold $^{40}\mathrm{Ca}^{+}$ ion, and more importantly, a counterintuitive effect of perfect quantum coherence occurring at the EP. Excellent agreement between experimental results and theoretical predictions is identified. In view of the versatile role of cold ions in building quantum memory or processor, our experiment provides a new platform to explore and utilize pure quantum EP effects, with diverse applications in quantum engineering of trapped ions.

A disordered system of interacting particles exhibits localized behavior when the disorder is large compared to the interaction strength. Studying this phenomenon on a quantum computer without error correction is challenging because even weak coupling to a thermal environment destroys most signatures of localization. Fortunately, spectral functions of local operators are known to contain features that can survive the presence of noise. In these spectra, discrete peaks and a soft gap at low frequencies compared to the thermal phase indicate localization. Here, we present the computation of spectral functions on a trapped-ion quantum computer for a one-dimensional Heisenberg model with disorder. Further, we design an error-mitigation technique which is effective at removing the noise from the measurement allowing clear signatures of localization to emerge as the disorder increases. Thus, we show that spectral functions can serve as a robust and scalable diagnostic of many-body localization on the current generation of quantum computers.

Extracting tomographic information about quantum states is a crucial task in the quest towards devising high-precision quantum devices. Current schemes typically require measurement devices for tomography that are a priori calibrated to high precision. Ironically, the accuracy of the measurement calibration is fundamentally limited by the accuracy of state preparation, establishing a vicious cycle. Here, we prove that this cycle can be broken and the dependence on the measurement device's calibration significantly relaxed. We show that exploiting the natural low-rank structure of quantum states of interest suffices to arrive at a highly scalable `blind' tomography scheme with a classically efficient post-processing algorithm. We further improve the efficiency of our scheme by making use of the sparse structure of the calibrations. This is achieved by relaxing the blind quantum tomography problem to the de-mixing of a sparse sum of low-rank matrices. We prove that the proposed algorithm recovers a low-rank quantum state and the calibration provided that the measurement model exhibits a restricted isometry property. For generic measurements, we show that it requires a close-to-optimal number of measurement settings. Complementing these conceptual and mathematical insights, we numerically demonstrate that robust blind quantum tomography is possible in a practical setting inspired by an implementation of trapped ions.

Building blocks of quantum computers have been demonstrated in small to intermediate-scale systems. As one of the leading platforms, the trapped ion system has attracted wide attention. A significant challenge in this system is to combine fast high-fidelity gates with scalability and convenience in ion trap fabrication. Here we propose an architecture for large-scale quantum computing with a two-dimensional array of atomic ions trapped at such large distance which is convenient for ion-trap fabrication but usually believed to be unsuitable for quantum computing as the conventional gates would be too slow. Using gate operations far outside of the Lamb-Dicke region, we show that fast and robust entangling gates can be realized in any large ion arrays. The gate operations are intrinsically parallel and robust to thermal noise, which, together with their high speed and scalability of the proposed architecture, makes this approach an attractive one for large-scale quantum computing.

We report the electromagnetically-induced-transparency (EIT) cooling of a large trapped $^{171}$Yb$^+$ ion chain to the quantum ground state. Unlike conventional EIT cooling, we engage a four-level tripod structure and achieve fast sub-Doppler cooling over all motional modes. We observe simultaneous ground-state cooling across the complete transverse mode spectrum of up to $40$ ions, occupying a bandwidth of over $3$ MHz. The cooling time is observed to be less than $300\,μ$s, independent of the number of ions. Such efficient cooling across the entire spectrum is essential for high-fidelity quantum operations using trapped ion crystals for quantum simulators or quantum computers.

We theoretically and experimentally investigate double electromagnetically induced transparency (double-EIT) cooling of two-dimensional ion crystals confined in a Paul trap. The double-EIT ground-state cooling is observed for \Yb ions with clock state, for which EIT cooling has not been realized like many other ions with a simple $Λ$-scheme. A cooling rate of $\dot{\bar n}=34~(\pm1.8)~\rm{ms}^{-1}$ and a cooling limit of $\bar n=0.06~(\pm 0.059)$ are observed for a single ion. The measured cooling rate and limit are consistent with theoretical predictions. We apply double-EIT cooling to the transverse modes of two-dimensional (2D) crystals with up to 12 ions. In our 2D crystals, the micromotion and the transverse mode directions are perpendicular, which makes them decoupled. Therefore, the cooling on transverse modes is not disturbed by micromotion, which is confirmed in our experiment. For the center of mass mode of a 12 ions crystal, we observe a cooling rate and a cooling limit that are consistent with those of a single ion, including heating rates proportional to the number of ions. This method can be extended to other hyperfine qubits, and near ground-state cooling of stationary 2D crystals with large numbers of ions may advance the field of quantum information sciences.

Driving an ion at a motional sideband transition induces the Jaynes--Cummings (JC) interaction. This JC interaction creates an anharmonic ladder of JC eigenstates, resulting in the suppression of phonon hopping due to energy conservation. Here, we realize phonon blockade in the presence of multiple local phonons in a trapped-ion chain. Our work establishes a key technological component for quantum simulation with multiple bosonic particles, which can simulate classically intractable problems.

We report sympathetic cooling of $^{113}$Cd$^+$ by laser-cooled $^{40}$Ca$^+$ in a linear Paul trap for microwave clocks. Long-term low-temperature confinement of $^{113}$Cd$^+$ ions was achieved. The temperature of these ions was measured at $90(10)$ mK, and the corresponding uncertainty arising from the second-order Doppler shifts was estimated to a level of $2\times10^{-17}$. Up to $4.2\times10^5$ Cd$^+$ ions were confined in the trap, and the confinement time constant was measured to be 84 hours. After three hours of confinement, there were still $10^5$ Cd$^+$ ions present, indicating that this Ca$^+$--Cd$^+$ dual ion system is surprisingly stable. The ac Stark shift was induced by the Ca$^+$ lasers and fluorescence, which was carefully estimated to an accuracy of $5.4(0.5)\times10^{-17}$ using a high-accuracy \textit{ab initio} approach. The Dick-effect-limited Allan deviation was also deduced because deadtimes were shorter. These results indicate that a microwave clock based on this sympathetic cooling scheme holds promise in providing ultra-high frequency accuracy and stability.

How a closed interacting quantum many-body system relaxes and dephases as a function of time is a fundamental question in thermodynamic and statistical physics. In this work, we analyse and observe the persistent temporal fluctuations after a quantum quench of a tunable long-range interacting transverse-field Ising Hamiltonian realized with a trapped-ion quantum simulator. We measure the temporal fluctuations in the average magnetization of a finite-size system of spin-$1/2$ particles. We experiment in a regime where the properties of the system are closely related to the integrable Hamiltonian with global spin-spin coupling, which enables analytical predictions even for the long-time non-integrable dynamics. The analytical expression for the temporal fluctuations predicts the exponential suppression of temporal fluctuations with increasing system size. Our measurement data is consistent with our theory predicting the regime of many-body dephasing.

We study the spin transport through a 1D quantum Ising-XY-Ising spin link that emulates a topological superconducting-normal-superconducting structure via Jordan-Wigner (JW) transformation. We calculate, both analytically and numerically, the spectrum of spin Andreev bound states and the resulting $\mathbb{Z}_2$ fractional spin Josephson effect (JE) pertaining to the emerging Majorana JW fermions. Deep in the topological regime, we identify an effective time-reversal symmetry that leads to $\mathbb{Z}_4$ fractional spin JE in the $\textit{presence}$ of interactions within the junction. Moreover, we uncover a hidden inversion time-reversal symmetry that protects the $\mathbb{Z}_4$ periodicity in chains with an odd number of spins, even in the $\textit{absence}$ of interactions. We also analyze the entanglement between pairs of spins by evaluating the concurrence in the presence of spin current and highlight the effects of the JW Majorana states. We propose to use a microwave cavity setup for detecting the aforementioned JEs by dispersive readout methods and show that, surprisingly, the $\mathbb{Z}_2$ periodicity is immune to $\textit{any}$ local magnetic perturbations. Our results are relevant for a plethora of spin systems, such as trapped ions, photonic lattices, electron spins in quantum dots, or magnetic impurities on surfaces.

Confinement is a ubiquitous mechanism in nature, whereby particles feel an attractive force that increases without bound as they separate. A prominent example is color confinement in particle physics, in which baryons and mesons are produced by quark confinement. Analogously, confinement can also occur in low-energy quantum many-body systems when elementary excitations are confined into bound quasiparticles. Here, we report the first observation of magnetic domain wall confinement in interacting spin chains with a trapped-ion quantum simulator. By measuring how correlations spread, we show that confinement can dramatically suppress information propagation and thermalization in such many-body systems. We are able to quantitatively determine the excitation energy of domain wall bound states from non-equilibrium quench dynamics. Furthermore, we study the number of domain wall excitations created for different quench parameters, in a regime that is difficult to model with classical computers. This work demonstrates the capability of quantum simulators for investigating exotic high-energy physics phenomena, such as quark collision and string breaking.

Laser-cooled and trapped atomic ions form an ideal standard for the simulation of interacting quantum spin models. Effective spins are represented by appropriate internal energy levels within each ion, and the spins can be measured with near-perfect efficiency using state-dependent fluorescence techniques. By applying optical fields that exert optical dipole forces on the ions, their Coulomb interaction can be modulated to produce long-range and tunable spin-spin interactions that can be reconfigured by shaping the spectrum and pattern of the laser fields, in a prototypical example of a quantum simulator. Here we review the theoretical mapping of atomic ions to interacting spin systems, the preparation of complex equilibrium states, the study of dynamical processes in these many-body interacting quantum systems, and the use of this platform for optimization and other tasks. The use of such quantum simulators for studying spin models may inform our understanding of exotic quantum materials and shed light on the behavior of interacting quantum systems that cannot be modeled with conventional computers.

We present a simple Paul trap that stably accommodates up to a couple of dozens of \ensuremath{^{171}\mathrm{Yb}^+~} ions in a stationary two-dimensional lattice. The trap is constructed on a single plate of gold-plated laser-machined alumina and can produce a pancake-like pseudo-potential that makes ions form a self-assembly two-dimensional crystal which locates on the plane composed of axial and one of the transverse axes with around 5 $μ$m spacing. We use Raman laser beams to coherently manipulate these ion-qubits where the net propagation direction is perpendicular to the plane of the crystal and micromotion. We perform the coherent operations and study the spectrum of vibrational modes through globally addressed Raman laser-beams on a dozen of ions in the two-dimensional crystal. We measure the amplitude of micro-motion by comparing the strengths of carrier and micro-motion sideband transitions with three ions, where the micro-motion amplitude is similar to that of a single ion. The spacings of ions are small enough for large coupling strengths, which is a favorable condition for two-dimensional quantum simulation.

String breaking is a central dynamical process in theories featuring confinement, where a string connecting two charges decays at the expense of the creation of new particle-antiparticle pairs. Here, we show that this process can also be observed in quantum Ising chains where domain walls get confined either by a symmetry-breaking field or by long-range interactions. We find that string breaking occurs, in general, as a two-stage process: First, the initial charges remain essentially static and stable. The connecting string, however, can become a dynamical object. We develop an effective description of this motion, which we find is strongly constrained. In the second stage, which can be severely delayed due to these dynamical constraints, the string finally breaks. We observe that the associated time scale can depend crucially on the initial separation between domain walls and can grow by orders of magnitude by changing the distance by just a few lattice sites. We discuss how our results generalize to one-dimensional confining gauge theories and how they can be made accessible in quantum simulator experiments such as Rydberg atoms or trapped ions.

With the long-term goal of studying models of quantum gravity in the lab, we propose holographic teleportation protocols that can be readily executed in table-top experiments. These protocols exhibit similar behavior to that seen in the recent traversable wormhole constructions of [1,2]: information that is scrambled into one half of an entangled system will, following a weak coupling between the two halves, unscramble into the other half. We introduce the concept of teleportation by size to capture how the physics of operator-size growth naturally leads to information transmission. The transmission of a signal through a semi-classical holographic wormhole corresponds to a rather special property of the operator-size distribution we call size winding. For more general systems (which may not have a clean emergent geometry), we argue that imperfect size winding is a generalization of the traversable wormhole phenomenon. In addition, a form of signalling continues to function at high temperature and at large times for generic chaotic systems, even though it does not correspond to a signal going through a geometrical wormhole, but rather to an interference effect involving macroscopically different emergent geometries. Finally, we outline implementations feasible with current technology in two experimental platforms: Rydberg atom arrays and trapped ions.

Recent studies indicated that contamination by adatoms on the surface ion trap can generate contact potential, leading to fluctuations in patch potential. By investigating contamination induced by surface adatoms during a loading process, a direct physical image of the contamination process and the relationship between the capacitance change and the contamination from surface adatoms is examined theoretically and experimentally. From the relationship, the contamination by surface adatoms and the effect of in situ treatment process can be monitored by the capacitance between electrodes in real time. This study is foundational to further research on anomalous heating with practical applications in quantum information processing from surface ion traps.

Long-range interacting spin systems are ubiquitous in physics and exhibit a variety of ground state disorder-to-order phase transitions. We consider a prototype of infinite-range interacting models known as the Lipkin-Meshkov-Glick (LMG) model describing the collective interaction of $N$ spins, and investigate the dynamical properties of fluctuations and correlations after a sudden quench of the Hamiltonian. Specifically, we focus on critical quenches, where the initial state and/or the quench Hamiltonian are critical. Depending on the type of quench, we identify three distinct behaviors where both the short-time dynamics and the stationary state at long times are effectively thermal, quantum, and genuinely non-equilibrium, characterized by distinct universality classes and static and dynamical critical exponents. These behaviors can be identified by an infrared effective temperature that is finite, zero, and infinite (the latter scaling with the system size as $N^{1/3}$), respectively. The quench dynamics is studied through a combination of exact numerics and analytical calculations utilizing the non-equilibrium Keldysh field theory. Our results are amenable to realization in experiments with trapped-ion experiments where long-range interactions naturally arise.

Gauge field theories play a central role in modern physics and are at the heart of the Standard Model of elementary particles and interactions. Despite significant progress in applying classical computational techniques to simulate gauge theories, it has remained a challenging task to compute the real-time dynamics of systems described by gauge theories. An exciting possibility that has been explored in recent years is the use of highly-controlled quantum systems to simulate, in an analog fashion, properties of a target system whose dynamics are difficult to compute. Engineered atom-laser interactions in a linear crystal of trapped ions offer a wide range of possibilities for quantum simulations of complex physical systems. Here, we devise practical proposals for analog simulation of simple lattice gauge theories whose dynamics can be mapped onto spin-spin interactions in any dimension. These include 1+1D quantum electrodynamics, 2+1D Abelian Chern-Simons theory coupled to fermions, and 2+1D pure Z2 gauge theory. The scheme proposed, along with the optimization protocol applied, will have applications beyond the examples presented in this work, and will enable scalable analog quantum simulation of Heisenberg spin models in any number of dimensions and with arbitrary interaction strengths.

Modern computation relies crucially on modular architectures, breaking a complex algorithm into self-contained subroutines. A client can then call upon a remote server to implement parts of the computation independently via an application programming interface (API). Present APIs relay only classical information. Here we implement a quantum API that enables a client to estimate the absolute value of the trace of a server-provided unitary $U$. We demonstrate that the algorithm functions correctly irrespective of what unitary $U$ the server implements or how the server specifically realizes $U$. Our experiment involves pioneering techniques to coherently swap qubits encoded within the motional states of a trapped \Yb ion, controlled on its hyperfine state. This constitutes the first demonstration of modular computation in the quantum regime, providing a step towards scalable, parallelization of quantum computation.

In trapped-ion quantum information processing, interactions between spins (qubits) are mediated by collective modes of motion of an ion crystal. While there are many different experimental strategies to design such interactions, they all face both technical and fundamental limitations to the achievable coherent interaction strength. In general, obtaining strong interactions and fast gates is an ongoing challenge. Here, we extend previous work [Phys. Rev. Lett. 112, 030501 (2019)] and present a general strategy for enhancing the interaction strengths in trapped-ion systems via parametric amplification of the ions' motion. Specifically, we propose a stroboscopic protocol using alternating applications of parametric amplification and spin-motion coupling. In comparison with the previous work, we show that the current protocol can lead to larger enhancements in the coherent interaction that increase exponentially with the gate time.

In statistical mechanics, a small system exchanges conserved quantities---heat, particles, electric charge, etc.---with a bath. The small system thermalizes to the canonical ensemble, or the grand canonical ensemble, etc., depending on the conserved quantities. The conserved quantities are represented by operators usually assumed to commute with each other. This assumption was removed within quantum-information-theoretic (QI-theoretic) thermodynamics recently. The small system's long-time state was dubbed ``the non-Abelian thermal state (NATS).'' We propose an experimental protocol for observing a system thermalize to the NATS. We illustrate with a chain of spins, a subset of which form the system of interest. The conserved quantities manifest as spin components. Heisenberg interactions push the conserved quantities between the system and the effective bath, the rest of the chain. We predict long-time expectation values, extending the NATS theory from abstract idealization to finite systems that thermalize with finite couplings for finite times. Numerical simulations support the analytics: The system thermalizes to the NATS, rather than to the canonical prediction. Our proposal can be implemented with ultracold atoms, nitrogen-vacancy centers, trapped ions, quantum dots, and perhaps nuclear magnetic resonance. This work introduces noncommuting conserved quantities from QI-theoretic thermodynamics into quantum many-body physics: atomic, molecular, and optical physics and condensed matter.

Finite-temperature phases of many-body quantum systems are fundamental to phenomena ranging from condensed-matter physics to cosmology, yet they are generally difficult to simulate. Using an ion trap quantum computer and protocols motivated by the Quantum Approximate Optimization Algorithm (QAOA), we generate nontrivial thermal quantum states of the transverse-field Ising model (TFIM) by preparing thermofield double states at a variety of temperatures. We also prepare the critical state of the TFIM at zero temperature using quantum-classical hybrid optimization. The entanglement structure of thermofield double and critical states plays a key role in the study of black holes, and our work simulates such nontrivial structures on a quantum computer. Moreover, we find that the variational quantum circuits exhibit noise thresholds above which the lowest depth QAOA circuits provide the best results.

Quantum computers and simulators may offer significant advantages over their classical counterparts, providing insights into quantum many-body systems and possibly improving performance for solving exponentially hard problems, such as optimization and satisfiability. Here we report the implementation of a low-depth Quantum Approximate Optimization Algorithm (QAOA) using an analog quantum simulator. We estimate the ground state energy of the Transverse Field Ising Model with long-range interactions with tunable range and we optimize the corresponding combinatorial classical problem by sampling the QAOA output with high-fidelity, single-shot individual qubit measurements. We execute the algorithm with both an exhaustive search and closed-loop optimization of the variational parameters, approximating the ground state energy with up to 40 trapped-ion qubits. We benchmark the experiment with bootstrapping heuristic methods scaling polynomially with the system size. We observe, in agreement with numerics, that the QAOA performance does not degrade significantly as we scale up the system size, and that the runtime is approximately independent from the number of qubits. We finally give a comprehensive analysis of the errors occurring in our system, a crucial step in the path forward towards the application of the QAOA to more general problem instances.

We introduce an approach to improve the accuracy and reduce the sample complexity of near term quantum-classical algorithms. We construct a simpler initial parameterized quantum state, or ansatz, based on the past causal cone of each observable, generally yielding fewer qubits and gates. We implement this protocol on a trapped ion quantum computer and demonstrate improvement in accuracy and time-to-solution at an arbitrary point in the variational search space. We report a $\sim 27\%$ improvement in the accuracy of the calculation of the deuteron binding energy and $\sim 40\%$ improvement in the accuracy of the quantum approximate optimization of the MAXCUT problem applied to the dragon graph $T_{3,2}$. When the time-to-solution is prioritized over accuracy, the former requires $\sim 71\%$ fewer measurements and the latter requires $\sim 78\%$ fewer measurements.

Various quantum applications can be reduced to estimating expectation values, which are inevitably deviated by operational and environmental errors. Although errors can be tackled by quantum error correction, the overheads are far from being affordable for near-term technologies. To alleviate the detrimental effects of errors, quantum error mitigation techniques have been proposed, which require no additional qubit resources. Here, we benchmark the performance of a quantum error mitigation technique based on probabilistic error cancellation in a trapped-ion system. Our results clearly show that effective gate fidelities exceed physical fidelities, i.e. we surpass the break-even point of eliminating gate errors, by programming quantum circuits. The error rates are effectively reduced from $(1.10\pm 0.12)\times10^{-3}$ to $(1.44\pm 5.28)\times10^{-5}$ and from $(0.99\pm 0.06)\times10^{-2}$ to $(0.96\pm 0.10)\times10^{-3}$ for single- and two-qubit gates, respectively. Our demonstration opens up the possibility of implementing high-fidelity computations on a near-term noisy quantum device.

Ion trap systems are a leading platform for large scale quantum computers. Trapped ion qubit crystals are fully-connected and reconfigurable, owing to their long range Coulomb interaction that can be modulated with external optical forces. However, the spectral crowding of collective motional modes could pose a challenge to the control of such interactions for large numbers of qubits. Here, we show that high-fidelity quantum gate operations are still possible with very large trapped ion crystals, simplifying the scaling of ion trap quantum computers. To this end, we present analytical work that determines how parallel entangling gates produce a crosstalk error that falls off as the inverse cube of the distance between the pairs. We also show experimental work demonstrating entangling gates on a fully-connected chain of seventeen $^{171}{\rm{Yb}}^{+}$ ions with fidelities as high as $97(1)\%$.

We report results for simulating an effective field theory to compute the binding energy of the deuteron nucleus using a hybrid algorithm on a trapped-ion quantum computer. Two increasingly complex unitary coupled-cluster ansaetze have been used to compute the binding energy to within a few percent for successively more complex Hamiltonians. By increasing the complexity of the Hamiltonian, allowing more terms in the effective field theory expansion and calculating their expectation values, we present a benchmark for quantum computers based on their ability to scalably calculate the effective field theory with increasing accuracy. Our result of $E_4=-2.220 \pm 0.179$MeV may be compared with the exact Deuteron ground-state energy $-2.224$MeV. We also demonstrate an error mitigation technique using Richardson extrapolation on ion traps for the first time. The error mitigation circuit represents a record for deepest quantum circuit on a trapped-ion quantum computer.

This paper reports the direct temperature determination of sympathetically cooled 113Cd+ ions with laser-cooled 24Mg+ in a linear Paul trap. The sympathetically cooled ion species distribute in the outer shell of the large ensembles, which contain up to 3.3E5 ions. With optimized parameters, the minimum temperature of the sympathetically cooled 113Cd+ ions was measured to be tens of mK. These results indicate promising performance for microwave atomic clocks. The second order Doppler frequency shift is two orders of magnitudes lower and the Dick effect is suppressed.

In this chapter, we illustrate how a trapped ion system can be used for the experimental study of quantum thermodynamics, in particular, quantum fluctuation of work. As technology of nano/micro scale develops, it becomes critical to understand thermodynamics at the quantum mechanical level. The trapped ion system is a representative physical platform to experimentally demonstrate quantum phenomena with excellent control and precision. We provide a basic introduction of the trapped ion system and present the theoretical framework for the experimental study of quantum thermodynamics. Then we bring out two concrete examples of the experimental demonstrations. Finally, we discuss the results and the future of the experimental study of quantum thermodynamics with trapped ion systems.

The output randomness from a random number generator can be certified by observing the violation of quantum contextuality inequalities based on the Kochen-Specker theorem. Contextuality can be tested in a single quantum system, which significantly simplifies the experimental requirements to observe the violation comparing to the ones based on nonlocality tests. However, it is not yet resolved how to ensure compatibilities for sequential measurements that is required in contextuality tests. Here, we employ a modified Klyachko-Can-Binicioğlu-Shumovsky contextuality inequality, which can ease the strict compatibility requirement on measurements. On a trapped single \Ba ion system, we experimentally demonstrate violation of the contextuality inequality and realize self-testing quantum random number expansion by closing detection loopholes. We perform $1.29 \times 10^8$ trials of experiments and extract the randomness of $8.06 \times 10^5$ bits with a speed of 270 bits s$^{-1}$. Our demonstration paves the way for the practical high-speed spot-checking quantum random number expansion and other secure information processing applications.

Scaling up and effective cooling of ions in surface ion trap are central challenges in quantum computing and quantum simulation with trapped ions. In this theoretical study, we propose a versatile surface ion trap. In the manipulation zone of our trap, a symmetric seven-wire geometry enables innate principle-axes rotation of two parallel linear ion chains, which facilitates the cooling of ions along all principle trap axes. To alleviate contaminating the manipulation zone during ion loading, a symmetric five-wire geometry is designed as the loading zone. And a "fork junction" connects the loading and manipulation zones, which also enables the shuttling and reordering of ions. A multi-objective optimization procedure suitable for arbitrary junction designs is described in detail, and we present the corresponding optimal results for the key components of our trap. The proposed versatile trap can be used in the construction of large-scale ion quantum processors. The trap also can be used as the multi-ion-mixer or the efficient ion beam splitter, which has the potential applications in quantum simulation and quantum computing, the research of 2D dimensional ion crystals and the guides of quantum microscope, like an electron beam splitter used for quantum matter-wave optics experiments. Interesting topics involving the spin-spin interactions between two ion chains can also be simulated in our trap.

We generalize past work on quantum sensor networks to show that, for $d$ input parameters, entanglement can yield a factor $\mathcal O(d)$ improvement in mean squared error when estimating an analytic function of these parameters. We show that the protocol is optimal for qubit sensors, and conjecture an optimal protocol for photons passing through interferometers. Our protocol is also applicable to continuous variable measurements, such as one quadrature of a field operator. We outline a few potential applications, including calibration of laser operations in trapped ion quantum computing.

A quantum algorithm can be decomposed into a sequence consisting of single qubit and 2-qubit entangling gates. To optimize the decomposition and achieve more efficient construction of the quantum circuit, we can replace multiple 2-qubit gates with a single global entangling gate. Here, we propose and implement a scalable scheme to realize the global entangling gates on multiple $\yb$ ion qubits by coupling to multiple motional modes through external fields. Such global gates require simultaneously decoupling of multiple motional modes and balancing of the coupling strengths for all the qubit-pairs at the gate time. To satisfy the complicated requirements, we develop a trapped-ion system with fully-independent control capability on each ion, and experimentally realize the global entangling gates. As examples, we utilize them to prepare the Greenberger-Horne-Zeilinger (GHZ) states in a single entangling operation, and successfully show the genuine multi-partite entanglements up to four qubits with the state fidelities over $93.4\%$.

Generative modeling is a flavor of machine learning with applications ranging from computer vision to chemical design. It is expected to be one of the techniques most suited to take advantage of the additional resources provided by near-term quantum computers. We implement a data-driven quantum circuit training algorithm on the canonical Bars-and-Stripes data set using a quantum-classical hybrid machine. The training proceeds by running parameterized circuits on a trapped ion quantum computer, and feeding the results to a classical optimizer. We apply two separate strategies, Particle Swarm and Bayesian optimization to this task. We show that the convergence of the quantum circuit to the target distribution depends critically on both the quantum hardware and classical optimization strategy. Our study represents the first successful training of a high-dimensional universal quantum circuit, and highlights the promise and challenges associated with hybrid learning schemes.

We examine the viability of quantum repeaters based on two-species trapped ion modules for long distance quantum key distribution. Repeater nodes comprised of ion-trap modules of co-trapped ions of distinct species are considered. The species used for communication qubits has excellent optical properties while the other longer lived species serves as a memory qubit in the modules. Each module interacts with the network only via single photons emitted by the communication ions. Coherent Coulomb interaction between ions is utilized to transfer quantum information between the communication and memory ions and to achieve entanglement swapping between two memory ions. We describe simple modular quantum repeater architectures realizable with the ion-trap modules and numerically study the dependence of the quantum key distribution rate on various experimental parameters, including coupling efficiency, gate infidelity, operation time and length of the elementary links. Our analysis suggests crucial improvements necessary in a physical implementation for co-trapped two-species ions to be a competitive platform in long-distance quantum communication.

In this paper, we explore the possibilities of realizing weak value amplification (WVA) using purely atomic degrees of freedom. Our scheme identifies the internal electronic states and external motional states of a single trapped $^{40}$Ca$^+$ ion as the system degree and pointer degree respectively, and their controllable weak coupling is provided by a bichromatic light field. In our experimental demonstration, by performing appropriate postselection on the internal states, a position displacement of 4 angstroms (in phase space) of the trapped ion is amplified to 10 nanometers. The sensitivity of the amplification effect to the relative phase of the quantum state is also demonstrated. The high operational flexibility of this procedure allows fully exploration of the peculiarities of WVA.

The circuit model of a quantum computer consists of sequences of gate operations between quantum bits (qubits), drawn from a universal family of discrete operations. The ability to execute parallel entangling quantum gates offers clear efficiency gains in numerous quantum circuits as well as for entire algorithms such as Shor's factoring algorithm and quantum simulations. In cases such as full adders and multiple-control Toffoli gates, parallelism can provide an exponential improvement in overall execution time. More importantly, quantum gate parallelism is essential for the practical fault-tolerant error correction of qubits that suffer from idle errors. The implementation of parallel quantum gates is complicated by potential crosstalk, especially between qubits fully connected by a common-mode bus, such as in Coulomb-coupled trapped atomic ions or cavity-coupled superconducting transmons. Here, we present the first experimental results for parallel 2-qubit entangling gates in an array of fully-connected trapped ion qubits. We demonstrate an application of this capability by performing a 1-bit full addition operation on a quantum computer using a depth-4 quantum circuit. These results exploit the power of highly connected qubit systems through classical control techniques, and provide an advance toward speeding up quantum circuits and achieving fault tolerance with trapped ion quantum computers.

We study the quasiparticle excitation and quench dynamics of the one-dimensional transverse-field Ising model with power-law ($1/r^α$) interactions. We find that long-range interactions give rise to a confining potential, which couples pairs of domain walls (kinks) into bound quasiparticles, analogous to mesonic bound states in high-energy physics. We show that these quasiparticles have signatures in the dynamics of order parameters following a global quench and the Fourier spectrum of these order parameters can be expolited as a direct probe of the masses of the confined quasiparticles. We introduce a two-kink model to qualitatively explain the phenomenon of long-range-interaction induced confinement, and to quantitatively predict the masses of the bound quasiparticles. Furthermore, we illustrate that these quasiparticle states can lead to slow thermalization of one-point observables for certain initial states. Our work is readily applicable to current trapped-ion experiments.

Quantum simulation of complex quantum systems and their properties often requires the ability to prepare initial states in an eigenstate of the Hamiltonian to be simulated. In addition, to compute the eigenvalues of a Hamiltonian is in general a non-trivial problem. Here, we propose a hybrid quantum-classical probabilistic method to compute eigenvalues and prepare eigenstates of Hamiltonians which are simulatable with a trapped-ion quantum processor.

The study of quantum phase transitions requires the preparation of a many-body system near its ground state, a challenging task for many experimental systems. The measurement of quench dynamics, on the other hand, is now a routine practice in most cold atom platforms. Here we show that quintessential ingredients of quantum phase transitions can be probed directly with quench dynamics in integrable and nearly integrable systems. As a paradigmatic example, we study global quench dynamics in a transverse-field Ising model with either short-range or long-range interactions. When the model is integrable, we discover a new dynamical critical point with a non-analytic signature in the short-range correlators. The location of the dynamical critical point matches that of the quantum critical point and can be identified using a finite-time scaling method. We extend this scaling picture to systems near integrability and demonstrate the continued existence of a dynamical critical point detectable at prethermal time scales. Therefore, our method can be used to approximately locate the quantum critical point. The scaling method is also relevant to experiments with finite time and system size, and our predictions are testable in near-term experiments with trapped ions and Rydberg atoms.

Trapped ions offer a pristine platform for quantum computation and simulation, but improving their coherence remains a crucial challenge. Here, we propose and analyze a new strategy to enhance the coherent interactions in trapped ion systems via parametric amplification of the ions' motion--by squeezing the collective motional modes (phonons), the spin-spin interactions they mediate can be significantly enhanced. We illustrate the power of this approach by showing how it can enhance collective spin states useful for quantum metrology, and how it can improve the speed and fidelity of two-qubit gates in multi-ion systems, important ingredients for scalable trapped ion quantum computation. Our results are also directly relevant to numerous other physical platforms in which spin interactions are mediated by bosons.

Quantum scrambling is the dispersal of local information into many-body quantum entanglements and correlations distributed throughout the entire system. This concept underlies the dynamics of thermalization in closed quantum systems, and more recently has emerged as a powerful tool for characterizing chaos in black holes. However, the direct experimental measurement of quantum scrambling is difficult, owing to the exponential complexity of ergodic many-body entangled states. One way to characterize quantum scrambling is to measure an out-of-time-ordered correlation function (OTOC); however, since scrambling leads to their decay, OTOCs do not generally discriminate between quantum scrambling and ordinary decoherence. Here, we implement a quantum circuit that provides a positive test for the scrambling features of a given unitary process. This approach conditionally teleports a quantum state through the circuit, providing an unambiguous litmus test for scrambling while projecting potential circuit errors into an ancillary observable. We engineer quantum scrambling processes through a tunable 3-qubit unitary operation as part of a 7-qubit circuit on an ion trap quantum computer. Measured teleportation fidelities are typically $\sim80\%$, and enable us to experimentally bound the scrambling-induced decay of the corresponding OTOC measurement.

Hydrogen exhibits qualitatively different charge states depending on the host material, as nicely explained by the state-of-the-art impurity-state calculation. Motivated by a recent experiment [Nature 546, 124 (2017)], we show that the complex oxide SrCoO$_{2.5}$ represents an interesting example, in which the interstitial H appears as a deep-level center according to the commonly-used transition level calculation, but no bound electron can be found around the impurity. Via a combination of charge difference analysis, density of states projection and constraint magnetization calculation, it turns out that the H-doped electron is spontaneously trapped by a nonunique Co ion and is fully spin-polarized by the onsite Hund's rule coupling. Consequently, the doped system remains insulating, whereas the antiferromagnetic exchange is slightly perturbed locally.

The optical manipulation of Ba$^{+}$ ions is mainly performed by a 493 nm laser for the S$_{1/2}$-P$_{1/2}$ transition and a 650 nm laser for the P$_{1/2}$-D$_{3/2}$ transition. Since the branching ratio between the 493 nm and 650 nm transitions of a single Ba$^{+}$ ion is comparable, stabilization systems of both lasers are equally important for Doppler cooling, sub-Doppler cooling, optical pumping and state detection. The stabilization system of a 493 nm laser to an absolute Te$_2$ reference has been well established. However, the stabilization of a 650 nm laser has not been presented before. Here we report twenty spectral lines of I$_{2}$ in the range of 0.9 GHz above the resonance of the P$_{1/2}$-D$_{3/2}$ transition. We stabilize the 650 nm laser through the optical cavity to the lowest one among these lines, which is about 350 MHz apart, as the absolute frequency reference. Furthermore, we measure the frequency differences between these iodine lines and the Ba$^+$ resonance through fluorescence excitation spectrum with well-resolved dark states, which is in agreement with the theoretical expectation. The presented stabilization scheme enables us to perform precise experiments with Ba$^{+}$ ions.

We reduce measurement errors in a quantum computer using machine learning techniques. We exploit a simple yet versatile neural network to classify multi-qubit quantum states, which is trained using experimental data. This flexible approach allows the incorporation of any number of features of the data with minimal modifications to the underlying network architecture. We experimentally illustrate this approach in the readout of trapped-ion qubits using additional spatial and temporal features in the data. Using this neural network classifier, we efficiently treat qubit readout crosstalk, resulting in a 30\% improvement in detection error over the conventional threshold method. Our approach does not depend on the specific details of the system and can be readily generalized to other quantum computing platforms.

We demonstrate a Bayesian quantum game on an ion trap quantum computer with five qubits. The players share an entangled pair of qubits and perform rotations on their qubit as the strategy choice. Two five-qubit circuits are sufficient to run all 16 possible strategy choice sets in a game with four possible strategies. The data are then parsed into player types randomly in order to combine them classically into a Bayesian framework. We exhaustively compute the possible strategies of the game so that the experimental data can be used to solve for the Nash equilibria of the game directly. Then we compare the payoff at the Nash equilibria and location of phase-change-like transitions obtained from the experimental data to the theory, and study how it changes as a function of the amount of entanglement.

The realization of high fidelity quantum gates in a multi-qubit system, with a typical target set at 99.9%, is a critical requirement for the implementation of fault-tolerant quantum computation. To reach this level of fidelity, one needs to carefully analyze the noises and imperfections in the experimental system and optimize the gate operations to mitigate their effects. Here, we consider one of the leading experimental systems for the fault-tolerant quantum computation, ions in an anharmonic linear Paul trap, and optimize entangling quantum gates using segmented laser pulses with the assistance of all the collective transverse phonon modes of the ion crystal. We present detailed analyses of the effects of various kinds of intrinsic experimental noises as well as errors from imperfect experimental controls. Through explicit calculations, we find the requirements on these relevant noise levels and control precisions to achieve the targeted high fidelity of 99.9% for the entangling quantum gates in a multi-ion crystal.

We present a cryogenic ion trapping system designed for large scale quantum simulation of spin models. Our apparatus is based on a segmented-blade ion trap enclosed in a 4 K cryostat, which enables us to routinely trap over 100 $^{171}$Yb$^+$ ions in a linear configuration for hours due to a low background gas pressure from differential cryo-pumping. We characterize the cryogenic vacuum by using trapped ion crystals as a pressure gauge, measuring both inelastic and elastic collision rates with the molecular background gas. We demonstrate nearly equidistant ion spacing for chains of up to 44 ions using anharmonic axial potentials. This reliable production and lifetime enhancement of large linear ion chains will enable quantum simulation of spin models that are intractable with classical computer modelling.

The efficient simulation of correlated quantum systems is the most promising near-term application of quantum computers. Here, we present a measurement of the second Renyi entropy of the ground state of the two-site Fermi-Hubbard model on a $5$-qubit programmable quantum computer based on trapped ions. Our work illustrates the extraction of a non-linear characteristic of a quantum state using a controlled-swap gate acting on two copies of the state. This scalable measurement of entanglement on a universal quantum computer will, with more qubits, provide insights into many-body quantum systems that are impossible to simulate on classical computers.

The Weyl equation describes chiral massless relativistic particles, called Weyl fermions, which have important relations to neutrinos. A direct observation of the dynamics of Weyl fermions in an experiment is difficult to achieve. This study investigates a method of simulating the Weyl equation in 1+2-dimension by a single trapped ion. The predictions about a two-dimensional Zitterbewegung and an especially interesting phenomenon of Weyl fermions can be tested by the future trapped ion experiment, which might enhance our understanding of neutrinos.

The NOT gate that flips a classical bit is ubiquitous in classical information processing. However its quantum analogue, the universal NOT (UNOT) gate that flips a quantum spin in any alignment into its antipodal counterpart is strictly forbidden. Here we explore the connection between this discrepancy and how UNOT gates affect classical and quantum correlations. We show that while a UNOT gate always preserves classical correlations between two spins, it can non-locally increase or decrease their shared discord in ways that allow violation of the data processing inequality. We experimentally illustrate this using a multi-level trapped \Yb ion that allows simulation of anti-unitary operations.

The quantum Rabi model, involving a two-level system and a bosonic field mode, is arguably the simplest and most fundamental model describing quantum light-matter interactions. Historically, due to the restricted parameter regimes of natural light-matter processes, the richness of this model has been elusive in the lab. Here, we experimentally realize a quantum simulation of the quantum Rabi model in a single trapped ion, where the coupling strength between the simulated light mode and atom can be tuned at will. The versatility of the demonstrated quantum simulator enables us to experimentally explore the quantum Rabi model in detail, including a wide range of otherwise unaccessible phenomena, as those happening in the ultrastrong and deep strong coupling regimes. In this sense, we are able to adiabatically generate the ground state of the quantum Rabi model in the deep strong coupling regime, where we are able to detect the nontrivial entanglement between the bosonic field mode and the two-level system. Moreover, we observe the breakdown of the rotating-wave approximation when the coupling strength is increased, and the generation of phonon wave packets that bounce back and forth when the coupling reaches the deep strong coupling regime. Finally, we also measure the energy spectrum of the quantum Rabi model in the ultrastrong coupling regime.

The local phonon modes in a Coulomb crystal of trapped ions can represent a Hubbard system of coupled bosons. We selectively prepare single excitations at each site and observe free hopping of a boson between sites, mediated by the long-range Coulomb interaction between ions. We then implement phonon blockades on targeted sites by driving a Jaynes-Cummings interaction on individually addressed ions to couple their internal spin to the local phonon mode. The resulting dressed states have energy splittings that can be tuned to suppress phonon hopping into the site. This new experimental approach opens up the possibility of realizing large-scale Hubbard systems from the bottom up with tunable interactions at the single-site level.

We demonstrate quantum entanglement of two trapped atomic ion qubits using a sequence of ultrafast laser pulses. Unlike previous demonstrations of entanglement mediated by the Coulomb interaction, this scheme does not require confinement to the Lamb-Dicke regime and can be less sensitive to ambient noise due to its speed. To elucidate the physics of an ultrafast phase gate, we generate a high entanglement rate using just 10 pulses, each of $\sim20$ ps duration, and demonstrate an entangled Bell-state with $(76\pm1)$% fidelity. These results pave the way for entanglement operations within a large collection of qubits by exciting only local modes of motion.

In an ion trap quantum computer, collective motional modes are used to entangle two or more qubits in order to execute multi-qubit logical gates. Any residual entanglement between the internal and motional states of the ions results in loss of fidelity, especially when there are many spectator ions in the crystal. We propose using a frequency-modulated (FM) driving force to minimize such errors. In simulation, we obtained an optimized FM two-qubit gate that can suppress errors to less than 0.01\% and is robust against frequency drifts over $\pm$1 kHz. Experimentally, we have obtained a two-qubit gate fidelity of $98.3(4)\%$, a state-of-the-art result for two-qubit gates with 5 ions.

Although nonequilibrium work and fluctuation relations have been studied in detail within classical statistical physics, extending these results to open quantum systems has proven to be conceptually difficult. For systems that undergo decoherence but not dissipation, we argue that it is natural to define quantum work exactly as for isolated quantum systems, using the two-point measurement protocol. Complementing previous theoretical analysis using quantum channels, we show that the nonequilibrium work relation remains valid in this situation, and we test this assertion experimentally using a system engineered from an optically trapped ion. Our experimental results reveal the work relation's validity over a variety of driving speeds, decoherence rates, and effective temperatures and represent the first confirmation of the work relation for non-unitary dynamics.

A quantum simulator is a restricted class of quantum computer that controls the interactions between quantum bits in a way that can be mapped to certain difficult quantum many-body problems. As more control is exerted over larger numbers of qubits, the simulator can tackle a wider range of problems, with the ultimate limit being a universal quantum computer that can solve general classes of hard problems. We use a quantum simulator composed of up to 53 qubits to study a non-equilibrium phase transition in the transverse field Ising model of magnetism, in a regime where conventional statistical mechanics does not apply. The qubits are represented by trapped ion spins that can be prepared in a variety of initial pure states. We apply a global long-range Ising interaction with controllable strength and range, and measure each individual qubit with near 99% efficiency. This allows the single-shot measurement of arbitrary many-body correlations for the direct probing of the dynamical phase transition and the uncovering of computationally intractable features that rely on the long-range interactions and high connectivity between the qubits.

Searching large databases is an important problem with broad applications. The Grover search algorithm provides a powerful method for quantum computers to perform searches with a quadratic speedup in the number of required database queries over classical computers. It is an optimal search algorithm for a quantum computer, and has further applications as a subroutine for other quantum algorithms. Searches with two qubits have been demonstrated on a variety of platforms and proposed for others, but larger search spaces have only been demonstrated on a non-scalable NMR system. Here, we report results for a complete three-qubit Grover search algorithm using the scalable quantum computing technology of trapped atomic ions, with better-than-classical performance. The algorithm is performed for all 8 possible single-result oracles and all 28 possible two-result oracles. Two methods of state marking are used for the oracles: a phase-flip method employed by other experimental demonstrations, and a Boolean method requiring an ancilla qubit that is directly equivalent to the state-marking scheme required to perform a classical search. All quantum solutions are shown to outperform their classical counterparts. We also report the first implementation of a Toffoli-4 gate, which is used along with Toffoli-3 gates to construct the algorithms; these gates have process fidelities of 70.5% and 89.6%, respectively.

Exactly solvable models have played an important role in establishing the sophisticated modern understanding of equilibrium many-body physics. And conversely, the relative scarcity of solutions for non-equilibrium models greatly limits our understanding of systems away from thermal equilibrium. We study a family of non-equilibrium models, some of which can be viewed as dissipative analogues of the transverse-field Ising model, in that an effectively classical Hamiltonian is frustrated by dissipative processes that drive the system toward states that do not commute with the Hamiltonian. Surprisingly, a broad and experimentally relevant subset of these models can be solved efficiently in any number of spatial dimensions. We leverage these solutions to prove a no-go theorem on steady-state phase transitions in a many-body model that can be realized naturally with Rydberg atoms or trapped ions, and to compute the effects of decoherence on a canonical trapped-ion-based quantum computation architecture.

Molecules are the most demanding quantum systems to be simulated by quantum computers because of their complexity and the emergent role of quantum nature. The recent theoretical proposal of Huh et al. (Nature Photon., 9, 615 (2015)) showed that a multi-photon network with a Gaussian input state can simulate a molecular spectroscopic process. Here, we report the first experimental demonstration of molecular vibrational spectroscopy of SO$_{2}$ with a trapped-ion system. In our realization, the molecular scattering operation is decomposed to a series of elementary quantum optical operations, which are implemented through Raman laser beams, resulting in a multimode Gaussian (Bogoliubov) transformation. The molecular spectroscopic signal is reconstructed from the collective projection measurements on phonon modes of the trapped-ion system. Our experimental demonstration would pave the way to large-scale molecular quantum simulations, which are classically intractable.

We run a selection of algorithms on two state-of-the-art 5-qubit quantum computers that are based on different technology platforms. One is a publicly accessible superconducting transmon device with limited connectivity, and the other is a fully connected trapped-ion system. Even though the two systems have different native quantum interactions, both can be programmed in a way that is blind to the underlying hardware, thus allowing the first comparison of identical quantum algorithms between different physical systems. We show that quantum algorithms and circuits that employ more connectivity clearly benefit from a better connected system of qubits. While the quantum systems here are not yet large enough to eclipse classical computers, this experiment exposes critical factors of scaling quantum computers, such as qubit connectivity and gate expressivity. In addition, the results suggest that co-designing particular quantum applications with the hardware itself will be paramount in successfully using quantum computers in the future.

We demonstrate experimental implementation of robust phase estimation (RPE) to learn the phases of X and Y rotations on a trapped $\textrm{Yb}^+$ ion qubit. We estimate these phases with uncertainties less than $4\cdot10^{-4}$ radians using as few as 176 total experimental samples per phase, and our estimates exhibit Heisenberg scaling. Unlike standard phase estimation protocols, RPE neither assumes perfect state preparation and measurement, nor requires access to ancillae. We cross-validate the results of RPE with the more resource-intensive protocol of gate set tomography.

A standard method to obtain information on a quantum state is to measure marginal distributions along many different axes in phase space, which forms a basis of quantum state tomography. We theoretically propose and experimentally demonstrate a general framework to manifest nonclassicality by observing a single marginal distribution only, which provides a novel insight into nonclassicality and a practical applicability to various quantum systems. Our approach maps the 1-dim marginal distribution into a factorized 2-dim distribution by multiplying the measured distribution or the vacuum-state distribution along an orthogonal axis. The resulting fictitious Wigner function becomes unphysical only for a nonclassical state, thus the negativity of the corresponding density operator provides an evidence of nonclassicality. Furthermore, the negativity measured this way yields a lower bound for entanglement potential---a measure of entanglement generated using a nonclassical state with a beam splitter setting that is a prototypical model to produce continuous-variable (CV) entangled states. Our approach detects both Gaussian and non-Gaussian nonclassical states in a reliable and efficient manner. Remarkably, it works regardless of measurement axis for all non-Gaussian states in finite-dimensional Fock space of any size, also extending to infinite-dimensional states of experimental relevance for CV quantum informatics. We experimentally illustrate the power of our criterion for motional states of a trapped ion confirming their nonclassicality in a measurement-axis independent manner. We also address an extension of our approach combined with phase-shift operations, which leads to a stronger test of nonclassicality, i.e. detection of genuine non-Gaussianity under a CV measurement.

A long-time quantum memory capable of storing and measuring quantum information at the single-qubit level is an essential ingredient for practical quantum computation and com-munication. Recently, there have been remarkable progresses of increasing coherence time for ensemble-based quantum memories of trapped ions, nuclear spins of ionized donors or nuclear spins in a solid. Until now, however, the record of coherence time of a single qubit is on the order of a few tens of seconds demonstrated in trapped ion systems. The qubit coherence time in a trapped ion is mainly limited by the increasing magnetic field fluctuation and the decreasing state-detection efficiency associated with the motional heating of the ion without laser cooling. Here we report the coherence time of a single qubit over $10$ minutes in the hyperfine states of a \Yb ion sympathetically cooled by a \Ba ion in the same Paul trap, which eliminates the heating of the qubit ion even at room temperature. To reach such coherence time, we apply a few thousands of dynamical decoupling pulses to suppress the field fluctuation noise. A long-time quantum memory demonstrated in this experiment makes an important step for construction of the memory zone in scalable quantum computer architectures or for ion-trap-based quantum networks. With further improvement of the coherence time by techniques such as magnetic field shielding and increase of the number of qubits in the quantum memory, our demonstration also makes a basis for other applications including quantum money.

Multi-party entangled states have important applications in quantum metrology and quantum computation. Experimental preparation of large entangled state, in particular, the NOON states, however, remains challenging as the particle number $N$ increases. Here we develop a deterministic method to generate arbitrarily high NOON states for phonons and experimentally create the states up to $N=9$ phonons in two radial modes of a single trapped \Yb ion. We demonstrate that the fidelity of the NOON states are significantly above the classical limit by measuring the interference contrast and the population through the projective phonon measurement of two motional modes. We also measure the quantum Fisher information of the generated NOON state and observe the Heisenberg scaling in the lower bounds of the phase sensitivity as the $N$ increases. Our scheme is generic and applicable to other photonic or phononic systems.

Quantum computers will eventually reach a size at which quantum error correction becomes imperative. Quantum information can be protected from qubit imperfections and flawed control operations by encoding a single logical qubit in multiple physical qubits. This redundancy allows the extraction of error syndromes and the subsequent detection or correction of errors without destroying the logical state itself through direct measurement. Here we show the encoding and syndrome measurement of a fault-tolerant logical qubit via an error detection protocol on four physical qubits, represented by trapped atomic ions. This demonstrates for the first time the robustness of a fault-tolerant qubit to imperfections in the very operations used to encode it. The advantage persists in the face of large added error rates and experimental calibration errors.

Quantum field theories describe a wide variety of fundamental phenomena in physics. However, their study often involves cumbersome numerical simulations. Quantum simulators, on the other hand, may outperform classical computational capacities due to their potential scalability. Here, we report an experimental realization of a quantum simulation of fermion-antifermion scattering mediated by bosonic modes, using a multilevel trapped ion, which is a simplified model of fermion scattering in both perturbative and nonperturbative quantum electrodynamics. The simulated model exhibits prototypical features in quantum field theory including particle pair creation and annihilation, as well as self-energy interactions. These are experimentally observed by manipulating four internal levels of a $^{171}\mathrm{Yb}^{+}$ trapped ion, where we encode the fermionic modes, and two motional degrees of freedom that simulate the bosonic modes. Our experiment establishes an avenue towards the efficient implementation of fermionic and bosonic quantum field modes, which may prove useful in scalable studies of quantum field theories in perturbative and nonperturbative regimes.

We theoretically investigate the many-body localization phase transition in a one-dimensional Ising spin chain with random long-range spin-spin interactions, $V_{ij}\propto\left|i-j\right|^{-α}$, where the exponent of the interaction range $α$ can be tuned from zero to infinitely large. By using exact diagonalization, we calculate the half-chain entanglement entropy and the energy spectral statistics and use them to characterize the phase transition towards the many-body localization phase at infinite temperature and at sufficiently large disorder strength. We perform finite-size scaling to extract the critical disorder strength and the critical exponent of the divergent localization length. With increasing $α$, the critical exponent experiences a sharp increase at about $α=1$ and then gradually decreases to a value found earlier in a disordered short-ranged interacting spin chain. For $α<1$, we find that the system is mostly localized and the increase in the disorder strength may drive a transition between two many-body localized phases. In contrast, for $α>1$, the transition is from a thermalized phase to the many-body localization phase. Our predictions could be experimentally tested with ion-trap quantum emulator with programmable random long-range interactions, or with randomly distributed Rydberg atoms or polar molecules in lattices.

Optomechanical systems show tremendous promise for high sensitivity sensing of forces and modification of mechanical properties via light. For example, similar to neutral atoms and trapped ions, laser cooling of mechanical motion by radiation pressure can take single mechanical modes to their ground state. Conventional optomechanical cooling is able to introduce additional damping channel to mechanical motion, while keeping its thermal noise at the same level, and as a consequence, the effective temperature of the mechanical mode is lowered. However, the ratio of temperature to quality factor remains roughly constant, preventing dramatic advances in quantum sensing using this approach. Here we propose an approach for simultaneously reducing the thermal load on a mechanical resonator while improving its quality factor. In essence, we use the optical interaction to dynamically modify the dominant damping mechanism, providing an optomechanically-induced effect analogous to a phononic band gap. The mechanical mode of interest is assumed to be weakly coupled to its heat bath but strongly coupled to a second mechanical mode, which is cooled by radiation pressure coupling to a red detuned cavity field. We also identify a realistic optomechanical design that has the potential to realize this novel cooling scheme.

The first generation of quantum computers are on the horizon, fabricated from quantum hardware platforms that may soon be able to tackle certain tasks that cannot be performed or modelled with conventional computers. These quantum devices will not likely be universal or fully programmable, but special-purpose processors whose hardware will be tightly co-designed with particular target applications. Trapped atomic ions are a leading platform for first generation quantum computers, but are also fundamentally scalable to more powerful general purpose devices in future generations. This is because trapped ion qubits are atomic clock standards that can be made identical to a part in 10^15, and their quantum circuit connectivity can be reconfigured through the use of external fields, without modifying the arrangement or architecture of the qubits themselves. In this article we show how a modular quantum computer of any size can be engineered from ion crystals, and how the wiring between ion trap qubits can be tailored to a variety of applications and quantum computing protocols.

The application of adiabatic protocols in quantum technologies is severely limited by environmental sources of noise and decoherence. Shortcuts to adiabaticity by counterdiabatic driving constitute a powerful alternative that speed up time-evolution while mimicking adiabatic dynamics. Here we present the first experimental implementation of counterdiabatic driving in a continuous variable system, a shortcut to the adiabatic transport of a trapped ion in the phase space. The resulting dynamics is equivalent to a "fast-motion video" of the adiabatic trajectory. The robustness of this protocol is shown to surpass that of competing schemes based on classical local controls and Fourier optimization methods. Our results demonstrate that shortcuts to adiabaticity provide a robust speedup of quantum protocols of wide applicability in quantum technologies.

We present accurate ab initio and quantum scattering calculations on a prototypical hybrid ion-atom system Yb$^+$-Rb, recently suggested as a promising candidate for the experimental study of open quantum systems, quantum information processing, and quantum simulation. We identify the second-oder spin-orbit (SO) interaction as the dominant source of hyperfine relaxation and decoherence in cold Yb$^+$-Rb collisions. Our results are in good agreement with recent experimental observations [L. Ratschbacher et al., Phys. Rev. Lett. 110, 160402 (2013)] of hyperfine relaxation rates of trapped Yb$^+$ immersed in an ultracold Rb gas. The calculated rates are 4 times smaller than predicted by the Langevin capture theory and display a weak $T^{-0.3}$ temperature dependence, indicating significant deviations from statistical behavior. Our analysis underscores the deleterious nature of the SO interaction and implies that light ion-atom combinations such as Yb$^+$-Li should be used to minimize hyperfine relaxation and decoherence of trapped ions in ultracold atomic gases.

We consider a chain of trapped ions to interact with each other via long-range interactions. This system can be used to simulate the long-range Ising model. We study the dynamics of quantum coherence of a single spin in the chain, where the spins are initially prepared in their upper states. The relaxation dynamics exhibits due to the genuine long-range interaction. The degree of quantum coherence of a single spin rapidly decreases and vanishes in the steady state. However, our numerical result suggests that the conventional spin chain model, which truncates the interactions between the distant spins, cannot show the relaxation dynamics. This implies that the usual truncation in approximating the long-range interaction is not applicable to describing the non-equilibrium dynamics. The effect of the interaction range on the relaxation dynamics is studied. The higher relaxation rate will show if a system has a longer range of interaction. However, it takes a longer relaxation time in the vicinity of infinite interaction range. We also examine the dynamics of quantum coherence of a block of spins. Our result may shed light on the relationship between long-range interaction and the coherence dynamics of a quantum many-body system.

We analyze the dynamics and steady state of a linear ion array when some of the ions are continuously laser cooled. We calculate the ions' local temperature measured by its position fluctuation under various trapping and cooling configurations, taking into account background heating due to the noisy environment. For a large system, we demonstrate that by arranging the cooling ions evenly in the array, one can suppress the overall heating considerably. We also investigate the effect of different cooling rates and find that the optimal cooling efficiency is achieved by an intermediate cooling rate. We discuss the relaxation time for the ions to approach the steady state, and show that with periodic arrangement of the cooling ions, the cooling efficiency does not scale down with the system size.

Theoretical models of spins coupled to bosons provide a simple setting for studying a broad range of important phenomena in many-body physics, from virtually mediated interactions to decoherence and thermalization. In many atomic, molecular, and optical systems, such models also underlie the most successful attempts to engineer strong, long-ranged interactions for the purpose of entanglement generation. Especially when the coupling between the spins and bosons is strong---such that it cannot be treated perturbatively---the properties of such models are extremely challenging to calculate theoretically. Here, exact analytical expressions for nonequilibrium spin-spin correlation functions are derived for a specific model of spins coupled to bosons. The spatial structure of the coupling between spins and bosons is completely arbitrary, and thus the solution can be applied to systems in any number of dimensions. The explicit and nonperturbative inclusion of the bosons enables the study of entanglement generation (in the form of spin squeezing) even when the bosons are driven strongly and near-resonantly, and thus provides a quantitative view of the breakdown of adiabatic elimination that inevitably occurs as one pushes towards the fastest entanglement generation possible. The solution also helps elucidate the effect of finite temperature on spin squeezing. The model considered is relevant to a variety of atomic, molecular, and optical systems, such as atoms in cavities or trapped ions. As an explicit example, the results are used to quantify phonon effects in trapped ion quantum simulators, which are expected to become increasingly important as these experiments push towards larger numbers of ions.

Motivated by recent trapped-ion quantum simulation experiments, we carry out a comprehensive study of the phase diagram of a spin-1 chain with XXZ-type interactions that decay as $1/r^α$, using a combination of finite and infinite-size DMRG calculations, spin-wave analysis, and field theory. In the absence of long-range interactions, varying the spin-coupling anisotropy leads to four distinct phases: a ferromagnetic Ising phase, a disordered XY phase, a topological Haldane phase, and an antiferromagnetic Ising phase. If long-range interactions are antiferromagnetic and thus frustrated, we find primarily a quantitative change of the phase boundaries. On the other hand, ferromagnetic (non-frustrated) long-range interactions qualitatively impact the entire phase diagram. Importantly, for $α\lesssim3$, long-range interactions destroy the Haldane phase, break the conformal symmetry of the XY phase, give rise to a new phase that spontaneously breaks a $U(1)$ continuous symmetry, and introduce an exotic tricritical point with no direct parallel in short-range interacting spin chains. We show that the main signatures of all five phases found could be observed experimentally in the near future.

Continuous symmetry breaking (CSB) in low-dimensional systems, forbidden by the Mermin-Wagner theorem for short-range interactions, may take place in the presence of slowly decaying long-range interactions. Nevertheless, there is no stringent bound on how slowly interactions should decay to give rise to CSB in 1D quantum systems at zero temperature. Here, we study a long-range interacting spin chain with $U(1)$ symmetry and power-law interactions $V(r)\sim1/r^α$, directly relevant to ion-trap experiments. Using bosonization and renormalization group theory, we find CSB for $α$ smaller than a critical exponent $α_c(\le 3)$ depending on the microscopic parameters of the model. Furthermore, the transition from the gapless XY phase to the gapless CSB phase is mediated by the breaking of conformal symmetry due to long-range interactions, and is described by a new universality class akin to the Berezinskii-Kosterlitz-Thouless transition. Our analytical findings are in good agreement with a numerical calculation. Signatures of the CSB phase should be accessible in existing trapped-ion experiments.

Quantum theory is based on a mathematical structure totally different from conventional arithmetic. Due to the symmetric nature of bosonic particles, annihilation or creation of single particles translates a quantum state depending on how many bosons are already in the given quantum system. This proportionality results in a variety of non-classical features of quantum mechanics including the bosonic commutation relation. The annihilation and creation operations have recently been implemented in photonic systems. However, this feature of quantum mechanics does not preclude the possibility of realizing conventional arithmetic in quantum systems. We implement conventional addition and subtraction of single phonons for a trapped \Yb ion in a harmonic potential. In order to realize such operations, we apply the transitionless adiabatic passage scheme on the anti-Jaynes-Cummings coupling between the internal energy states and external motion states of the ion. By performing the operations on superpositions of Fock states, we realize the hybrid computation of classical arithmetic in quantum parallelism, and show that our operations are useful to engineer quantum states. Our single-phonon operations are nearly deterministic and robust against parameter changes, enabling handy repetition of the operations independently from the initial state of the atomic motion. We demonstrate the transform of a classical state to a nonclassical one of highly sub-Poissonian phonon statistics and a Gaussian state to a non-Gaussian state, by applying a sequence of the operations. The operations implemented here are the Susskind-Glogower phase operators, whose non-commutativity is also demonstrated.

In classical computational chemistry, the coupled-cluster ansatz is one of the most commonly used $ab~initio$ methods, which is critically limited by its non-unitary nature. The unitary modification as an ideal solution to the problem is, however, extremely inefficient in classical conventional computation. Here, we provide the first experimental evidence that indeed the unitary version of the coupled cluster ansatz can be reliably performed in physical quantum system, a trapped ion system. We perform a simulation on the electronic structure of a molecular ion (HeH$^+$), where the ground-state energy surface curve is probed, energies of excited-states are studied and the bond-dissociation is simulated non-perturbatively. Our simulation takes advantages from quantum computation to overcome the intrinsic limitations in classical computation and our experimental results indicate that the method is promising for preparing molecular ground-states for quantum simulation.

We propose to experimentally explore the Haldane phase in spin-one XXZ antiferromagnetic chains using trapped ions. We show how to adiabatically prepare the ground states of the Haldane phase, demonstrate their robustness against sources of experimental noise, and propose ways to detect the Haldane ground states based on their excitation gap and exponentially decaying correlations, nonvanishing nonlocal string order, and doubly-degenerate entanglement spectrum.

Topological phases of matter are primarily studied in systems with short-range interactions. In nature, however, non-relativistic quantum systems often exhibit long-range interactions. Under what conditions topological phases survive such interactions, and how they are modified when they do, is largely unknown. By studying the symmetry-protected topological phase of an antiferromagnetic spin-1 chain with $1/r^α$ interactions, we show that two very different outcomes are possible, depending on whether or not the interactions are frustrated. While non-frustrated long-range interactions can destroy the topological phase for $α\lesssim3$, the topological phase survives frustrated interactions for all $α>0$. Our conclusions are based on strikingly consistent results from large-scale matrix-product-state simulations and effective-field-theory calculations, and we expect them to hold for more general interacting spin systems. The models we study can be naturally realized in trapped-ion quantum simulators, opening the prospect for experimental investigation of the issues confronted here.

Besides the conventional transverse couplings between superconducting qubits (SQs) and electromagnetic fields, there are additional longitudinal couplings when the inversion symmetry of the potential energies of the SQs is broken. We study nonclassical-state generation in a SQ which is driven by a classical field and coupled to a single-mode microwave field. We find that the classical field can induce transitions between two energy levels of the SQs, which either generate or annihilate, in a controllable way, different photon numbers of the cavity field. The effective Hamiltonians of these classical-field-assisted multiphoton processes of the single-mode cavity field are very similar to those for cold ions, confined to a coaxial RF-ion trap and driven by a classical field. We show that arbitrary superpositions of Fock states can be more efficiently generated using these controllable multiphoton transitions, in contrast to the single-photon resonant transition when there is only a SQ-field transverse coupling. The experimental feasibility for different SQs is also discussed.

We report on experiments with a microfabricated surface trap designed for trapping a chain of ions in a ring. Uniform ion separation over most of the ring is achieved with a rotationally symmetric design and by measuring and suppressing undesired electric fields. After minimizing these fields the ions are confined primarily by an rf trapping pseudo-potential and their mutual Coulomb repulsion. The ring-shaped crystal consists of approximately 400 Ca$^+$ ions with an estimated average separation of 9 $μm$.

Experimental realization of a universal set of quantum logic gates is the central requirement for implementation of a quantum computer. An all-geometric approach to quantum computation offered a paradigm for implementation where all the quantum gates are achieved based on the Berry phases and their non-abelian extensions, the holonomies, from geometric transformation of quantum states in the Hilbert space. Apart from its fundamental interest and rich mathematical structure, the geometric approach has some built-in noise-resilient features. On the experimental side, geometric phases and holonomies have been observed using nuclear magnetic resonance with thermal ensembles of liquid molecules, however, such systems are known to be non-scalable for quantum computing. There are proposals to implement geometric quantum computation in scalable experimental platforms such as trapped ions, superconducting qubits, or quantum dots, and a recent experiment has realized geometric single-bit gates with the superconducting system. Here, we report the experimental realization of a universal set of geometric quantum gates with solid-state spins of the diamond defects. The diamond defects provide a scalable experimental platform with the potential for room-temperature quantum computing, which has attracted strong interest in recent years. Based on advance of coherent control in this system, our experiment shows that all-geometric and potentially robust quantum computation can be realized with solid-state spin qubits.

The past two decades witnessed important developments in the field of non-equilibrium statistical mechanics. Among these developments, the Jarzynski equality, being a milestone following the landmark work of Clausius and Kelvin, stands out. The Jarzynski equality relates the free energy difference between two equilibrium states and the work done on the system through far from equilibrium processes. While experimental tests of the equality have been performed in classical regime, the verification of the quantum Jarzynski equality has not yet been fully demonstrated due to experimental challenges. Here, we report an experimental test of the quantum Jarzynski equality with a single \Yb ion trapped in a harmonic potential. We perform projective measurements to obtain phonon distributions of the initial thermal state. Following that we apply the laser induced force on the projected energy eigenstate, and find transition probabilities to final energy eigenstates after the work is done. By varying the speed of applying the force from equilibrium to far-from equilibrium regime, we verified the quantum Jarzynski equality in an isolated system.

The understanding of symmetry operations has brought enormous advancements in physics, ranging from elementary particle to condensed matter systems. In quantum mechanics, symmetry operations are described by either unitary or antiunitary operators, where the latter are unphysical transformations that cannot be realized in physical systems. So far, quantum simulators of unitary and dissipative processes, the only allowed physical dynamics, have been realized in key experiments. Here, we present an embedding quantum simulator able to encode unphysical operations in a multilevel single trapped ion. In this sense, we experimentally observe phenomena associated with the nonunitary Majorana dynamics and implement antiunitary symmetry operations, i.e., time reversal and charge conjugation, at arbitrary evolution times. These experiments enhance the toolbox of quantum simulations towards applications involving unphysical operations.

We propose a scheme to realize scalable quantum computation in a planar ion crystal confined by a Paul trap. We show that the inevitable in-plane micromotion affects the gate design via three separate effects: renormalization of the equilibrium positions, coupling to the transverse motional modes, and amplitude modulation in the addressing beam. We demonstrate that all of these effects can be taken into account and high-fidelity gates are possible in the presence of micromotion. This proposal opens the prospect to realize large-scale fault-tolerant quantum computation within a single Paul trap.

Instabilities due to extrinsic interference are routinely faced in systems engineering, and a common solution is to rely on a broad class of $\textit{filtering}$ techniques in order to afford stability to intrinsically unstable systems. For instance, electronic systems are frequently designed to incorporate electrical filters composed of, $\textit{e.g.}$ RLC components, in order to suppress the effects of out-of-band fluctuations that interfere with desired performance. Quantum coherent systems are now moving to a level of complexity where challenges associated with realistic time-dependent noise are coming to the fore. Unfortunately, standard control solutions involving feedback are generally impossible due to the strictures of quantum mechanics, and existing error-suppressing gate constructions generally rely on unphysical bang-bang controls or quasi-static error models that do not reflect realistic laboratory environments. In this work we use the theory of quantum control engineering and experiments with trapped $^{171}$Yb$^{+}$ ions to demonstrate the construction of novel $\textit{noise filters}$ which are specifically designed to mitigate the effect of realistic time-dependent fluctuations on qubits \emph{during useful operations}. Starting with desired filter characteristics and the Walsh basis functions, we use a combination of analytic design rules and numeric search to construct time-domain noise filters tailored to a desired state transformation. Our results validate the generalized filter-transfer function framework for arbitrary quantum control operations, and demonstrate that it can be leveraged as an effective and efficient tool for developing novel robust control protocols.

We present a theoretical study of noise-induced quantum coherences in a model three-level V-type system interacting with incoherent radiation, an important prototype for a wide range of physical systems ranging from trapped ions to biomolecules and quantum dots. By solving the quantum optical equations of motion for the V-type system, we obtain analytical expressions for the noise-induced coherences and show that they exhibit an oscillating behavior in the limit of large excited level spacing $Δ$ ($Δ/γ\gg 1$, where $γ$ is the radiative decay rate). Most remarkably, we find that in the opposite limit of small level spacing $Δ/γ\ll 1$, appropriate for large molecules, (a) the coherences can survive for an arbitrarily long time $τ=(2/γ) (Δ/γ)^{-2}$ before eventually decaying to zero, and (b) coherences at short times can be substantial. We further show that the long-lived coherences can be robust against environmental relaxation and decoherence, and discuss implications to the design of quantum heat engines and incoherent light excitation of biological systems.

We develop and demonstrate a technique to engineer universal unitary baths in quantum systems. Using the correspondence between unitary decoherence due to ambient environmental noise and errors in a control system for quantum bits, we show how a wide variety of relevant classical error models may be realized through In-Phase/Quadrature modulation on a vector signal generator producing a resonant carrier signal. We demonstrate our approach through high-bandwidth modulation of the 12.6 GHz carrier appropriate for trapped $^{171}$Yb$^{+}$ ions. Experiments demonstrate the reduction of coherent lifetime in the system in the presence of an engineered bath, with the observed $T_{2}$ scaling as predicted by a quantitative model described herein. These techniques form the basis of a toolkit for quantitative tests of quantum control protocols, helping experimentalists characterize the performance of their quantum coherent systems.

We study the precision limits of detecting a linear magnetic-field gradient by using W-states in the presence of different types of noises. We consider to use an atomic spin chain for probing the magnetic-field gradient, where a W-state is prepared. We compare this method with the measurement of using two uncorrelated atoms. For pure states, W-states can provide an improvement over uncorrelated states in determining the magnetic-field gradient up to four particles. We examine the effects of local dephasing and dissipations on the performances of detections. In presence of dephasing, the uncorrelated atoms can give a higher precision than using W-states. But W-states provide a better performance in the presence of dissipation for a few particles. We briefly discuss the implementation of the detection methods with cold atoms and trapped ions.

Over the last few decades, quantum chemistry has progressed through the development of computational methods based on modern digital computers. However, these methods can hardly fulfill the exponentially-growing resource requirements when applied to large quantum systems. As pointed out by Feynman, this restriction is intrinsic to all computational models based on classical physics. Recently, the rapid advancement of trapped-ion technologies has opened new possibilities for quantum control and quantum simulations. Here, we present an efficient toolkit that exploits both the internal and motional degrees of freedom of trapped ions for solving problems in quantum chemistry, including molecular electronic structure, molecular dynamics, and vibronic coupling. We focus on applications that go beyond the capacity of classical computers, but may be realizable on state-of-the-art trapped-ion systems. These results allow us to envision a new paradigm of quantum chemistry that shifts from the current transistor to a near-future trapped-ion-based technology.

Ultracold atoms in optical lattices are an important platform for quantum information science, lending itself naturally to quantum simulation of many-body physics and providing a possible path towards a scalable quantum computer. To realize its full potential, atoms at individual lattice sites must be accessible to quantum control and measurement. This challenge has so far been met with a combination of high-resolution microscopes and resonance addressing that have enabled both site-resolved imaging and spin-flips. Here we show that methods borrowed from the field of inhomogeneous control can greatly increase the performance of resonance addressing in optical lattices, allowing us to target arbitrary single-qubit gates on desired sites, with minimal crosstalk to neighboring sites and greatly improved robustness against uncertainty in the lattice position. We further demonstrate the simultaneous implementation of different gates at adjacent sites with a single global control waveform. Coherence is verified through two-pulse Ramsey interrogation, and randomized benchmarking is used to measure an average gate fidelity of ~95%. Our control-based approach to reduce crosstalk and increase robustness is broadly applicable in optical lattices irrespective of geometry, and may be useful also on other platforms for quantum information processing, such as ion traps and nitrogen-vacancy centers in diamond.

We propose an experimental scheme to observe prethermalization and dynamical transition in one-dimensional XY spin chain with long range interaction and inhomogeneous lattice spacing, which can be readily implemented with the recently developed trapped-ion quantum simulator. Local physical observables are found to relax to prethermal values at intermediate time scale, followed by complete relaxation to thermal values at much longer time. The physical origin of prethermalization is explained by spotting a non-trivial structure in lower half of the energy spectrum. The dynamical behavior of the system is shown to cross difference phases when the interaction range is continuously tuned, indicating the existence of dynamical phase transition.

We report the first experimental realization of the geometric Landau-Zener-Stückelberg (LZS) interferometry proposed by [Phys. Rev. Lett. 107, 207002 (2011)] in a single trapped ion system. Different from a conventional LZS interferometer, the interference fringes of our geometric interferometer originate solely from geometric phase. We also observe the robustness of the interference contrast against noise or fluctuation in the experimental parameters. Our scheme can be applied to other complex systems subject to relatively large errors in system control.

This is a reply to the comment from Patrick Bruno (arXiv:1211.4792) on our paper (Phys. Rev. Lett. 109, 163001 (2012)).

I propose a method to remove the axial motional excitation from an ion qubit within a few oscillation periods of a harmonic trap. The principle is to prepare another coolant ion in its motional ground state, and then apply a phonon beam splitter to swap the motional state between the ions without affecting the internal state. This method requires only the precise control of the trapping potential, thus its performance does not suffer from the limitations of current laser-cooling schemes. With sufficient coolant ions pre-prepared, this method can rapidly re-cool ion qubits during quantum information processing.

We propose to synthesize arbitrary nonclassical motional states in optomechanical systems by using sideband excitations and photon blockade. We first demonstrate that the Hamiltonian of the optomechanical systems can be reduced, in the strong single-photon optomechanical coupling regime when the photon blockade occurs, to one describing the interaction between a driven two-level trapped ion and the vibrating modes, and then show a method to generate target states by using a series of classical pulses with desired frequencies, phases, and durations. We further analyze the effect of the photon leakage, due to small anharmonicity, on the fidelity of the expected motional state, and study environment induced decoherence. Moreover, we also discuss the experimental feasibility and provide operational parameters using the possible experimental data.

We experimentally observed state-independent violations of Kochen-Specker inequalities for the simplest indivisible quantum system manifesting quantum contextuality, a three-level (qutrit) system. We performed the experiment with a single trapped ^{171}Yb^{+} ion, by mapping three ground states of the ^{171}Yb^{+} ion to a qutrit system and carrying out quantum operatations by applying microwaves resonant to the qutrit transition frequencies. Our results are free from the detection loophole and cannot be explained by the non-contextual hidden variable models.

The practical construction of scalable quantum computer hardware capable of executing non-trivial quantum algorithms will require the juxtaposition of different types of quantum systems. We analyze a modular ion trap quantum computer architecture with a hierarchy of interactions that can scale to very large numbers of qubits. Local entangling quantum gates between qubit memories within a single register are accomplished using natural interactions between the qubits, and entanglement between separate registers is completed via a probabilistic photonic interface between qubits in different registers, even over large distances. We show that this architecture can be made fault-tolerant, and demonstrate its viability for fault-tolerant execution of modest size quantum circuits.

Spontaneous symmetry breaking can lead to the formation of time crystals, as well as spatial crystals. Here we propose a space-time crystal of trapped ions and a method to realize it experimentally by confining ions in a ring-shaped trapping potential with a static magnetic field. The ions spontaneously form a spatial ring crystal due to Coulomb repulsion. This ion crystal can rotate persistently at the lowest quantum energy state in magnetic fields with fractional fluxes. The persistent rotation of trapped ions produces the temporal order, leading to the formation of a space-time crystal. We show that these space-time crystals are robust for direct experimental observation. We also study the effects of finite temperatures on the persistent rotation. The proposed space-time crystals of trapped ions provide a new dimension for exploring many-body physics and emerging properties of matter.

We describe a possible architecture to implement a universal bosonic simulator (UBS) using trapped ions. Single ions are confined in individual traps, and their motional states represent the bosonic modes. Single-mode linear operators, nonlinear phase-shifts, and linear beam splitters can be realized by precisely controlling the trapping potentials. All the processes in a bosonic simulation, except the initialization and the readout, can be conducted beyond the Lamb-Dicke regime. Aspects of our proposal can also be applied to split adiabatically a pair of ions in a single trap.

A collection of trapped atomic ions represents one of the most attractive platforms for the quantum simulation of interacting spin networks and quantum magnetism. Spin-dependent optical dipole forces applied to an ion crystal create long-range effective spin-spin interactions and allow the simulation of spin Hamiltonians that possess nontrivial phases and dynamics. Here we show how appropriate design of laser fields can provide for arbitrary multidimensional spin-spin interaction graphs even for the case of a linear spatial array of ions. This scheme uses currently existing trap technology and is scalable to levels where classical methods of simulation are intractable.

We show how to realize, by means of non-abelian quantum holonomies, a set of universal quantum gates acting on decoherence-free subspaces and subsystems. In this manner we bring together the quantum coherence stabilization virtues of decoherence-free subspaces and the fault-tolerance of all-geometric holonomic control. We discuss the implementation of this scheme in the context of quantum information processing using trapped ions and quantum dots.

We propose two protocols to encode a logical qubit into physical qubits relying on common types of qubit-qubit interactions in as simple forms as possible. We comment on its experimental implementation in several quantum computing architectures, e.g. with trapped atomic ion qubits, atomic qubits inside a high Q optical cavity, solid state Josephson junction qubits, and Bose-Einstein condensed atoms.

The "pushing gate" proposed by Cirac and Zoller in 2000 for quantum logic in ion traps is discussed, in which a force is used to give a controlled push to a pair of trapped ions and thus realize a phase gate. The original proposal had a weakness in that it involved a hidden extreme sensitivity to the size of the force. Also, the physical origin of this force was not fully addressed. Here, we discuss the sensitivity and present a way to avoid it by choosing the spatial form of the pushing force in an optimal way. We also analyse the effect of imperfections in a pair of pi pulses which are used to implement a "spin-echo" to cancel correlated errors. We present a physical model for the force, namely the dipole force, and discuss the impact of unwanted photon scattering, and of finite temperature of the ions. The main effect of the temperature is to blur the phase of the gate owing to the ions exploring a range of values of the force. When the distance scale of the force profile is smaller than the ion separation, this effect is more important than the high-order terms in the Coulomb repulsion which were originally discussed. Overall, we find that whereas the "pushing gate" is not as resistant to imperfections as was supposed, it remains a significant candidate for ion trap quantum computing since it does not require ground state cooling, and in some cases it does not require the Lamb-Dicke limit, while the gate rate is fast, close to (rather than small compared to) the trap vibrational frequency.