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Exponential suppression of bit or phase flip errors with repetitive error correction

Zijun Chen, Kevin J. Satzinger, Juan Atalaya, Alexander N. Korotkov, Andrew Dunsworth, Daniel Sank, Chris Quintana, Matt McEwen, Rami Barends, Paul V. Klimov, Sabrina Hong, Cody Jones, Andre Petukhov, Dvir Kafri, Sean Demura, Brian Burkett, Craig Gidney, Austin G. Fowler, Harald Putterman, Igor Aleiner, Frank Arute, Kunal Arya, Ryan Babbush, Joseph C. Bardin, Andreas Bengtsson, Alexandre Bourassa, Michael Broughton, Bob B. Buckley, David A. Buell, Nicholas Bushnell, Benjamin Chiaro, Roberto Collins, William Courtney, Alan R. Derk, Daniel Eppens, Catherine Erickson, Edward Farhi, Brooks Foxen, Marissa Giustina, Jonathan A. Gross, Matthew P. Harrigan, Sean D. Harrington, Jeremy Hilton, Alan Ho, Trent Huang, William J. Huggins, L. B. Ioffe, Sergei V. Isakov, Evan Jeffrey, Zhang Jiang, Kostyantyn Kechedzhi, Seon Kim, Fedor Kostritsa, David Landhuis, Pavel Laptev, Erik Lucero, Orion Martin, Jarrod R. McClean, Trevor McCourt, Xiao Mi, Kevin C. Miao, Masoud Mohseni, Wojciech Mruczkiewicz, Josh Mutus, Ofer Naaman, Matthew Neeley, Charles Neill, Michael Newman, Murphy Yuezhen Niu, Thomas E. O'Brien, Alex Opremcak, Eric Ostby, Bálint Pató, Nicholas Redd, Pedram Roushan, Nicholas C. Rubin, Vladimir Shvarts, Doug Strain, Marco Szalay, Matthew D. Trevithick, Benjamin Villalonga, Theodore White, Z. Jamie Yao, Ping Yeh, Adam Zalcman, Hartmut Neven, Sergio Boixo, Vadim Smelyanskiy, Yu Chen, Anthony Megrant, Julian Kelly

arXiv:2102.06132v1quant-ph

TL;DR

Quantum error correction must reduce physical error rates while preserving performance over repeated cycles and handling nonlocal error mechanisms. This paper implements repetition and surface-code experiments on Sycamore, finding stable, exponentially suppressed logical errors and agreement with depolarizing-noise simulations.

  • Problem

    QEC needs sufficiently low physical error rates, local errors, and stable performance across many correction rounds, but crosstalk and cyclic-operation effects complicate experimental validation.

  • Method

    The paper implements repetition and surface codes on Sycamore and compares experimental logical errors with circuit simulations using component-informed Pauli-error channels.

  • Results

    Logical errors are exponentially suppressed from 5 to 21 qubits, while detection-event correlations remain localized to the 3 × 10^-3 level and results agree with depolarizing-model simulations.

  • Takeaways & Limitations

    A 21-qubit superconducting processor can sustain repetitive stabilizer measurement cycles with exponentially suppressed repetition-code logical errors.

Abstract

from arXiv · show

Realizing the potential of quantum computing will require achieving sufficiently low logical error rates. Many applications call for error rates in the $10^{-15}$ regime, but state-of-the-art quantum platforms typically have physical error rates near $10^{-3}$. Quantum error correction (QEC) promises to bridge this divide by distributing quantum logical information across many physical qubits so that errors can be detected and corrected. Logical errors are then exponentially suppressed as the number of physical qubits grows, provided that the physical error rates are below a certain threshold. QEC also requires that the errors are local and that performance is maintained over many rounds of error correction, two major outstanding experimental challenges. Here, we implement 1D repetition codes embedded in a 2D grid of superconducting qubits which demonstrate exponential suppression of bit or phase-flip errors, reducing logical error per round by more than $100\times$ when increasing the number of qubits from 5 to 21. Crucially, this error suppression is stable over 50 rounds of error correction. We also introduce a method for analyzing error correlations with high precision, and characterize the locality of errors in a device performing QEC for the first time. Finally, we perform error detection using a small 2D surface code logical qubit on the same device, and show that the results from both 1D and 2D codes agree with numerical simulations using a simple depolarizing error model. These findings demonstrate that superconducting qubits are on a viable path towards fault tolerant quantum computing.

I. INTRODUCTION

Stabilizer QEC encodes information across data qubits and uses repeated parity measurements to detect and decode errors. Exponential logical-error suppression requires sufficiently low physical error rates, while crosstalk and cyclic-operation effects remain challenges.

  • QEC background: Stabilizer codes encode one quantum-information bit across many data qubits and use measure qubits to track parity changes.A decoder uses the history of parity measurements to determine the most likely correction.
  • QEC background: Below a decoder-dependent threshold, logical error per correction round scales exponentially with code distance.The suppression factor is Λ, while d increases with the number of physical qubits.
  • Experimental challenge: Exponential suppression is not guaranteed because typical QEC error models may omit crosstalk.Cyclic stabilizer measurements also introduce leakage, heating, and data-qubit decoherence mechanisms.
  • Code architectures: Repetition codes protect against either bit or phase flips, whereas surface codes use alternating X- and Z-type checks to protect against both.The repetition code uses a one-dimensional chain; the surface code uses a two-dimensional checkerboard layout.

II. QEC WITH THE SYCAMORE PROCESSOR

The experiment implements repetition and surface-code circuits on Sycamore’s two-dimensional transmon array. Repeated stabilizer measurements produce detection events whose fraction remains stable across 50 rounds in the 21-qubit phase-flip experiment.

  • Processor and layouts: Sycamore is a two-dimensional transmon array with nearest-neighbor connectivity suitable for surface-code layouts, although at most 21 of its 54 qubits were used.The experiment includes a distance-11 repetition code and a distance-2 surface code.
  • Component performance: 0.62% median Pauli error was measured for CZ gates, with 0.50% average error under simultaneous cross-entropy benchmarking.The device also uses a reset protocol that reaches the ground state within 280 ns with typical error below 0.5%.
  • Repetition-code circuit: The phase-flip circuit initializes data qubits in X-basis states, repeatedly measures XX stabilizers, and finishes with X-basis data measurements.Measure qubits are reset during the repeated circuit.
  • Detection events: Detection events are obtained by comparing each measure qubit’s stabilizer outcomes between adjacent rounds.Each spacetime location of a possible event is called a detection node.
  • Experimental stability: 11% of measurements signaled detection events overall in the 50-round, 21-qubit phase-flip experiment.The first and last rounds differ because they use initialization and final-measurement values, while intermediate rounds show stable detection fractions.

III. CORRELATIONS IN ERROR DETECTION EVENTS

Detection-event correlations largely follow the expected local spacetime structure of repetition-code errors. High-precision analysis additionally reveals short-range crosstalk, leakage-related long-lived correlations, and rare device-wide transients.

  • Expected correlations: Errors normally produce paired detection events classified as spacelike, spacetimelike, or timelike according to their spatial and temporal separation.These expected relationships form the detection-event graph used to interpret correlations.
  • Correlation analysis: The measured correlation matrix shows that visible correlations are predominantly spacelike or timelike, indicating mostly expected device behavior.The matrix compares arbitrary pairs of detection nodes using pairwise correlation probabilities.
  • Unexpected correlations: Crosstalk links non-adjacent measure qubits in the one-dimensional code that are spatially close in the embedded two-dimensional array.The observed crosstalk remains short range and local in the physical layout.
  • Unexpected correlations: Leakage produces excess correlations across measurement rounds separated by more than one round.The paper attributes these long-lived correlations to leakage on data qubits, which can arise from gates, measurement, or heating.
  • Rare correlated events: 0.5% of the data contained sharp error spikes followed by exponential decay, attributed to high-energy particles striking the processor.These events were removed when computing typical logical-error probabilities.

IV. LOGICAL ERRORS IN THE REPETITION CODE

Logical error probabilities are decoded from detection events and evaluated across code sizes and correction durations. Increasing the repetition-code size produces exponential suppression for both phase- and bit-flip errors.

  • Decoding procedure: A minimum-weight perfect-matching decoder identifies likely errors and corrects the final data-qubit state in post-processing.A logical error occurs when the corrected final state differs from the initial state.
  • Measurement strategy: The experiment varies detection rounds from 1 to 50 at 21 qubits and estimates smaller code sizes from spatial subsets of the same data.Logical error per round is extracted by fitting exponential decay in logical fidelity versus the number of rounds.
  • Exponential suppression: More than 100× suppression occurred in logical error per round, decreasing from 8.7 × 10^-3 at 5 qubits to 6.7 × 10^-5 at 21 qubits.The result is reported for the repetition-code experiments spanning phase- and bit-flip protection.
  • Exponential suppression: ΛX = 3.18 ± 0.08 for the phase-flip code and ΛZ = 2.99 ± 0.09 for the bit-flip code.These suppression factors are obtained by fitting logical error per round versus code distance.

V. ERROR BUDGETING AND PROJECTING QEC PERFORMANCE

The authors use a depolarizing-error model to budget repetition-code errors and project surface-code performance. Distance-2 surface-code measurements agree well with simulations, while larger-code suppression requires improved operation fidelity.

  • Modeling approach: A depolarizing noise model injects random X, Y, or Z errors after each operation, using measured operation-specific error rates.The model is compared with experimental data through direct and Monte Carlo simulations.
  • Error budgets: The repetition-code error budgets slightly underestimate experimental errors, with the discrepancy attributed to stray error.
  • Surface-code projection: Improving overall performance is necessary to observe error suppression in a distance-5 surface code compared with distance 3.
  • Surface-code experiment: The distance-2 surface code uses seven qubits to implement one weight-4 X stabilizer and two weight-2 Z stabilizers.Because correction assignments are ambiguous after detection, runs with detection events are discarded.
  • Surface-code experiment: Approximately 2 × 10^-3 logical error probability occurs per round after post-selection, while the model slightly underestimates the experimental error.The model’s agreement with the experiment supports surface-code projections up to small crosstalk and leakage corrections.

VI. CONCLUSION AND OUTLOOK

The study demonstrates stable repetitive stabilizer measurements and exponentially suppressed repetition-code logical errors in a 21-qubit superconducting system. It also identifies remaining improvements needed for scalable quantum error correction.

  • Conclusion: A 21-qubit superconducting system remains stable during many repetitive stabilizer measurement cycles.
  • Conclusion: Repetition-code logical errors are exponentially suppressed as the number of qubits increases from 5 to 21, even after 50 rounds.
  • Conclusion: Detected physical errors are localized in space and time to the 3 × 10^-3 level.
  • Outlook: Reducing CZ-gate errors and data-qubit errors during measurement and reset is identified as necessary for reaching the surface-code threshold.
  • Outlook: Practical quantum computation requires Λ ∼10 for reasonable physical-qubit overhead, alongside further mitigation of mechanisms such as high-energy particles.

VII. AUTHOR CONTRIBUTIONS

The paper reports distinct contributions spanning experiment design, device operations, simulation, decoding, calibration, and manuscript preparation. All authors contributed to the experimental and theoretical infrastructure.

  • Experiment: Z. Chen, K. Satzinger, H. Putterman, A. Fowler, A. Korotkov, and J. Kelly designed the experiment.
  • Experiment: Z. Chen, K. Satzinger, and J. Kelly performed the experiment and analyzed the data.
  • Modeling and software: C. Jones developed the 1/Λ model, while A. Fowler and C. Gidney wrote the decoder and interface software.
  • Manuscript: Z. Chen, K. Satzinger, and J. Kelly wrote the manuscript, and all authors contributed to revising it and writing the supplementary information.
  • Infrastructure: All authors contributed to the experimental and theoretical infrastructure that enabled the experiment.

VIII. DATA AVAILABILITY

The paper’s plots and other study findings are available from the corresponding authors upon reasonable request.

  • Data availability: The data supporting the plots within the paper are available from the corresponding authors upon reasonable request.
  • Data availability: Other findings of the study are also available from the corresponding authors upon reasonable request.
  • Data availability: The data-availability statement directs requests to the paper’s corresponding authors.

I. DATA FOR BIT FLIP CODE

The bit-flip repetition-code implementation reverses the phase-flip setup in basis, stabilizers, decoupling, and data-qubit pulses, while supplementary analyses examine correlations, logical errors, surface-code structure, and simulations.

  • Bit-flip implementation: Bit-flip experiments use Z-basis initialization and measurement with Z-type stabilizers.Data qubits omit the phase-flip code’s Hadamards, and parity is measured in the Z basis.
  • Bit-flip implementation: π pulses before each round prevent data qubits from collapsing into the ground state and artificially lowering logical error probabilities.
  • Bit-flip results: The supplementary bit-flip data include detection fractions, two-point correlations, and logical error probabilities through 50 rounds.The main-text logical-error analysis excludes device-wide correlated events attributed to high-energy particles; fitted ΛX and ΛZ remain unchanged when those data are retained, within uncertainty.
  • Surface-code structure: The distance-2 surface-code logical qubit uses four data qubits and three stabilizers: Z0Z1, X0X1X2X3, and Z2Z3.Its logical operators are XL = X0X1 and ZL = Z0Z2; the supplementary material also depicts a distance-3 analogue with 9 data qubits and 8 stabilizers.
  • Surface-code results: Surface-code post-selection removes about 27% of runs each round, while the remaining logical error probability is about 0.002 per round.Measured values are 0.0016 ± 0.0001 in the X basis and 0.0027 ± 0.0001 in the Z basis.
  • Simulation model: Circuit simulations use component-informed noise parameters and simplified Pauli or depolarizing channels to compare simulated and experimental logical-error performance.The simulator uses six parameters collectively denoted x, including component-specific error probabilities and Λ, the improvement ratio when distance increases by 2.

B. Comparing Component-Error Simulations to the Experiments

The simulations use component-error benchmarks to model repetition-code experiments, compare logical-error scaling with code distance, and construct an error budget for Λ. The simplified model reproduces experimental trends imperfectly because it omits several physical effects.

  • Component-error model: The model includes separate benchmarked error channels for single- and two-qubit gates, idle operations, reset, and measurement.Idle noise is modeled with memoryless depolarizing channels using T1 decay for bit-flip experiments and T2 decay for phase-flip experiments.
  • Model scope: The error model is limited to independent, identically distributed Markovian Pauli channels and excludes leakage, crosstalk, cosmic rays, parameter drift, and other non-Markovian noise.Its restricted form was chosen because it scales to large problem sizes and supports surface-code forecasts.
  • Simulation conditions: 160,000 trials were run for each syndrome-round count from 1 to 50 while simulating bit-flip and phase-flip repetition codes.The simulations used component-error probabilities from the main-text experiments and treated a differing final logical measurement as an error.
  • Scaling analysis: Logical-error rates are fitted for code distances d ∈ {3, 5, 7, 9, 11}, and Λ is calculated from the improvement produced by increasing distance by 2.The fitted ansatz gives zero error at zero rounds, saturates at 0.5 for infinitely many rounds, and identifies the one-round error with ϵlogical.
  • Comparison with experiment: Simulated logical-error rates match experiments well but not perfectly; simulations give lower errors and higher Λ values than experiments.The discrepancy is attributed to assumptions that may fail experimentally, including possible crosstalk and long-time correlations.
  • Error budget: More than 50% of the logical-error budget comes from idling during measurement and reset, with CZ gates and combined reset-measurement errors accounting for most of the remainder.The reported idle contribution is associated with T1 times around 15 µs and 880 ns idle intervals, producing 4–5% error per such operation.

A. Error graph and correlation matrix pij

The error graph represents detection events across space and time, while the correlation matrix pij estimates pairwise physical-error probabilities from detection-event statistics. The method supports decoder weights and reveals small but nonzero edge correlations.

  • Graph construction: An error graph for Nmq = 4 measure qubits and Nr = 8 rounds contains spacelike, timelike, and spacetimelike edges connecting detection nodes.Nodes are indexed by measure-qubit position and round, with boundaries represented by boundary spacelike edges.
  • Detection events: Detection events are defined by XOR differences between consecutive stabilizer measurements, with special initialization and final-readout rules at the time boundaries.The final node column represents the end-of-circuit comparison rather than an additional physical syndrome round.
  • Decoding: The minimum-weight perfect-matching decoder connects detection events pairwise or to a space boundary to infer the initial logical state.The decoder assumes conventional Pauli errors generate paired detection events corresponding to error-graph edges.
  • Correlation estimation: Pairwise probabilities pij are inferred from joint and marginal detection-event frequencies under an initial assumption that different edges are uncorrelated.The method treats arbitrary node pairs as possible edges, not only conventional graph neighbors.
  • Correlation estimation: Equation-based corrections make pij more accurate than the simple covariance approximation, whose denominator is about 0.6 when both detection-event fractions are approximately 0.11.The approximation slightly overestimates the exact expression, with correction factor roughly (1 − 3pij).
  • Boundary edges: Individual S, T, and ST edge probabilities support minimum-weight decoding, while boundary-edge probabilities are recovered from the node’s remaining detection-event fraction.Boundary estimation combines connected edge probabilities through the color-flip operation rather than a simple arithmetic sum.
  • Correlation validation: The uncorrelated-edge assumption leaves median relative inaccuracies of about 4% for phase-flip data and 9% for bit-flip data.The authors interpret these residuals as evidence of small but nonzero correlations, potentially involving processes that flip four or more nodes.
  • Statistical precision: Additional averaging over rounds reduces the correlation-matrix noise floor below 2 × 10^-4.The estimate concerns statistical fluctuations from finite experimental samples and is reported for the main-text matrix analysis.

C. Experimental results for pij

The correlation matrix is dominated by expected S, T, and ST error-graph edges, while leakage and crosstalk produce smaller but diagnostically important unconventional correlations. These measurements identify temporal persistence, leakage signatures, and chip-local crosstalk relevant to decoding.

  • Expected error-graph edges: S, T, and ST edges are the main correlation features, with typical S and T values around 0.03 and ST values around 0.004.S and T lines correspond to spacelike and timelike edges, while ST edges are diagonal spacetime correlations.
  • Unconventional edges: Temporal correlations can survive for over 5 rounds, including unconventional 2T, 3T, and related edges.Among unconventional edges, 2T edges have median probability 1.7 × 10^-3, more than twice smaller than typical ST-edge probability.
  • Leakage correlations: Leakage to data-qubit state |2⟩ explains multi-round detection-event correlations and can significantly impair minimum-weight-matching decoding.For T1 ≃ 15 µs and a 960 µs round duration, |2⟩ is expected to survive about 8 rounds; the largest averaged 2T-edge value is 3.6 × 10^-3.
  • Crosstalk correlations: Crosstalk creates correlations between measure qubits that are distant in the repetition-code chain but physically close on the chip.Time-shifting crosstalk pairs by one round yields crudely twice smaller edge probabilities.
  • Crosstalk correlations: Crosstalk is small and local in physical chip distance, although long-range code correlations can effectively reduce the code distance.The authors therefore expect crosstalk not to present a serious problem for future surface-code operation in this device.
  • Decoding and weighting: Minimum-weight perfect matching uses correlated detection-node probabilities to weight error-graph edges, while sophisticated weighting preserves Λ better than uniform weighting.Uniform weighting reduces Λx to 2.7 and Λz to 2.5, whereas methods 2, 3, and 4 agree within fitting uncertainty.

X. OVERVIEW OF ERROR CORRECTION EXPERIMENTS

Table S6 surveys experimental error-correction and error-detection implementations, distinguishing classical codes from non-classical quantum codes and clarifying the scope of listed experiments.

  • Notation and scope: Classical codes that detect only phase flips or bit flips use [n, k, d] notation rather than quantum [[n, k, d]] notation.The notation distinguishes one-error-type experiments from non-classical quantum-code experiments.
  • Notation and scope: Entries marked N/A concern embedding error correction into physical qubits rather than layering it on top of the physical qubits.This separates embedded implementations from layered QEC experiments in the comparison table.
  • Experimental coverage: No experiment listed in the table explores both a range of rounds and a range of code distances using a non-classical code.The caption states this as the remaining coverage boundary of the surveyed experiments.
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