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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

arXiv:1805.09072v1quant-ph

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.

Abstract

from arXiv · show

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

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.Source: MEASURED SYSTEM PARAMETERS, COHERENCE… · p. 11
  • 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.Source: Figure S1 · p. 9, MEASURED SYSTEM PARAMETERS, COHERENCE… · p. 10

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.Source: MEASURED SYSTEM PARAMETERS, COHERENCE… · pp. 10–11

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.Source: EXPERIMENTAL SEQUENCES · p. 11
  • 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.Source: EXPERIMENTAL SEQUENCES · p. 11
  • 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.Source: EXPERIMENTAL SEQUENCES · p. 12
  • 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.Source: EXPERIMENTAL SEQUENCES · p. 12

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.Source: T GATE FIDELITY USING REPEATED… · p. 12
  • 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.Source: T GATE FIDELITY USING REPEATED… · p. 12

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

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.Source: THEORETICAL ANALYSIS AND ERROR MODEL… · p. 13

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.Source: THEORETICAL ANALYSIS AND ERROR MODEL… · pp. 12–13

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.Source: Intrinsic Errors from Oscillator… · pp. 14, 16

QEC trade-off

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.Source: Analytical Model and Error Propagation · p. 18
  • 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.Source: Figure S12 · p. 17, Analytical Model and Error Propagation · p. 18
  • 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.Source: Analytical Model and Error Propagation · p. 18
  • 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.Source: Analytical Model and Error Propagation · p. 18

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.Source: Towards Break-Even Point · p. 18

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.

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

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