Quantum error correction below the surface code threshold

I. INTRODUCTION

• Below the critical physical-noise threshold, surface-code logical error is expected to decrease exponentially with code distance; no processor had definitively shown below-threshold performance.Source: §I · p. 1
• Fault-tolerant computing also requires stability over hours-long timescales and active removal of correlated errors such as leakage, beyond low physical error rates.Source: §I · p. 1
• Λ > 2 was demonstrated through distance-5 and distance-7 on two superconducting processors, with an integrated real-time decoder for distance-5.The distance-5 code used a 72-qubit processor, while the distance-7 code used a 105-qubit processor.Source: §I · p. 1
• Distance-7 preserved quantum information for more than twice as long as its best constituent physical qubit, and distance-5 memories were also beyond break-even.Source: §I · p. 1
• Once-an-hour correlated error events dominated high-distance repetition-code error rates and set a current error floor of 10^-10.The origins of these errors were not yet understood.Source: §I · p. 1
• Below-threshold operation was maintained with real-time decoding on the 72-qubit processor at its 1.1 µs cycle duration.The introduction identifies decoding syndrome information as fast as it is generated as a fault-tolerance requirement.Source: §I · p. 1

II. A SURFACE CODE MEMORY BELOW THRESHOLD

• Λ = 2.14 ± 0.02 with neural-network decoding and ε7 = (1.43 ± 0.03) × 10−3; matching synthesis yields Λ = 2.04±0.02 and ε7 = (1.71 ± 0.03) × 10−3.Logical error per cycle was fit across distances 3, 5, and 7, averaging over X and Z bases and the distance-3 and distance-5 subgrids.Source: Figure 1 · p. 2, §II · pp. 2–3
• Two offline decoders are used: a data-fine-tuned neural network and a harmonized ensemble of correlated matching decoders with matching synthesis and reinforcement-learning-optimized graph weights.They run on different classical hardware and offer potential paths toward real-time decoding.Source: §II · p. 2
• The distance-7 memory uses 49 data qubits, 48 measure qubits, and 4 leakage-removal qubits; syndrome extraction is repeated, with data-qubit leakage removal after each cycle.The logical state is initialized in either the XL or ZL basis, then measured through individual data-qubit measurements and decoder-corrected outcomes.Source: §II · p. 2
• Bulk detection probabilities are pdet = (7.7%, 8.5%, 8.7%) for d = (3, 5, 7), respectively; the increase is attributed to finite-size effects and parasitic qubit couplings.The authors expect both effects to saturate at larger processor sizes.Source: §II · p. 2
• Simulations using measured component error rates, leakage, and stray interactions agree reasonably with experiment and show decisive error suppression, affirming below-threshold operation.The simulations extend to higher code distances and are decoded with ensembled matching synthesis.Source: Figure 1 · p. 2, §II · p. 3
• The distance-7 logical lifetime is 291 ± 6 µs, 2.4±0.3 times the best constituent physical-qubit lifetime of 119±13 µs.The logical lifetime exceeds the median constituent physical-qubit lifetime of 85±7 µs as well.Source: §II · p. 3

III. LOGICAL ERROR SENSITIVITY

• 20% detection probability is the approximate crossing point for distance-3, -5, and -7 logical-error curves, whose below-crossing power-law fits reach about 80% of the ideal (d+1)/2 exponent.The crossing demonstrates threshold behavior in situ; detection probability is a good proxy for 1/Λ, but excess device correlations may explain the deviation from the ideal exponent.Source: §III · p. 3, Figure 2 · p. 3
• 17% of the estimated 1/Λ error budget comes from correlated errors, while the budget overpredicts Λ by 20%; local and correlated CZ-gate errors are the largest contributors.The estimate uses simulations of a detailed model of the 72-qubit processor; improving coherence and calibration is identified as crucial to further reducing logical error.Source: §III · pp. 3–4, Figure 2 · p. 3
• 35% increase in Λ results from data-qubit leakage removal for distance-5, despite only a 12% decrease in detection probability; distance-3 performance is not strongly affected.This contrast shows that detection probability is a good proxy for logical error suppression only when errors are uncorrelated.Source: Figure 2 · p. 3, §III · p. 4
• Average Λ = 2.18 ± 0.07 and best Λ = 2.31 ± 0.02 were observed across repeated runs over 15 hours; distance-5 performance was less affected by a transient TLS fluctuation than the worst distance-3 quadrant.Recalibration did not appreciably improve logical error rates, indicating robustness to the observed qubit-frequency and readout drift.Source: Figure 2 · p. 3, §III · p. 4

IV. PROBING THE ULTRA-LOW ERROR REGIME WITH REPETITION CODES

• On a 72-qubit processor, distance-29 runs used 2 × 10^7 shots split evenly between bit- and phase-flip codes, totaling 2 × 10^10 cycles over 5.5 hours.Source: §IV · p. 4
• At d ≥ 15, error suppression deviated from exponential scaling toward an apparent 10^-10 logical-error floor, although no errors were observed at d = 29.The authors suggest the d = 29 observation may reflect randomly decoding correctly on the few most damaging bursts.Source: §IV · p. 4, Figure 3 · p. 5
• High-distance failures showed either one or two detectors spiking by over a factor of 3 before recovering, or simultaneous larger spikes across many detectors.The less damaging mode may involve transient TLS’s or coupler excitations; the cause of the more catastrophic bursts remains unknown.Source: §IV · pp. 4–5
• Λ = 8.4±0.1 from d = 5–11; logical error per cycle fell far below 10^-6, breaking the floor previously observed in repetition codes [17].The suppression factor is fitted to logical error per cycle versus code distance.Source: §IV · p. 4, Figure 3 · p. 5
• At d = 15, halving detection probability produced a 250-fold reduction in logical error, consistent with O(p^8) suppression.Across the injected-error tests, the observed scaling agreed well with O(p^(d+1)/2).Source: Figure 3 · p. 5, §IV · p. 5
• Six large, spatially localized bursts affecting neighborhoods of roughly 30 qubits occurred in 2 × 10^10 cycles, about once an hour, and decayed with a time constant around 400 µs.These bursts differ from earlier high-energy impact events, which occurred every few seconds and decayed over tens of milliseconds.Source: §IV · pp. 4–5

V. REAL-TIME DECODING

• The decoder must keep syndrome transmission, processing, and decoding pace with the 1.1 µs cycle time to avoid backlogs that can cause exponential growth in computation time.This requirement is especially challenging for fast superconducting processors; real-time decoding is needed because some logical operations depend on interpreting measurement outcomes on the fly.Source: §V · p. 5
• A constant-sized graph buffer supports streaming decoding across experiments of varying cycle counts, using parallelized Sparse Blossom [42] and a strategy similar to Ref. [43].The system classifies measurements into bits, transmits them to a workstation, converts them into detections, and streams them to the decoder; greedy edge reweighting accounts for Y-type error correlations [47].Source: §V · pp. 5–6
• Distance-5 decoder latency averaged 63 ± 17 µs and remained roughly constant as experiments increased to 1.1 seconds, indicating real-time processing.Latency is measured from receipt of the final syndrome-measurement cycle to return of the correction; it does not include feedback into the logical circuit.Source: §V · p. 6
• Real-time decoding achieved ε5 = 0.35% ± 0.01% and Λ = 2.0 ± 0.1, versus offline neural-network decoding at ε5 = 0.269% ± 0.008% and Λ = 2.18 ± 0.09.The real-time decoder uses a device-data-independent prior; the offline neural network processes at 24 µs per cycle, compared with the real-time decoder’s required throughput of less than 1.1 µs per cycle.Source: §V · p. 6

VI. OUTLOOK

• Increasing code distance by two reduces logical error per cycle by more than half, and the distance-7 logical lifetime exceeds its best constituent physical qubit’s lifetime by more than twofold.This exponential suppression is presented as a foundation for large-scale quantum algorithms with error correction.Source: §VI · p. 6
• Performance remained repeatable over several hours and up to 106 cycles without deterioration, while real-time decoding incurred only a modest accuracy reduction versus offline decoders.Source: §VI · p. 6
• A 10^-6 error rate is projected to require a distance-27 logical qubit using 1457 physical qubits, and syndrome measurements per cycle will grow quadratically with code distance.Source: §VI · p. 6
• Repetition-code experiments identify a 10^-10 noise floor caused by correlated error bursts, which must be identified and mitigated for larger quantum algorithms.Source: §VI · pp. 6–7
• Scaling to large algorithms will require addressing logical computation and scaling calibration, real-time decoding, and compiler software for multi-surface-code operations [58] [59].The work demonstrates performance that can scale in principle, but that must now be scaled in practice.Source: §VI · p. 7

VII. AUTHOR CONTRIBUTIONS

• The Google Quantum AI team conceived and designed the experiment, developed analysis and metrology tools, built the system, calibrated it, and collected data.All authors wrote and revised the manuscript and Supplementary Information.Source: §VII · p. 7

I. Gates and Readout

• On the 105-qubit processor, XY rotations last 25 ns and CZ gates 42 ns; on the 72-qubit processor, π and π/2 rotations last 35 ns and 18 ns, and CZ gates 37 ns.Readout pulse duration, qubit frequency, and power are jointly optimized for simultaneous high-fidelity mid-circuit readout.Source: §I · p. 11

A. Frequency Optimization

• The frequency optimizer models the error-correction circuit directly and uses up to 9 dimensions to trade off data- and measurement-qubit coherence.To mitigate time-varying TLS collisions, stability strategies use historical T1 data to penalize frequencies associated with reduced T1 and exclude forecast collision frequencies.Source: A. Frequency Optimization · pp. 11–12

A. List of Decoders

• The decoder set comprises a neural network, Libra matching synthesis with ensemble size 51, Harmony ensembled matching with size 101, correlated matching, and a real-time decoder.Source: A. List of Decoders · p. 12

B. Decoder Priors

• Training decoder priors improved MWPM LER by 10% on average across distances relative to SI1000.For distance-7, distance-3 and distance-5 patches act as local sensors; for distance-29 repetition code, 25 subsampled distance-5 sensors avoid optimizing with the computationally infeasible target-code LER.Source: B. Decoder Priors · p. 12
• 1.14 times lower LER on average was achieved by ensembling than by correlated matching with SI1000 prior on the distance-5 code.The neural network decoder achieved the highest accuracy overall on the first dataset; among the tested matching-based decoders, Harmony was most performant, while the main text uses Libra.Source: B. Decoder Priors · p. 12

C. Neural Network Decoder Details

• The recurrent attention-based neural network predicts logical error from stabilizer measurements while maintaining a state vector for each stabilizer.Separate X- and Z-basis models are pretrained on synthetic SI1000 data and fine-tuned first on a fitted detector error model, then on experimental samples.Source: C. Neural Network Decoder Details · pp. 12–13
• A strict temporal split for the stability experiment achieved similar accuracy to an even/odd split.Gated recurrence was added to stabilize training on longer experiments; networks periodically sampled examples up to 200 cycles to support generalization beyond the usual synthetic training examples of up to 25 cycles.Source: C. Neural Network Decoder Details · pp. 13–14

1. Technical Requirements

• Fault-tolerant feed-forward requires real-time decoding; a system must provide accurate logical outcomes with low latency and enough throughput to prevent syndrome backlog and exponential slowdown.The text identifies gate teleportation as an example of feed-forward and defines accuracy by the per-cycle probability that the decoder's logical measurement outcome is correct.Source: §1 · pp. 14–15
• Tinput < 10 μs; software latency stays roughly constant across circuit depth, recovers quickly from brief spikes, and processes syndrome information as fast as it is generated.Adding Toutput and Tcontrol of a few microseconds would put reaction time within an order of magnitude of that assumed in Ref.; scaling latency remains an engineering challenge.Source: §1 · p. 15, Figure S3 · p. 16, Figure S4 · p. 16
• An optimized, parallel C++ software decoder can run in real time with exclusive access to CPUs, offering flexibility for research and support for broken components and customized constructions.The text contrasts software flexibility with the greater performance promised by hardware decoders.Source: §1 · p. 15

2. Correlated Parallel Blossom

• The decoder preweights correlations by identifying edges with detection events at both endpoints or isolated boundary events, then conditioning on candidate errors to update related edge weights.This reweighting addresses hyperedge mechanisms, including Y errors, that graphlike matching alone ignores.Source: §2 · p. 16
• It parallelizes matching by assigning syndrome-cycle blocks to threads, processing each block with Sparse Blossom, and fusing neighboring blocks in stages; M is 10 for distance-3/5 codes and 90 for distance-29 repetition.After fusion, regions still matched to a block boundary trigger a heralded error; the chosen block sizes make these failures negligible relative to other logical-error contributions.Source: §2 · pp. 16–17
• A separate grapher thread maintains a constant-sized buffer emulating the full matching graph while storing only 128 contiguous blocks at a time.It exploits the experiment's repeating graph pattern, with significant buffer work mainly at the experiment's beginning and end.Source: §2 · pp. 15, 17

1. Surface Code Simulation Details

• Simulated circuits are dressed with characterized noise channels for decoherence, readout/reset, leakage, crosstalk, and excess gate or idle errors before generating experimental-data samples.The model also includes transport of leakage between CZ-gate qubits through higher-excitation transitions.Source: §1 · p. 17
• Imperfect data-qubit leakage removal is modeled phenomenologically to match observed reset fidelity for state |2⟩; generalized Pauli twirling then makes noise channels, including leakage, compatible with Clifford simulation.The resulting noisy circuit is passed to the Pauli+ simulator to generate simulated experimental data.Source: §1 · p. 17

b. Performance forecast for devices with twice better component errors

• With twice-better component errors, forecast Λ≈4.5 and LER=10^-6 at d=13 (337 qubits), versus d=27 (1457 qubits) at demonstrated fidelity.The study identifies CZ errors and data-qubit idle errors as the most important component errors to improve.Source: Figure S5 · p. 18, §2 · p. 18

a. Effect of non-uniform component errors on surface code performance

• With DQLR, Λ3/5=2.16, Λ5/7=2.2, Λ7/9=2.16, Λ9/11=2.12, and Λ11/13=2.01, showing suppression persists across non-uniform errors.Ten bootstrapped error models at each distance from d=3 to 13 showed a small spread; their geometric-mean LERs agreed well with nearly uniform-error simulations.Source: Figure S5 · p. 18, §2 · p. 18

c. Importance of DQLR in surface codes

• Without DQLR, Λd/d+2 saturates near 1 at large distances, whereas DQLR preserves exponential suppression; leakage persisting across cycles creates decoder-confusing temporal correlations.The simulations use component errors that are uniform except for stray-coupling crosstalk.Source: §2 · p. 18

d. The crossover regime

• For Λ3/5 ≤1.3, initial finite-distance suppression diminishes rapidly and may turn around; for Λ3/5 ≥1.5, below-threshold-like suppression continues.The passage notes that threshold behavior is better behaved when error rate is reported per d rounds.Source: Figure S6 · p. 19, §2 · p. 19

3. Surface Code 1/Λ Error Budget

• Λ3/5 is 2.25 from the error-budget prediction and 2.17 from the simulated distance-3-to-distance-5 logical-error-rate ratio.The budget weights represent sensitivities to uniform increases in each error channel from the baseline operation point.Source: §3 · p. 19

B. Comparison to Physical Qubit Lifetime

• Average channel fidelity to identity is axis-neutral and can be experimentally extracted by averaging over the six Pauli eigenstates.The passage describes this as a protocol for extracting the fidelity of an arbitrary channel to identity.Source: B. Comparison to Physical Qubit Lifetime · p. 19
• At short times, fidelity decay defines an effective lifetime of 1/Γ, with Γ dominated by the largest error rate, enabling comparison across different physical-qubit error channels.Different channels decay differently over time, complicating direct comparisons without this short-time approximation.Source: B. Comparison to Physical Qubit Lifetime · pp. 19–20

3. Surface Code 1/Λ Error Budget

• CZ-gate errors contribute about 60% of the 1/Λ budget, versus about 20% from data-qubit idles, 12% from readout and reset, and 9% from single-qubit gates.The sensitivities assume perfect DQLR and a correlated matching decoder.Source: §3 · p. 19, Table S4 · p. 20

B. Comparison to Physical Qubit Lifetime

• Logical Pauli error probabilities per cycle are modeled as continuous rates proportional to 1/tc, and simulations suggest pY is suppressed relative to pX and pZ.The passage notes that Y-eigenstate lifetime is not measured because its preparation is more complicated.Source: B. Comparison to Physical Qubit Lifetime · p. 20
• A gain of 2.4 ± 0.3 over the best measured physical qubit places the multi-qubit logical memory beyond break-even.Gain is defined as improvement in effective lifetime, with G = 1 corresponding to break-even; it does not account for state preparation, measurement, or gate costs.Source: B. Comparison to Physical Qubit Lifetime · p. 20

A. Low Probability Events in the Repetition Code

• Λ > 8 and an apparent logical-error-per-cycle floor of 10^-10 were observed at large distances in the distance-29 repetition code, a four-orders-of-magnitude improvement over previous experiments.The paper notes that even this floor could limit larger fault-tolerant circuits.Source: A. Low Probability Events in the… · p. 20
• Roughly hourly, correlated detection-probability bursts caused all observed distance-27 errors and half of distance-21 to distance-25 errors; fitted decay times were 400–700 µs.These bursts occurred in bit- and phase-flip codes, unlike prior quasiparticle events, which occurred every 10 seconds and recovered in 25–30 ms.Source: A. Low Probability Events in the… · pp. 20–21, Figure S7 · p. 21
• A single noisy detector can cause a high-distance error (d > 19) roughly once per hour, with its firing likelihood elevated for 1–2 ms.The noisy detector varies between events.Source: A. Low Probability Events in the… · p. 21, Figure S7 · p. 21

B. Grid and Circuit Details

• The surface-code experiments place distance-3 and distance-5 grids on both processors and average several minimally overlapping smaller codes to compare with larger codes, extending Ref..Source: B. Grid and Circuit Details · p. 21
• Λ = 2.31 ± 0.02 in a representative 72-qubit processor dataset collected before the time series in Fig. 2e.Source: B. Grid and Circuit Details · p. 21
• For d = (3, 5, 7), simulated detection probabilities were (8.0%, 8.5%, 8.6%), versus experimental (7.7%, 8.5%, 8.7%); the rise with size reflects reduced finite-size effects.The text relates this to corner, edge, and bulk data qubits participating in 2, 3, and 4 CZ gates, respectively.Source: B. Grid and Circuit Details · p. 21
• Experiments without data qubit leakage removal showed increased detections, attributed to leakage accumulation.Source: B. Grid and Circuit Details · p. 21
• Surface-code circuit operations are organized into single- and two-qubit gate layers, dynamical decoupling, and measurements; code, basis, and cycle-count datasets were interleaved with 50,000 repetitions each.Each dataset used 10 initialization bitstrings and 5,000 repetitions; the bitstrings were five random strings and their complements.Source: B. Grid and Circuit Details · p. 21

Logical Performance

• Logical error per cycle εd is fit from pL versus cycle count, with excess residuals used to account for performance drift beyond sampling uncertainty.The fit is linear in ln(1 − 2pL) versus t; estimates are averaged over logical basis and code for manuscript results.Source: A. Logical Performance · p. 22
• pL and εd depend on the decoder; the section reports neural-network decoder values and identifies Libra as an ensembled synthetic matcher.The specific decoder values are referred to Table S1, while per-grid and per-basis neural-network values are referred to Table S5.Source: A. Logical Performance · p. 22
• Λx = 8.27 ± 0.02 and Λz = 8.55 ± 0.02 for repetition codes fitted over distances 5 to 11.The reported overall Λ is the mean of the two basis-specific estimates, with uncertainty estimated as half their absolute difference.Source: A. Logical Performance · p. 23

• The single-point estimate treats each cycle as a binomial logical-error event, with cumulative error probability given by the probability of an odd number of errors.Error-injection experiments ran for 10 cycles to resolve high logical error, while repetition-code experiments ran for 1000 cycles to resolve ε3 ∼ 10^-3 and maximize experiment length.Source: B. Logical Error per Cycle From One… · p. 23
• Uncertainty is more noticeable in low-error experiments, with a statistical floor set by the dataset’s 2 × 10^10 total cycles.This is especially relevant for higher-distance points, where relatively few errors are observed.Source: B. Logical Error per Cycle From One… · p. 23
• One-point logical-error estimates remain relatively constant across cycle counts, indicating performance is not degrading at longer durations.The estimates are averaged over grid and basis.Source: B. Logical Error per Cycle From One… · pp. 23, 26