Exponential suppression of bit or phase errors with cyclic error correction

Abstract

Realizing the potential of quantum computing requires sufficiently low logical error rates¹. Many applications call for error rates as low as 10^-15 (refs. ^2-9), but state-of-the-art quantum platforms typically have physical error rates near 10^-3 (refs. ^10-14). Quantum error correction^15-17 promises to bridge this divide by distributing quantum logical information across many physical qubits in such a way that errors can be detected and corrected. Errors on the encoded logical qubit state can be exponentially suppressed as the number of physical qubits grows, provided that the physical error rates are below a certain threshold and stable over the course of a computation. Here we implement one-dimensional repetition codes embedded in a two-dimensional grid of superconducting qubits that demonstrate exponential suppression of bit-flip or phase-flip errors, reducing logical error per round more than 100-fold 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 analysing error correlations with high precision, allowing us to characterize error locality while performing quantum error correction. Finally, we perform error detection with a small logical qubit using the 2D surface code on the same device^18,19 and show that the results from both one- and two-dimensional codes agree with numerical simulations that use a simple depolarizing error model. These experimental demonstrations provide a foundation for building a scalable fault-tolerant quantum computer with superconducting qubits.

Fig. 1. Stabilizer circuits on Sycamore.

a, Layout of distance-11 repetition code and distance-2 surface code in the Sycamore processor. In the experiment, the two codes use overlapping sets of qubits, which are offset in the figure for clarity. b, Pauli error rates for single-qubit and CZ gates and identification error rates for measurement, each benchmarked in simultaneous operation. c, Circuit schematic for the phase-flip code. Data qubits are randomly initialized into $|+⟩$ or $|-⟩$, followed by repeated application of XX stabilizer measurements and finally X-basis measurements of the data qubits. Hadamard refers to the Hadamard gate, a quantum operation. d, Illustration of error detection events that occur when a measurement disagrees with the previous round. e, Fraction of measurements (out of 80,000) that detected an error versus measurement round for the d = 11 phase-flip code. The dark line is an average of the individual traces (grey lines) for each of the 10 measure qubits. The first (last) round also uses data qubit initialization (measurement) values to identify detection events.

Fig. 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. 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 (Δround) in rounds (time). For the purpose 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. A description of the uncertainties on the matrix values can be found in Supplementary Information section IX. The upper triangle shows the full scale, where only the expected spacelike and timelike correlations are apparent. The lower triangle shows a truncated colour scale, highlighting unexpected detection pairs due to crosstalk and leakage. Note that observed crosstalk errors occur between next-nearest neighbours 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 errors, and data that are removed when computing logical error probabilities.

Fig. 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(error probability) between the code sizes. Error bars are estimated standard error from binomial sampling given the total number of statistics over all subsamples. The lines are exponential fits to data for rounds greater than 10. c, Logical error per round (ε_L) versus number of qubits, showing exponential suppression of error rate for both bit-flip and phase-flip, with extracted Λ factors. The fits for Λ and uncertainties were obtained using a linear regression on the log of the logical error per round versus the code distance. The fit excludes n_qubits = 5 to reduce the influence of spatial boundary effects (Supplementary Information section VII).

Fig. 4. Error budgeting repetition and surface codes.

a, Depolarizing error probability (bit flip errors for M and R) for various operations in the stabilizer circuit, derived from averaging quantities in Fig. 1b. Note that the idle gate (I) and dynamical decoupling (DD) values depend on the code being run because the data qubits occupy different states. Op., operation; Rep., repetition. 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 labelled stray error. For the surface code, the estimated 1/Λ corresponds to the difference in ε_L between a d = 3 and d = 5 surface code, and is ~4 times higher than in the repetition codes owing to the more stringent threshold for the surface code. Rep., repetition c, For the d = 2 surface code, the 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, seven qubits embedded in a 2D array. d, Probability of logical error in the surface code among runs with no detection events versus number of rounds. Depolarizing model simulations that do not include leakage or crosstalk (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.