Quantum error correction below the surface code threshold

TL;DR

This work demonstrates quantum error correction below the surface-code threshold using superconducting qubits, implementing both a distance-5 memory with real-time decoding and a distance-7 memory with advanced leakage control. It shows exponential suppression of logical errors with code distance, achieving a Lambda greater than 2 and a logical error per cycle around 1.4×10^-3 for distance-7, with a logical lifetime exceeding the best physical qubits by a factor of about 2.4. By pushing repetition codes to high distances, the study reveals ultra-low error behavior and identifies rare correlated burst events that set a practical floor near 1e-10, highlighting remaining challenges for scaling. Importantly, the work demonstrates real-time decoding with latency on the order of tens of microseconds, signaling a viable path toward large-scale fault-tolerant quantum computation if these gains can be scaled and stabilized.

Abstract

Quantum error correction provides a path to reach practical quantum computing by combining multiple physical qubits into a logical qubit, where the logical error rate is suppressed exponentially as more qubits are added. However, this exponential suppression only occurs if the physical error rate is below a critical threshold. In this work, we present two surface code memories operating below this threshold: a distance-7 code and a distance-5 code integrated with a real-time decoder. The logical error rate of our larger quantum memory is suppressed by a factor of $Λ$ = 2.14 $\pm$ 0.02 when increasing the code distance by two, culminating in a 101-qubit distance-7 code with 0.143% $\pm$ 0.003% error per cycle of error correction. This logical memory is also beyond break-even, exceeding its best physical qubit's lifetime by a factor of 2.4 $\pm$ 0.3. We maintain below-threshold performance when decoding in real time, achieving an average decoder latency of 63 $μ$s at distance-5 up to a million cycles, with a cycle time of 1.1 $μ$s. To probe the limits of our error-correction performance, we run repetition codes up to distance-29 and find that logical performance is limited by rare correlated error events occurring approximately once every hour, or 3 $\times$ 10$^9$ cycles. Our results present device performance that, if scaled, could realize the operational requirements of large scale fault-tolerant quantum algorithms.

Figures (4)

• Figure 1: Surface code performance.a, Schematic of a distance-7 surface code on a 105-qubit processor. Each measure qubit (blue) is associated with a stabilizer (blue colored tile). Red outline: one of nine distance-3 codes measured for comparison ($3\times 3$ array). Orange outline: one of four distance-5 codes measured for comparison (4 corners). Black outline: distance-7 code. We remove leakage from each data qubit (gold) via a neighboring qubit below it, using additional leakage removal qubits at the boundary (green). b, Cumulative distributions of error probabilities measured on the 105-qubit processor. Red: Pauli errors for single-qubit gates. Black: Pauli errors for CZ gates. Blue: Average identification error for measurement. Gold: Pauli errors for data qubit idle during measurement and reset. Teal: weight-4 detection probabilities (distance-7, averaged over 250 cycles). c, Logical error probability, $p_L$, for a range of memory experiment durations. Each datapoint represents $10^5$ repetitions decoded with the neural network and is averaged over logical basis ($X_L$ and $Z_L$). Black and grey: data from Ref. google2023suppressing for comparison. Curves: exponential fits after averaging $p_L$ over code and basis. To compute $\varepsilon_d$ values, we fit each individual code and basis separately supplement. d, Logical error per cycle, $\varepsilon_d$, reducing with surface code distance, $d$. Uncertainty on each point is less than $5\times10^{-5}$. Symbols match panel c. Means for $d=3$ and $d=5$ are computed from the separate $\varepsilon_d$ fits for each code and basis. Line: fit to Eq. 1, determining $\Lambda$. Inset: simulations up to $d=11$ alongside experimental points, both decoded with ensembled matching synthesis for comparison. Line: fit to simulation, $\Lambda_\textrm{sim}=2.25\pm 0.02$.
• Figure 2: Error sensitivity in the surface code.a, One cycle of the surface code circuit, focusing on one data qubit and one measure qubit. Black bar: CZ, H: Hadamard, M: measure, R: reset, DD: dynamical decoupling. Orange: Injected coherent errors. Purple: Data qubit leakage removal (DQLR) miao2023overcoming. b, Error injection in the surface code. Distance-3 averages over 9 subset codes, and distance-5 averages over 4 subset codes, as in Fig. \ref{['surface_code']}. Logical performance is plotted against the mean weight-4 detection probability averaging over all codes, where increasing the error injection angle $\alpha$ increases detection probability. Each experiment is 10 cycles with $2\times 10^4$ total repetitions. Lines: power law fits for data points at or below where the codes cross. Inset: Inverse error suppression factor, $1/\Lambda$, versus detection probability. Line: fit to points with $1/\Lambda < 1$. c, Estimated error budget for the surface code based on component errors and simulations. CZ: CZ error, excluding leakage and stray interactions. CZ stray int.: CZ error from unwanted interactions. Data idle: Data qubit idle error during measurement and reset. Meas.: Measurement and reset error. Leakage: Leakage during CZs and due to heating. 1Q: Single-qubit gate error. d, Comparison of logical performance with and without data qubit leakage removal each cycle. Distance-3 points (red triangles) are averaged over four quadrants. Each experiment is $10^5$ repetitions. Curves: exponential fits. e, Repeating experiments to assess performance stability, comparing distance-3 and distance-5. Each point represents a sweep of logical performance versus experiment duration, up to 250 cycles.
• Figure 3: High-distance error scaling in repetition codes.a, Logical error per cycle, $\varepsilon_d$, versus code distance, $d$, when decoding with minimum-weight perfect matching. Repetition code points are from $d=29$, $10^3$-cycle experiments, $10^7$ repetitions for each basis $X$ and $Z$. We subsample smaller codes from the same $d=29$ dataset, averaging over subsamples. Line: fit of error suppression factor $\Lambda$. We include data from Rep. google2023suppressing for comparison. b, Logical error scaling with injected error. We inject a range of coherent errors on all qubits and plot against observed mean detection probability $p_{\text{det}}$. Each experiment is 10 cycles, and we average over $10^6$ repetitions. Smaller code distances are again subsampled from $d=29$. Lines: power law fits $\varepsilon_d = A_d p_\textrm{det}^{(d+1)/2}$ (one fit parameter, $A_d$), restricted to $\varepsilon_d > 10^{-7}$ and $p_\textrm{det} < 0.3$. c,$1/\Lambda$ scaling with injected error. Typical relative fit uncertainty is 2%. Line: fit. d, Example event causing elevated detection probabilities which decay exponentially with time constant $369 \pm 6$ µ s (gray dashed line). Three consecutive experimental shots are plotted, delimited by vertical gray lines. The 28 measure qubits are divided into four quartiles based on average detection probability in the gray-shaded window. Each trace represents the detection probability averaged over one quartile and a time window of 10 cycles. Inset: Average detection probability for each measure qubit (colored circle) within the gray-shaded window.
• Figure 4: Real-time decoding.a, Schematic of the streaming decoding algorithm. Decoding problems are subdivided into blocks, with different threads responsible for different blocks. b, Task graph for processing blocks. Detections are allowed to match to block boundaries, which will then be processed downstream during a fuse step. If a configuration of detection events cannot be resolved by a future fuse step, the decoder heralds failure. We use 10-cycle blocks to ensure that the heralded failure rate is negligible compared to the logical failure rate. c, Accuracy comparison for the surface code with three decoders. We include the real-time decoder (RT), ensembled matching synthesis (Ens.), and the neural network decoder (NN). Uncertainty on each point is less than $10^{-4}$supplement. d, Decoder latency versus experiment duration. The blue points correspond to end-of-shot latencies (10 shots per duration, horizontal bar: median, blue shading: violin plot). The yellow histograms represent sub-shot latencies obtained by checking how long after each 10-cycle block's data is received that the block is completed by the decoder. The sub-shot latencies tend to be slightly longer than end-of-shot latencies as the decoder may need to wait to fuse with detection events in future cycles in order to process up to the current cycle.