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

TL;DR

This work demonstrates exponential suppression of logical errors using 1D repetition codes embedded in a 2D superconducting-qubit array, achieving more than two orders of magnitude improvement in logical error per round when increasing qubits from 5 to 21 and maintaining stability over 50 rounds. By combining detailed error-detection-event analyses, a $p_{ij}$ correlation framework, and depolarizing-noise simulations, the authors show that errors remain predominantly local in space and time, with manageable leakage and crosstalk. They also implement a small $d=2$ surface code on the same device and find agreement with the repetition-code results and simple depolarizing models, underscoring progress toward fault-tolerant quantum computing on superconducting qubits. The study identifies CZ-gate errors and data-qubit decoherence during measurement as primary bottlenecks and provides an error-budgeting methodology to project surface-code performance, outlining concrete paths to reach practical quantum error correction thresholds. Overall, the results establish a viable path toward scalable fault tolerance on superconducting platforms and clarify the technical requirements for advancing QEC experiments.

Abstract

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.

Figures (27)

• Figure 1: Stabilizer circuits on Sycamore.a, Layout of distance-11 repetition code and distance-2 surface code in the Sycamore architecture. In the experiment, the two codes use overlapping sets of qubits, which are offset in the figure for clarity. b, Pauli error rates for gates and identification error rates for measurement. All benchmarks are for simultaneous operation. c, Circuit schematic for the phase flip code. Data qubits are randomly initialized into $|{+}\rangle$ or $|{-}\rangle$, followed by repeated application of $XX$ stabilizer measurements and finally $X$-basis measurements of the data qubits. d, Illustration of error detection events which occur when a measurement disagrees with the previous round. e, Fraction of measurements which detected an error versus measurement round for the $d=11$ phase flip code. The dark line is an average of the individual traces (gray lines) for each of the 10 measure qubits. The first (last) round also uses data qubit initialization (measurement) values to identify parity errors and generate detection events.
• Figure 2: Analysis of error detections.a, Detection event graph. Errors in the code trigger two detections (except at the ends of the chain), each represented by a node, and edges represent the expected correlations due to data qubit errors (spacelike and spacetimelike) and measure qubit errors (timelike) b, Ordering of the measure qubits in the repetition code. c, Measured two point correlations ($p_{ij}$) between detection events represented as a symmetric matrix. The axes correspond to possible locations of detection events, with major ticks marking measure qubits (space) and minor ticks marking difference in rounds (time). For the purposes of illustration, we have averaged together the matrices for 4-round segments of the 50-round experiment shown in Fig. 1e, and also set $p_{ij} = 0$ if $i = j$. The upper triangle shows the full scale, where only the expected spacelike and timelike correlations are apparent. The lower triangle shows a truncated color scale, highlighting unexpected correlations due to crosstalk and leakage. Note that crosstalk errors are still local in the 2D array. d, (Top) Observed high energy event in a time series of repetition code runs. (Bottom) Zoom in on high energy event, showing rapid rise and exponential decay of device wide correlated errors, and data which is removed when computing logical error probabilities.
• Figure 3: Logical errors in the repetition code.a, Logical error probability versus number of detection rounds and number of qubits for the phase flip code. Smaller code sizes are subsampled from the 21 qubit code as shown in the inset; small dots are data from subsamples and large dots are averages. b, Semilog plot of the averages from a showing even spacing in $\log\!{(\text{error probability})}$ between the code sizes. Error bars are estimated standard error from binomial sampling. The lines are exponential fits to data for rounds greater than 10. c, Logical error per round ($\epsilon_L$) vs. number of qubits, showing exponential suppression of error rate for both bit and phase flip, with extracted $\Lambda$ factors. The fit excludes $n_\text{qubits}=3$ to reduce the influence of spatial boundary effects supplement.
• Figure 4: Error budgeting repetition and surface codes.a, Probability of depolarizing errors (bit flip errors for M and R) for various operations in the stabilizer circuit, derived from averaging quantities in Fig. 1b. Note the idle gate (I) and dynamical decoupling (DD) values depend on the code being run because the data qubits occupy different states. b, Estimated error budgets for the bit flip and phase flip codes, and projected error budget for the surface code, based on the depolarizing errors from a. The repetition code budgets slightly underestimate the experimental errors, and the discrepancy is labeled stray error. For the surface code, the estimated $1/\Lambda$ corresponds to difference in $\epsilon_L$ between a $d=3$ and $d=5$ surface code. c, For the $d=2$ surface code, fraction of runs that had no detection events versus number of rounds, plotted with the prediction from a similar error model as the repetition code (dashed line). Inset: physical qubit layout of the $d=2$ surface code, 7 qubits embedded in a 2D array. d, Surface code logical error probability among runs with no detection events versus number of rounds. Simulations from the same model as c (dashed lines) show good agreement. Error bars for c (not visible) and d are estimated standard error from binomial sampling with 240,000 experimental shots, minus the shots removed by post-selection in d.
• Figure S1: a, Detection event fraction for a 50 round bit flip code, similar to Fig. 1d of the main text. b,$p_{ij}$ correlation matrix for the 50 round bit flip code, similar to Fig. 2c of the main text