Suppressing quantum errors by scaling a surface code logical qubit

I. INTRODUCTION

• Billions of quantum operations are often needed for applications, while state-of-the-art processors have error rates around 10^-3 per gate, motivating quantum error correction despite its physical-qubit overhead.Quantum error correction can exponentially suppress operational error rates.Source: §I · p. 1
• A 49-qubit distance-5 surface code on a 72-qubit superconducting device narrowly outperforms its average 17-qubit distance-3 surface-code subset.The authors present this as a critical step toward scalable quantum error correction.Source: §I · p. 1
• Scaling code size should reduce logical errors when physical errors are sufficiently sparse, but demonstrating this requires correcting at least two errors without sacrificing state-of-the-art performance.Prior work had reported quantum error correction on small codes able to correct one error and in continuous-variable codes.Source: §I · p. 1

• Surface codes [26] [34] [35] [36] encode a logical qubit in a d × d grid of physical qubits, with non-local XL and ZL observables protecting it from local errors when those errors are detected and corrected.Source: §II · p. 1
• Stabilizer measurements of adjacent data-qubit clusters use d^2-1 measure qubits; unexpected parity changes reveal errors without disturbing the logical state.Source: §II · p. 1
• A decoder uses stabilizer-measurement history to infer physical errors, while most logical gates use boundary-measurement sequences that maintain logical memory [37] [38].Source: §II · p. 1
• The distance-5 code uses 25 data and 24 measure qubits on a 49-qubit subset of a 72-transmon Sycamore device, and is compared with four minimally overlapping distance-3 quadrant subgrids.Each distance-3 subgrid contains 9 data qubits and 8 measure qubits; the four subgrids capture average performance across the full distance-5 grid.Source: §II · p. 2, Figure 1 · p. 2
• Each experiment initializes a logical state, runs error-correction cycles, then measures the final state; it succeeds when the corrected result matches the known initial state.The first stabilizer-measurement cycle projects the prepared data into an entangled state, and the final cycle measures the data qubits in the Z basis.Source: §II · p. 2
• Circuit modifications include phase corrections, dynamical decoupling, and omitted Hadamard gates for the ZXXZ variant; randomized initialization avoids bias from even-parity outcomes [44] [46].Source: §II · p. 2

III. ERROR DETECTORS

• A detection event occurs when a parity measurement differs from its corresponding value in the preceding cycle, accounting for known circuit flips.The paper calls these cycle-to-cycle parity comparisons detectors.Source: §III · p. 2
• Weight-4 average detection probabilities were 0.185 ± 0.018 in distance-5 and 0.175±0.017 across distance-3 codes; weight-2 averages were 0.119 ± 0.012 and 0.115 ± 0.008, respectively.The relative consistency suggests that growing the lattice does not substantially increase component error rates during error correction.Source: §III · p. 3, Figure 2 · p. 3
• Average detection probabilities rose 12% for distance-5 and 8% for distance-3 over 25 cycles, with a typical characteristic risetime of roughly 5 cycles.The authors attribute the rise to data qubits leaking into non-computational excited states and hypothesize that the greater distance-5 increase reflects more stray interactions or leakage from simultaneously operating gates and measurements.Source: §III · p. 3
• In distance-5 simulation, average detection probabilities were 0.180 ± 0.013 for weight-4 and 0.116 ± 0.011 for weight-2 stabilisers, increasing 7% over 25 rounds.The simulation incorporated coherence information, leaked-state transitions, and crosstalk errors from unwanted coupling, and accurately predicted the measured averages.Source: §III · p. 3, Figure 2 · p. 3

IV. UNDERSTANDING ERRORS THROUGH CORRELATIONS

• Timelike pairs indicate measurement or reset errors, while spacelike pairs indicate data-qubit errors during measurement idling; CZ-gate errors can produce diagonal pairs.Y errors generate more complex clusters of detection events for both X and Z errors.Source: §IV · pp. 3–4
• Pauli simulation underpredicts detection-pair probabilities, while Pauli+ simulation is closer and predicts unexpected pairs surmised to relate to leakage and stray interactions.These errors can create distant detection events that a decoder may mistake for multiple independent component errors; mitigating them is expected to become more important as error rates decrease.Source: §IV · p. 4
• The analysis estimates normalized pairwise detection-event correlations, pij, and compares experimental and simulated distance-5 data across pair classes [42].It also quantifies pairs unexpected from component Pauli errors.Source: §IV · p. 4

V. DECODING AND LOGICAL ERROR PROBABILITIES

• Belief-matching updates hyperedge probabilities, then decodes separate X/Z graphs with MWPM; tensor-network decoding compares contracted likelihoods of logical-flipping and non-flipping error sets, near-optimal but slower.The tensor-network approximation is checked for convergence; both decoders use a physical-error hypergraph prior.Source: §V · p. 4
• Using logical error probabilities averaged between the X and Z bases, distance-5 had lower pL than the average of the subset distance-3 codes over all 25 error-correction cycles.Source: §V · p. 5
• The distance-5 code improved about twice as fast as distance-3 as physical error rates fell, eventually overtaking the smaller code.Improvements included optimizing single- and two-qubit gates, measurement, and data-qubit idling.Source: §V · p. 5
• CZ error and data-qubit decoherence during measurement and reset are dominant contributors to the inverse logical error-suppression factor.The error budget estimates component contributions by varying physical error rates in simulations of the distance-3 and distance-5 codes.Source: §V · p. 5

VI. ALGORITHMICALLY-RELEVANT ERROR RATES WITH REPETITION CODES

• (1.7 ± 0.3) × 10^-6 logical errors per cycle were measured without post-selection in a distance-25 repetition code; excluding one high-energy event (0.15% of trials) reduced this to (1.6 ± 0.8) × 10^-7.The code was decoded with minimum-weight perfect matching.Source: Figure 4 · p. 5, §VI · p. 6
• The bit-flip repetition code is a 1D version of the surface code that corrects bit-flip but not phase-flip errors, enabling lower logical error probabilities but making it unsuitable for quantum algorithms.Source: §VI · p. 5
• A high-energy impact can cause widespread correlated errors, identifiable through spikes in detection-event counts, and must be mitigated for scalable quantum error correction.The distance-25 results show that low logical error rates are possible in a superconducting system, while such events remain an important challenge.Source: §VI · p. 6

VII. TOWARDS LARGE-SCALE QUANTUM ERROR CORRECTION

• Simulations from distance-3 to 25 show high physical error rates worsen logical error as code size increases, while low rates yield exponential suppression.At intermediate rates, including the experiment's crossover regime, finite-size effects initially suppress logical error before it later increases with system size.Source: §VII · p. 6
• At least 20% component-performance improvement is estimated to be needed to move below threshold, with substantially more needed for practical scaling.The authors say reaching algorithmically relevant logical error rates with manageable resources requires an error-suppression factor Λ ≫1.Source: §VII · p. 6
• The projections use simplified Pauli-error models and require experimental validation with larger code sizes and longer durations.The work is described as a first step toward suppressing logical errors by scaling a quantum error-correcting code.Source: §VII · p. 6

• Google Quantum AI conceived and designed the experiment.Source: §VIII · p. 6
• Google Quantum AI developed data analysis, modeling, and metrological tools, built the system, performed calibrations, and collected data; modeling was done jointly with outside collaborators.Source: §VIII · p. 6
• All authors wrote and revised the manuscript and Supplementary Information.Source: §VIII · p. 6

• The main-text distance-5 surface-code diagram uses (x, y) coordinates, while supplementary heatmaps use a rotated (row, column) system.Source: Figure S5 · p. 11

A. Idling errors

• XY-4 dynamical decoupling with phase cycling mitigates dephasing from low-frequency flux noise during measure-qubit readout and reset.The sequence compensates first-order pulse errors, protects arbitrary states equally, and is tailored per qubit to its noise environment.Source: A. Idling errors · pp. 11–12

B. Optimizing gate parameters

• The Snake optimizer’s graph-optimization and dynamic-programming strategy outperforms local and global optimization in convergence rate and error on the distance-5 objective.It splits high-dimensional optimization into lower-dimensional subproblems, with the subproblem dimension trading optimization complexity for accessible solutions.Source: B. Optimizing gate parameters · p. 12
• When random error events make gates unusable, the Snake locally reoptimizes failing gates and stitches solutions rather than reoptimizing all gates.The optimizer’s frequency configurations change as error mechanisms are enabled; higher-dimensional optimization and historical data are used to improve solution quality and longevity.Source: B. Optimizing gate parameters · p. 12, Figure S6 · p. 13

C. Single qubit gates

• Single-qubit gates combine microwave XY rotations with virtual Z rotations, and DRAG pulse shaping mitigates leakage from off-resonant driving of the |1⟩↔|2⟩ transition.Performance is assessed through randomized benchmarking with all qubits operated simultaneously.Source: C. Single qubit gates · p. 12

D. Two qubit gates

• CZ-gate performance is assessed with simultaneous cross-entropy benchmarking on qubit pairs in their respective surface-code stabilizer layers.This benchmarking arrangement is intended to represent CZ performance in a surface-code error-correction round more accurately.Source: D. Two qubit gates · p. 12

E. Measurement and reset

• Readout frequency, pulse length, and pulse power are optimized per qubit using models of readout and reset errors, while total readout time is kept at 500 ns per qubit.The procedure includes resonator ringdown before measure-qubit reset and benchmarks readout by preparing and measuring random qubit states.Source: E. Measurement and reset · pp. 12, 14, Figure S9 · p. 15

F. Microwave crosstalk induced leakage mitigation

• Microwave crosstalk near a receiver’s f21 can drive |1⟩→|2⟩ leakage; an out-of-phase compensation tone is calibrated with a Ramsey error filter and pairwise scans to identify the source.The filter amplifies coherent leakage through constructive interference between successive pulses; compensation magnitude r and phase θ are optimized at the delay that maximizes leakage population p2.Source: F. Microwave crosstalk induced leakage… · p. 14

1. Surface code circuits

• Virtual Zα corrections before Hadamard gates are empirically tuned to minimize affected-detector rates, with ten compatible correction groups optimized in parallel.Corrections are typically smaller than 0.1; measure qubits have one correction per cycle, while data qubits have two and receive additional first- and last-round adjustments.Source: §1 · p. 15

2. Choice of d = 3 grids

• The four selected distance-3 grids minimized overlap, but shared boundary and center qubits remained; center-qubit performance was within the experiment’s standard range, so the Λ measurement was not considered skewed.The center qubit is included in all four smaller codes and is important because distance-3 codes are more susceptible to outlier qubits.Source: §2 · pp. 15–16

3. Order of experiments

• Datasets were acquired in shuffled cycle-count order, with 50,000 runs per dataset and frequency updates every five cycles; random initial bitstrings avoided a first-round measurement bias.Ten bitstrings were used, five for each logical value, to avoid the warmup asymmetry that would arise if all data qubits started in |0⟩.Source: §3 · p. 16, §4 · p. 16

A. Setting the prior distribution

• Device-level detector-event correlations set decoder priors, while decoding disjoint trial subsets with cross-computed pij values limited fitting to the data being decoded.For parameter scans, a two-pass correlated minimum-weight perfect matching variant is used; repetition-code decoding uses matching without the two-pass strategy.Source: A. Setting the prior distribution · p. 18, B. Correlated minimum-weight perfect… · p. 19
• Using earlier device data increased belief-matching logical error per cycle from 3.118% to 3.129% for distance-3 and from 3.056% to 3.059% for distance-5.Belief-matching combines belief propagation estimates of hyperedge faults with minimum-weight perfect matching; the implementation uses a serial schedule with at most five iterations.Source: A. Setting the prior distribution · pp. 18–19, C. Belief-matching · p. 19

D. Tensor network decoding

• The tensor-network decoder’s logical error probability converged at χ = 30 for the largest distance-5 Z-basis experiment, which was the bond dimension used for decoding.The contraction is approximated as matrix-product-state evolution, whose complexity grows with χ^3; the decoder uses device-level pij correlations as input.Source: D. Tensor network decoding · p. 20, Figure S16 · p. 21

XIII. FITTING ERROR VERSUS ROUNDS TO EXTRACT LOGICAL ERROR PER CYCLE

• An extended fit model produced near-identical distance-3 versus distance-5 logical-error-per-cycle separation, supporting robustness to out-of-model errors and fitting choices.The fitting procedure discards the first round and fits a weighted linear model to log-transformed logical fidelities; residuals about three times larger than binomial sampling expectations inflate uncertainty.Source: §XIII · p. 21

XIV. SIMULATION OF LOGICAL MEMORY EXPERIMENT

• Logical-memory simulations used Pauli-frame depolarizing noise, a polynomial-cost Pauli+ model for leakage and parallel-gate crosstalk, and an exact quantum-trajectories simulation for distance-3.The quantum simulation uses explicit quantum channels derived from physical considerations, unlike the Pauli-channel approximations used by the other approaches.Source: §XIV · p. 21
• The rounds 3–25 extended-model fit separates distance-3 and distance-5 logical error per cycle by 5.7σ.Fits beginning at round 1 disproportionately favor distance-5 and are excluded from the quoted results; the 5.7σ separation is consistent with 5.8σ from the original fits after scaling error bars for excess residuals.Source: Figure S18 · p. 22
• The experiment is simulated by mapping a noiseless circuit to a noisy circuit, optionally converting noise channels to GPCs, then sampling and processing measurements like experimental data.Noise channels are represented with Kraus operators; the Generalized Pauli Twirling Approximation enables the Pauli+ simulator to include leakage.Source: §XIV · p. 22

A. Noise models

• Noise models use Kraus-operator quantum channels applied to qubits after gates or idle operations, converted to GPCs for Pauli+ simulation.The models include decay, dephasing, readout error, leakage, crosstalk, and added depolarizing errors.Source: A. Noise models · pp. 22–23

1. Decay, dephasing, and leakage heating

• Idle-transmon models include decay, white-noise dephasing, and heating leakage; T1 and Tφ are qubit-specific and derived from experimental decay and CPMG data.The assumed |2⟩ decay rate is twice the |1⟩ rate, and leakage heating uses Γ1→2 = 1/(700µs).Source: §1 · p. 23

2. Readout and reset error

• Readout errors are added by a Markov transition matrix using calibrated fidelities, while leaked |2⟩ and |3⟩ states are assumed to read as |0⟩ or |1⟩ equally.Measure-qubit reset error is added to average readout error because it has the same effect as a measurement error at the end of the round.Source: §2 · p. 23

3. Leakage

• Leakage models include passive heating, CZ dephasing transitions, CZ crosstalk, leakage-conditioned Z-phase errors, and higher-level transitions that can transport leakage between qubits.CZ schedules are numerically calibrated to extract phase values and transition amplitudes for the modeled processes.Source: §3 · pp. 23–24

4. Crosstalk

• Crosstalk models simulate neighboring or diagonal pairs of simultaneous CZ or idle operations, including unwanted couplings and leakage-related effects using four-level qubits.The resulting interaction-picture unitaries are included as noisy-circuit error channels.Source: §4 · p. 24

5. Depolarizing errors

• Additional depolarizing errors are appended after gates to compensate for modeled errors underpredicting calibrated fidelity decay.Single-qubit gates, CZ gates, and data-qubit idles are characterized using parallel randomized benchmarking, parallel cross-entropy benchmarking, and interleaved single-qubit XEB, respectively.Source: §5 · p. 24

2. Pauli+ simulator implementation

• The Pauli+ simulator includes reset and generalized-Pauli-channel operations, updates leakage and stabilizer state during transitions, and records measurement results for each sample.For devices transitioning upward, it updates generator state and leakage labels as part of the channel operation.Source: §2 · p. 26

3. Generalized twirling approximation

• The generalized Pauli channel has exactly the same statistics as the physical process of preparing entangled pairs, applying the channel, measuring leakage, and reading stabilizers on devices remaining computational.The process associates stabilizer outcomes with Pauli errors on the devices that remain in the computational subspace.Source: §3 · p. 27
• The conditional probabilities sum to one for each input state, and when there are no leakage transitions the generalized approximation reduces to the typical Pauli twirling approximation.The probabilities can be determined from the channel’s Kraus operators and their decomposition on computational devices.Source: §3 · pp. 27–28
• For coherent rotation between |1⟩ and |2⟩, downward transitions occur with probability sin^2(θ), reverse transitions have half that probability, and computational-subspace transitions acquire dephasing.The reverse transition is halved because a computational-subspace device is in |1⟩ on average only half the time; the dephasing is interpreted as measurement back-action.Source: §3 · p. 28

C. Comparison of Pauli+ and quantum simulations

• Pauli+ and quantum simulations show generally good agreement in logical error and typical detection-event statistics for the distance-3 experiments.The comparison covers logical-error probabilities and detection-edge probabilities, and the West X-basis simulation shows detection-event fractions slowly increasing over rounds, attributed to data-qubit leakage accumulation.Source: Figure S19 · p. 28, Figure S20 · p. 28, C. Comparison of Pauli+ and quantum… · p. 28
• 200,000 samples were used for each 25-round distance-3 Z- and X-basis simulation; Pauli+ took about 160 seconds on 16 CPU cores, versus 11 hours for quantum simulation.Decoding used the algorithm in Ref. [63], with detection-cluster probabilities inferred from the pij matrix.Source: C. Comparison of Pauli+ and quantum… · p. 28
• The agreement is partly consistent with circuit-induced twirling: without leakage, the approximation is exact for a single data-qubit error, non-neighboring circuit-location errors, and some Pauli-described channels.At roughly 1% error per gate across about 20 qubits, the authors expect around one error per cycle.Source: C. Comparison of Pauli+ and quantum… · p. 28
• Mixed-ZXXZ stabilizers and inserted X gates on idle qubits reduce coherent-error effects, and the simulations indicate coherently added errors are not large enough to observe.Unitary crosstalk rotations may add coherently in quantum Monte Carlo simulations.Source: C. Comparison of Pauli+ and quantum… · pp. 28–29

XV. SENSITIVITY OF LOGICAL ERROR PER CYCLE TO PHYSICAL ERRORS

• The sensitivity analysis uses the Pauli+ simulator and correlated minimum-weight perfect matching decoder to assess εL sensitivity to CZ gates, single-qubit gates, data-qubit idle, readout, reset, leakage, and crosstalk.This analysis is separate from the main-text 1/Λ budget, whose sensitivities were evaluated at the half-operation point.Source: §XV · p. 29
• A component-type sensitivity varies the error probability of all components of that type and measures the resulting change in εL; for single-qubit gates, dynamical-decoupling gates are excluded.The changes are denoted δperror and δεL, respectively.Source: §XV · p. 29
• Because εL is nonlinear in component error probabilities, its sensitivity coefficients depend on the operation point, and this analysis evaluates them at the experimental operation point.The supplied passages define the analysis but do not provide numerical sensitivity coefficients.Source: §XV · pp. 29–30

A. Experimental operation point

• Individual circuit-element error probabilities were measured for the d = 5 experiment and used in the simulator, while Table I reports averages by component type.The experimental operation point is specified by the error probabilities of all surface-code circuit components.Source: A. Experimental operation point · p. 30
• SQ gate errors were measured by RB, CZ errors by XEB, data-qubit idle errors by interleaved RB, and readout and reset errors independently.The idle measurements include the dynamical-decoupling sequence used during readout and reset.Source: A. Experimental operation point · p. 30
• Leakage is modeled at high-frequency qubits during CZ gates with p_t = 8 × 10^-4 for all CZs, using a 0.25 probability of state |11⟩ before a CZ.Leakage is also attributed to a nonequilibrium heating process at each qubit.Source: A. Experimental operation point · p. 30
• Heating uses Γ1→2 = 1/(700 µs), a 896 ns QEC cycle, and a 0.5 probability of state |1⟩; the simulator considers heating after each gate or idle.The rate is assumed to be the same for all qubits.Source: A. Experimental operation point · p. 30
• Modeled crosstalk contributes CZ Pauli errors averaging roughly 15% of the observed total CZ error, and this contribution is subtracted to obtain the CZ depolarizing error.Crosstalk is modeled individually for circuit elements and occurs between simultaneous single-qubit idle or CZ gates.Source: A. Experimental operation point · p. 30

B. Logical error rate sensitivities for d = 3, 5 surface codes

• The d = 5 surface code is more sensitive to component errors than the d = 3 surface code.The sensitivities include errors in the main circuit components, leakage, and crosstalk.Source: B. Logical error rate sensitivities… · p. 30
• Sensitivities were computed at the experimental operation point, with the single-qubit gate error variation assumed equal across gates except those in dynamical decoupling.The change in logical error per cycle was fairly linear in the single-qubit gate error variation; the fitted slopes are reported as sensitivities.Source: B. Logical error rate sensitivities… · p. 30, Figure S22 · p. 31, Figure S23 · p. 31

XVI. pij DIAGNOSTICS OF SURFACE CODES

• The pij method quantifies the probability of simultaneous detection events at nodes i and j, following Ref..It can characterize any pair of detection nodes.Source: §XVI · p. 30
• The diagnostics focus on dominant timelike, spacelike, and spacetimelike (diagonal) pairs and compare experimental and Pauli+ simulation correlation matrices.The analysis refers to these pairs as edges.Source: §XVI · pp. 30–31

B. Logical error rate sensitivities for d = 3, 5 surface codes

• νsq = 6.2 ± 0.3 for d = 3 and 8.4 ± 0.2 for d = 5 single-qubit gate errors.These coefficients were obtained from linear fits to simulation data; the plotted logical error rates are averages over four d = 3 grids for Fig. S22.Source: B. Logical error rate sensitivities… · p. 30, Figure S22 · p. 31, Figure S23 · p. 31

XVI. pij DIAGNOSTICS OF SURFACE CODES

• The 600 × 600 pij matrix comes from a 25-round d = 5 surface code experiment with data qubits prepared in the Z basis; its main experiment–simulation discrepancies were attributed to excess leakage between data qubits 2_6 and 2_4.The differences are shown in Fig. S24.Source: Figure S24 · p. 31, §XVI · p. 31

A. Experiment vs. Pauli+ correlation matrices pij

• Broad correlations indicate long-lived leakage effects, and Pauli+ misses stronger-than-simulated leakage near data qubits 2_4 and 2_6.Conventional Pauli errors produce narrow correlations, while broader features indicate long-lived temporal correlations. The stronger experimental correlations are consistent with leakage accumulation in these data qubits.Source: A. Experiment vs. Pauli+ correlation… · p. 32

B. pij probability of main error edges in the surface code; 1. Timelike edges

• Timelike-edge discrepancies are generally small, with the largest near measure qubits 1_4, 3_4, 2_5, and 2_9, close to the more-leaky data qubits.Timelike edges mainly reflect readout/reset errors, though leakage and CZ errors can also contribute.Source: §1 · p. 33, Figure S26 · p. 33

2. Spacelike edges

• At data qubit 5_5, the median SX-edge probability is 0.043 and the SZ-edge probability is 0.031; experimental–simulation differences are relatively small.Spacelike edges can arise from idle data-qubit errors, including bit flips and phase flips; other errors may also contribute.Source: §2 · pp. 33–34, Figure S27 · p. 34, Figure S28 · p. 35

3. Spacetimelike edges

• The expected ST-edge probability is 0.8×10^-2, while the experiment exceeds it by 0.5×10^-2 and Pauli+ by 0.3×10^-2; ST’-edge probabilities are 0.6×10^-2 and 0.35×10^-2, respectively.The excess ST probability is similar to the unexpected ST’ probability, and the largest ST’ discrepancies occur near data qubits 2_4 and 2_6, where experimental leakage is higher.Source: §3 · pp. 34–35, Figure S30 · p. 36

C. Probability of clusters of detection events using generalized pij

• The generalized pij method estimates cluster probabilities by solving seven nonlinear equations for seven process probabilities when analyzing three detection nodes.It is used to diagnose higher-order processes such as Y data-qubit errors and to set decoder weights.Source: C. Probability of clusters of… · p. 36

XVII. SURFACE CODE 1/Λ ERROR BUDGET

• The error-budget contributions sum to 0.90, close to the inverse simulated value 1/Λ3/5 = 1.10.Contributions are calculated using component-error sensitivities at the half-operation point.Source: §XVII · p. 37

XVIII. OVERVIEW OF ERROR CORRECTION EXPERIMENTS

• Table IV summarizes error-correction and error-detection experiments, distinguishing classical-code notation and experiments that embed correction into physical qubits.Source: Table IV · p. 38

XIX. ADDITIONAL PLOTS OF SURFACE CODE DATA

• Additional plots show detection probabilities and distance-5 and distance-3 logical errors and fidelities across experimental and Pauli+ decoding scenarios.The distance-5 and four distance-3 plots compare belief matching and tensor-network decoding for experimental data and Pauli+ simulation.Source: Figure S31 · p. 39, Figure S32 · p. 40, Figure S33 · p. 41