Jing-Ning Zhang

Trapped-ion systems have emerged as a leading platform for scalable quantum information processing owing to their high-fidelity operations and long-range entangling capabilities. As the number of ions in a trap increases, the growing density of collective motional modes makes the synthesis of multimode entangling gates increasingly challenging. Designing large-scale gates requires simultaneously realizing the desired spin-spin interactions, suppressing residual spin-motion entanglement, and limiting experimental control resources, leading to a high-dimensional non-convex optimization problem. Here we develop a numerical framework for multi-tone gate synthesis that directly searches for control fields satisfying these competing requirements. By employing an alternating-minimization strategy, the framework improves numerical stability and remains effective for large systems with many motional modes and target interactions. As representative demonstrations, we synthesize gates implementing all-to-all and nearest-neighbor interaction patterns in ion chains of up to N = 1000, using only global laser control. Across the parameter regimes explored here, the control resources required to maintain high-fidelity interactions do not exhibit rapid growth with system size. We extend the framework to individual addressing using a structured qLDPC target at N = 512 as an example. These results identify multimode gate synthesis as a viable route toward programmable interaction engineering in large-scale trapped-ion quantum processors.

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.

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.

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 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.

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.

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.

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.

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.

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 $93.4\%$.

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 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.

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 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.

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.

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.

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.