M. J. Biercuk

The realisation of a universal quantum computer at scale promises to deliver a paradigm shift in information processing, providing the capability to solve problems that are intractable with conventional computers. A key limiting factor of realising fault-tolerant quantum information processing (QIP) is the large ratio of physical-to-logical qubits that outstrip device sizes available in the near future. An alternative approach proposed by Gottesman, Kitaev, and Preskill (GKP) encodes a single logical qubit into a single harmonic oscillator, alleviating this hardware overhead in exchange for a more complex encoding. Owing to this complexity, current experiments with GKP codes have been limited to single-qubit encodings and operations. Here, we report on the experimental demonstration of a universal gate set for the GKP code, which includes single-qubit gates and -- for the first time -- a two-qubit entangling gate between logical code words. Our scheme deterministically implements energy-preserving quantum gates on finite-energy GKP states encoded in the mechanical motion of a trapped ion. This is achieved by a novel optimal control strategy that dynamically modulates an interaction between the ion's spin and motion. We demonstrate single-qubit gates with a logical process fidelity as high as 0.960 and a two-qubit entangling gate with a logical process fidelity of 0.680. We also directly create a GKP Bell state from the oscillators' ground states in a single step with a logical state fidelity of 0.842. The overall scheme is compatible with existing hardware architectures, highlighting the opportunity to leverage optimal control strategies as a key accelerant towards fault tolerance.

Accurate simulation of dynamical processes in molecules and reactions is among the most challenging problems in quantum chemistry. Quantum computers promise efficient chemical simulation, but the existing quantum algorithms require many logical qubits and gates, placing practical applications beyond existing technology. Here, we carry out the first quantum simulations of chemical dynamics by employing a more hardware-efficient encoding scheme that uses both qubits and bosonic degrees of freedom. Our trapped-ion device accurately simulates the dynamics of non-adiabatic chemical processes, which are among the most difficult problems in computational chemistry because they involve strong coupling between electronic and nuclear motions. We demonstrate the programmability and versatility of our approach by simulating the dynamics of three different molecules as well as open-system dynamics in the condensed phase, all with the same quantum resources. Our approach requires orders of magnitude fewer resources than equivalent qubit-only quantum simulations, demonstrating the potential of using hybrid encoding schemes to accelerate quantum simulations of complex chemical processes, which could have applications in fields ranging from energy conversion and storage to biology and drug design.

Encoding logical qubits in bosonic modes provides a potentially hardware-efficient implementation of fault-tolerant quantum information processing. Here, we demonstrate high-fidelity and deterministic preparation of highly non-classical bosonic states in the mechanical motion of a trapped ion. Our approach implements error-suppressing pulses through optimized dynamical modulation of laser-driven spin-motion interactions to generate the target state in a single step. We demonstrate logical fidelities for the Gottesman-Kitaev-Preskill (GKP) state as high as $\bar{\mathcal{F}}=0.940(8)$, a distance-3 binomial state with an average fidelity of $\mathcal{F}=0.807(7)$, and a 12.91(5) dB squeezed vacuum state.

Qubits encoded in hyperfine states of trapped ions are ideal for quantum computation given their long lifetimes and low sensitivity to magnetic fields, yet they suffer from off-resonant scattering during detection often limiting their measurement fidelity. In ${}^{171}$Yb$^{+}$ this is exacerbated by a low fluorescence yield, which leads to a need for complex and expensive hardware - a problematic bottleneck especially when scaling up the number of qubits. We demonstrate a detection routine based on electron shelving to address this issue in ${}^{171}$Yb$^{+}$ and achieve a 5.6$\times$ reduction in single-ion detection error on an avalanche photodiode to $1.8(2)\times10^{-3}$ in a 100 $μ$s detection period, and a 4.3$\times$ error reduction on an electron multiplying CCD camera, with $7.7(2)\times10^{-3}$ error in 400 $μ$s. We further improve the characterization of a repump transition at 760 nm to enable a more rapid reset of the auxiliary $^2$F$_{7/2}$ states populated after shelving. Finally, we examine the detection fidelity limit using the long-lived $^2$F$_{7/2}$ state, achieving a further 300$\times$ and 12$\times$ reduction in error to $6(7)\times10^{-6}$ and $6.3(3)\times10^{-4}$ in 1 ms on the respective detectors. While shelving-rate limited in our setup, we suggest various techniques to realize this detection method at speeds compatible with quantum information processing, providing a pathway to ultra-high fidelity detection in ${}^{171}$Yb$^{+}$.

Quantum error correction provides a path to large-scale quantum computers, but is built on challenging assumptions about the characteristics of the underlying errors. In particular, the mathematical assumption of statistically independent errors in quantum logic operations is at odds with realistic environments where error sources may exhibit strong temporal and spatial correlations. We present experiments using trapped ions to demonstrate that the use of dynamically corrected gates (DCGs), generally considered for the reduction of error magnitudes, can also suppress error correlations in space and time throughout quantum circuits. We present a first-principles analysis of the manifestation of error correlations in randomized benchmarking, and validate this model through experiments performed using engineered errors. We find that standard DCGs can reduce error correlations by $\sim50\times$, while increasing the magnitude of uncorrelated errors by a factor scaling linearly with the extended DCG duration compared to a primitive gate. We then demonstrate that the correlation characteristics of intrinsic errors in our system are modified by use of DCGs, consistent with a picture in which DCGs whiten the effective error spectrum induced by external noise.

Quantum error correction provides a path to large-scale quantum computers, but is built on challenging assumptions about the characteristics of the underlying errors. In particular, the mathematical assumption of independent errors in quantum logic operations is at odds with realistic environments where error-sources may exhibit strong temporal correlations. We present experiments enabling the identification of error correlations between operations in quantum circuits, using only projective measurements at the end of the circuit. Using a single trapped ion qubit and engineered noise with tunable temporal correlations, we identify a clear signature of error correlations between sequential gates in randomly composed quantum circuits, and extract quantitative measures linked to the underlying noise correlation length. By replacing all gates in these circuits with "virtual" dynamically corrected gates (DCGs), we demonstrate that even in the presence of strongly correlated noise the signatures of error correlations between sequential gates appear similar to standard gates exposed to uncorrelated noise. A theoretical model applied to our experiments reveals that common DCGs suppress the correlated error component by over $270\times$ with $95\%$ confidence. Using block-correlated noise, we explore the scaling of the effective error correlation length at the virtual level, and show that DCGs exhibit error correlations indistinguishable from those arising from uncorrelated noise.

Growth in the complexity and capabilities of quantum information hardware mandates access to practical techniques for performance verification that function under realistic laboratory conditions. Here we experimentally characterise the impact of common temporally correlated noise processes on both randomised benchmarking (RB) and gate-set tomography (GST). We study these using an analytic toolkit based on a formalism mapping noise to errors for arbitrary sequences of unitary operations. This analysis highlights the role of sequence structure in enhancing or suppressing the sensitivity of quantum verification protocols to either slowly or rapidly varying noise, which we treat in the limiting cases of quasi-DC miscalibration and white noise power spectra. We perform experiments with a single trapped $^{171}$Yb$^{+}$ ion as a qubit and inject engineered noise ($\propto σ^z$) to probe protocol performance. Experiments on RB validate predictions that the distribution of measured fidelities over sequences is described by a gamma distribution varying between approximately Gaussian for rapidly varying noise, and a broad, highly skewed distribution for the slowly varying case. Similarly we find a strong gate set dependence of GST in the presence of correlated errors, leading to significant deviations between estimated and calculated diamond distances in the presence of correlated $σ^z$ errors. Numerical simulations demonstrate that expansion of the gate set to include negative rotations can suppress these discrepancies and increase reported diamond distances by orders of magnitude for the same error processes. Similar effects do not occur for correlated $σ^x$ or $σ^y$ errors or rapidly varying noise processes, highlighting the critical interplay of selected gate set and the gauge optimisation process on the meaning of the reported diamond norm in correlated noise environments.

Industrial, metrological, and medical applications provide a strong technological pull for advanced nanoscale sensors exploiting the unique sensitivity of quantum coherent systems to their environments. Essential to the functionality of these devices is the availability of control protocols which shape the sensor's response to the environment in frequency space. However, a key challenge in these applications is that common control routines result in out-of-band spectral leakage which complicates interpretation of the sensor's signal. In this work we demonstrate provably optimal narrowband control protocols ideally suited to quantum sensing. Our results, based on experiments with trapped ions using modulation in the form of discrete prolate spheroidal sequences (aka Slepian functions), demonstrate reduction of spectral leakage by orders of magnitude over conventional controls. We tune the narrowband sensitivity using concepts from RF engineering and experimentally reconstruct complex noise spectra using engineered noise for quantitative performance evaluation. We then deploy these techniques to identify previously immeasurable frequency-resolved amplitude noise in our qubit synthesis chain with calibrated sensitivity better than 0.001 dB.

Experimentalists seeking to improve the coherent lifetimes of quantum bits have generally focused on mitigating decoherence mechanisms through, for example, improvements to qubit designs and materials, and system isolation from environmental perturbations. In the case of the phase degree of freedom in a quantum superposition, however, the coherence that must be preserved is not solely internal to the qubit, but rather necessarily includes that of the qubit relative to the "master clock" (e.g. a local oscillator) that governs its control system. In this manuscript we articulate the impact of instabilities in the master clock on qubit phase coherence, and provide tools to calculate the contributions to qubit error arising from these processes. We first connect standard oscillator phase-noise metrics to their corresponding qubit dephasing spectral densities. We then use representative lab-grade and performance-grade oscillator specifications to calculate operational fidelity bounds on trapped-ion and superconducting qubits with relatively slow and fast operation times. We discuss the relevance of these bounds for quantum error correction in contemporary experiments and future large-scale quantum information systems, and discuss potential means to improve master clock stability.

We present a scheme designed to suppress the dominant source of infidelity in entangling gates between quantum systems coupled through intermediate bosonic oscillator modes. Such systems are particularly susceptible to residual qubit-oscillator entanglement at the conclusion of a gate period which reduces the fidelity of the target entangling operation. We demonstrate how the exclusive use of discrete phase shifts in the field moderating the qubit-oscillator interaction - easily implemented with modern synthesizers - is sufficient to both ensure multiple oscillator modes are decoupled and to suppress the effects of fluctuations in the driving field. This approach is amenable to a wide variety of technical implementations including geometric phase gates in superconducting qubits and the Molmer-Sorensen gate for trapped ions. We present detailed example protocols tailored to trapped-ion experiments and demonstrate that our approach allows multiqubit gate implementation with a significant reduction in technical complexity relative to previously demonstrated protocols.

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 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 describe a high-power, frequency-tunable, external cavity diode laser (ECDL) system near 626 nm useful for laser cooling of trapped $^9$Be$^+$ ions. A commercial single-mode laser diode with rated power output of 170 mW at 635 nm is cooled to $\approx - 31$ C, and a single longitudinal mode is selected via the Littrow configuration. In our setup, involving multiple stages of thermoelectric cooling, we are able to obtain $\approx$130 mW near 626 nm, sufficient for efficient frequency doubling to the required Doppler cooling wavelengths near 313 nm in ionized Beryllium. In order to improve nonlinear frequency conversion efficiency, we achieve larger useful power via injection locking of a slave laser. In this way the entirety of the slave output power is available for frequency doubling, while analysis may be performed on the master output. We believe that this simple laser system addresses a key need in the ion trapping community and dramatically reduces the cost and complexity associated with Beryllium ion trapping experiments.

We have demonstrated transport of Be+ ions through a 2D Paul-trap array that incorporates an X-junction, while maintaining the ions near the motional ground-state of the confining potential well. We expand on the first report of the experiment [1], including a detailed discussion of how the transport potentials were calculated. Two main mechanisms that caused motional excitation during transport are explained, along with the methods used to mitigate such excitation. We reduced the motional excitation below the results in Ref. [1] by a factor of approximately 50. The effect of a mu-metal shield on qubit coherence is also reported. Finally, we examined a method for exchanging energy between multiple motional modes on the few-quanta level, which could be useful for cooling motional modes without directly accessing the modes with lasers. These results establish how trapped ions can be transported in a large-scale quantum processor with high fidelity.

We report phase-coherent Doppler detection of optical dipole forces using large ion crystals in a Penning trap. The technique is based on laser Doppler velocimetry using a cycling transition in $^{9}$Be$^{+}$ near 313 nm and the center-of-mass (COM) ion motional mode. The optical dipole force is tuned to excite the COM mode, and measurements of photon arrival times synchronized with the excitation potential show oscillations with a period commensurate with the COM motional frequency. Experimental results compare well with a quantitative model for a driven harmonic oscillator. This technique permits characterization of motional modes in ion crystals; the measurement of both frequency and phase information relative to the driving force is a key enabling capability -- comparable to lockin detection -- providing access to a parameter that is typically not available in time-averaged measurements. This additional information facilitates discrimination of nearly degenerate motional modes.

The ability to detect extremely small forces is vital for a variety of disciplines including precision spin-resonance imaging, microscopy, and tests of fundamental physical phenomena. Current force-detection sensitivity limits have surpassed 1 $aN/\sqrt{Hz}$ (atto $=10^{-18}$) through coupling of micro or nanofabricated mechanical resonators to a variety of physical systems including single-electron transistors, superconducting microwave cavities, and individual spins. These experiments have allowed for probing studies of a variety of phenomena, but sensitivity requirements are ever-increasing as new regimes of physical interactions are considered. Here we show that trapped atomic ions are exquisitely sensitive force detectors, with a measured sensitivity more than three orders of magnitude better than existing reports. We demonstrate detection of forces as small as 174 $yN$ (yocto $=10^{-24}$), with a sensitivity 390$\pm150$ $yN/\sqrt{Hz}$ using crystals of $n=60$ $^{9}$Be$^{+}$ ions in a Penning trap. Our technique is based on the excitation of normal motional modes in an ion trap by externally applied electric fields, detection via and phase-coherent Doppler velocimetry, which allows for the discrimination of ion motion with amplitudes on the scale of nanometers. These experimental results and extracted force-detection sensitivities in the single-ion limit validate proposals suggesting that trapped atomic ions are capable of detecting of forces with sensitivity approaching 1 $yN/\sqrt{Hz}$. We anticipate that this demonstration will be strongly motivational for the development of a new class of deployable trapped-ion-based sensors, and will permit scientists to access new regimes in materials science.