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Demonstration of quantum error correction and universal gate set on a binomial bosonic logical qubit
Ling Hu, Yuwei Ma, Weizhou Cai, Xianghao Mu, Yuan Xu, Weiting Wang, Yukai Wu, Haiyan Wang, Yipu Song, Changling Zou, S. M. Girvin, L-M. Duan, Luyan Sun
TL;DR
Quantum error correction depends on syndrome measurements that do not introduce additional photon-jump errors. This work characterizes parity-based QEC operations and system imperfections, finding near-QND measurement while identifying ancilla-assisted operations as a dephasing constraint.
Problem
Quantum error correction requires syndrome measurements that do not introduce extra photon-jump errors.
Method
The experiment initializes the qubit–oscillator system by post-selection and characterizes oscillator parity measurements through repeated monitoring and parity-oscillation fits.
Results
99.92% QND parity measurement and 0.972 averaged parity readout fidelity were measured, with a 0.08% demolition probability per measurement.
Takeaways & Limitations
Improving ancilla coherence is identified as the most direct route to extending QEC lifetime, potentially enabling more frequent error detection and correction.
Takeaways & Limitations
Ancilla-qubit excitation during ancilla-assisted operations can introduce significant dephasing and is included only within combined numerical and experimental estimates.
Abstract
from arXiv · showhide
Logical qubit encoding and quantum error correction (QEC) have been experimentally demonstrated in various physical systems with multiple physical qubits, however, logical operations are challenging due to the necessary nonlocal operations. Alternatively, logical qubits with bosonic-mode-encoding are of particular interest because their QEC protection is hardware efficient, but gate operations on QEC protected logical qubits remain elusive. Here, we experimentally demonstrate full control on a single logical qubit with a binomial bosonic code, including encoding, decoding, repetitive QEC, and high-fidelity (97.0% process fidelity on average) universal quantum gate set on the logical qubit. The protected logical qubit has shown 2.8 times longer lifetime than the uncorrected one. A Ramsey experiment on a protected logical qubit is demonstrated for the first time with two times longer coherence than the unprotected one. Our experiment represents an important step towards fault-tolerant quantum computation based on bosonic encoding.
EXPERIMENTAL DEVICE AND SETUP
The experiment uses a cQED device with a transmon ancilla coupled to readout and storage cavities, supported by FPGA control and characterized system parameters. Measurements establish cavity coherence, Kerr nonlinearities, parity-readout fidelity, and QND performance.
- Hardware and measured parameters: The cQED device couples a transmon ancilla dispersively to separate readout and storage cavities used for measurement and logical-state encoding.The storage oscillator has a single-photon lifetime of 143 µs and coherence time of 252 µs.
- Hardware and measured parameters: Three FPGA boards control the ancilla, readout cavity, and oscillator, while distributing digitized readout results for coordinated feedback.Each board integrates DACs, ADCs, and digital I/O ports for pulse generation, measurement, and inter-board communication.
- Measurement properties: A single parity measurement is 99.92% QND, with a fitted demolition probability of pd = 0.08% per measurement.The monitored parity decay is modeled as the parallel combination of natural oscillator decay and measurement-induced demolition.
- Measurement properties: The ancilla readout fidelity exceeds 0.999 for |g⟩ and is 0.989 for |e⟩ using optimized dispersive readout with a JPA.The lower excited-state fidelity is attributed to ancilla decay during measurement.
- Hardware and measured parameters: The oscillator self-Kerr coefficient is measured as Ks/2π = 4.23 kHz from parity-oscillation frequencies after subtracting the known detuning.Higher-order Kerr corrections are included when designing GRAPE control pulses and selecting waiting intervals.
EXPERIMENTAL SEQUENCES
The experimental sequences initialize the ancilla and oscillator by post-selection, use the ancilla for syndrome detection and nonlinear control, and implement repetitive QEC with low-latency feedback.
- EXPERIMENTAL SEQUENCES: System initialization post-selects the ancilla ground state and oscillator vacuum, removing about 1.4% of the data.The protocol uses an ancilla ground-state measurement followed by an oscillator parity measurement.
- EXPERIMENTAL SEQUENCES: The ancilla provides both error-syndrome detection and the nonlinearity required for oscillator encoding, decoding, error correction, and universal logical operations.These operations are implemented through control pulses based on the ancilla–oscillator dispersive interaction.
- EXPERIMENTAL SEQUENCES: The repetitive QEC sequence balances photon-loss correction against operation and photon-loss errors using nested bottom-layer and top-layer correction steps.The bottom layer conserves parity in a deformed code space, while the top layer restores information to the code space.
- EXPERIMENTAL SEQUENCES: The adaptive-control latency is 336 ns, approximately 1% of the ancilla qubit lifetime.The latency is measured from the end of readout acquisition to the beginning of the control signal, including signal travel time.
T GATE FIDELITY USING REPEATED MEASUREMENTS
Because the T gate is non-Clifford, its fidelity is measured through repeated applications rather than randomized benchmarking. The repeated-gate procedure yields a fidelity of 0.987.
- T GATE FIDELITY USING REPEATED MEASUREMENTS: The T gate has fidelity 0.987 when extracted from repeated-gate measurements.The result is converted to match the fidelity definition used in randomized benchmarking.
- T GATE FIDELITY USING REPEATED MEASUREMENTS: Randomized benchmarking is not applicable to the T gate because it does not belong to the Clifford group.The repeated-gate χ-matrix decay instead gives normalized process fidelity F = 1−p for a depolarization channel.
THEORETICAL ANALYSIS AND ERROR MODEL OF THE QEC PERFORMANCE
The QEC analysis separates intrinsic oscillator errors from ancilla-induced imperfections and models how their competition determines performance. Experimentally, parity monitoring is strongly QND, while shorter correction intervals suppress intrinsic errors but increase ancilla-related errors.
- Error sources: Ancilla decoherence limits QEC because it can corrupt detection, recovery, encoding, decoding, and logical gate operations.Too-frequent QEC increases ancilla-induced errors, whereas infrequent QEC permits higher-order photon loss and gain errors and photon-loss-induced dephasing.
- Parity monitoring: 99.92% QND parity-measurement fidelity was obtained, with a measured single-parity-induced extra-change probability of 0.08%.The monitored parity-decay model uses 1/τtot = 1/τs + pd/τrep and yields τs = 143 µs.
- Modeling approach: The simulations treat the ancilla as a two-level system, omit the readout cavity, and use calibrated experimental parameters in Lindblad-based numerical modeling.Analytical and numerical approaches are used to analyze QEC sub-processes and compare their predictions with experiment.
- Error sources: The binomial code corrects a single photon loss, but multiple losses, photon gain, and Kerr-induced dephasing remain intrinsic error channels.After a photon loss, noncommutation between photon loss and self-Kerr evolution causes unknown-time-dependent dephasing; three-loss and one-gain leakage errors are smaller.
- QEC trade-off: Shorter QEC intervals drive intrinsic errors toward zero but require more frequent detection and correction, which introduces additional ancilla-decoherence errors.This trade-off produces an optimal QEC interval rather than favoring arbitrarily frequent correction.
Analytical Model and Error Propagation
The authors model QEC fidelity by propagating probabilistic no-loss and photon-loss branches through detection, recovery, and ancilla operations. The analytical model agrees with numerical simulations and the experiment, while the error budget identifies an optimized operating interval.
- Analytical Model and Error Propagation: The QEC fidelity model combines branches for no photon loss and single photon loss, including recovery-gate and ancilla-excitation fidelities.Protocol II extends this propagation to four branches corresponding to the two detection rounds.
- Analytical Model and Error Propagation: The analytical model and numerical simulations agree for QEC process-fidelity decay across QEC intervals and ancilla coherence times.The experimental result with T1 = 30 µs and T2 = 40 µs also agrees with the simulation.
- Analytical Model and Error Propagation: The π-pulse fidelity is Fπ = 0.984, with infidelity attributed mainly to finite gate bandwidth and pulse-calibration fluctuations.In one-error branches, this pulse adds an error source before recovery gates.
- Analytical Model and Error Propagation: An optimized QEC interval is required for the longest logical-qubit lifetime because operation errors and oscillator-loss errors compete.The recovery-gate fidelity FU is identified as a critical experimental parameter that is difficult to calibrate separately.
Towards Break-Even Point
The achieved logical-qubit lifetime remains close to that of the uncorrected Fock encoding, so further experiments are needed to reach the break-even point.
- Towards Break-Even Point: The logical-qubit lifetime is close to that of the Fock {|0⟩,|1⟩} encoding, leaving routes toward break-even for future experimental investigation.The passage presents this as an open performance boundary rather than a demonstrated result.
Improve the ancilla properties
Improving ancilla coherence is the most direct route to extending the QEC lifetime because the ancilla performs error detection and recovery. Nearly doubling T1 is predicted to reach break-even with unchanged Tφ.
- Ancilla coherence is the most direct lever for extending QEC lifetime because the ancilla performs error detection and all recovery operations.
- Nearly doubling T1 is predicted to achieve the break-even point while keeping Tφ unchanged.
Better parameter calibration and GRAPE pulses
Parameter fluctuations reduce parity-measurement and recovery fidelities, while additional bottom-layer QEC steps could improve performance by limiting unnecessary recovery operations. Reaching break-even would require four bottom-layer steps but would demand adaptive gate recalibration.
- Better parameter calibration and GRAPE pulses: Parity measurement reaches C0,1 ∼0.987 and recovery gates reach 0.976 under parameter fluctuations.The model also analyzes infidelities from finite ancilla T1 and Tφ.
- Better parameter calibration and GRAPE pulses: Increasing bottom-layer QEC steps limits unnecessary recovery-gate errors when no error is detected.
- Better parameter calibration and GRAPE pulses: Four bottom-layer QEC steps are estimated to achieve break-even with the current experimental system.
- Better parameter calibration and GRAPE pulses: The four-step strategy was not implemented because it requires more adaptive gate calibrations.