Suppressing quantum errors by scaling a surface code logical qubit

1Since Feynman’s proposal to compute using quantum mechanics, many potential applications have emerged, including factoring, optimization, machine learning, quantum simulation and quantum chemistry. These applications often require billions of quantum operations– and state-of-the-art quantum processors typically have error rates around 10−3 per gate–, far too high to execute such large circuits. Fortunately, quantum error correction can exponentially suppress the operational error rates in a quantum processor, at the expense of temporal and qubit overhead,.

2Several works have reported quantum error correction on codes able to correct a single error, including the distance-3 Bacon–Shor, colour, five-qubit, heavy-hexagon and surface, codes, as well as continuous variable codes–. However, a crucial question remains of whether scaling up the error-correcting code size will reduce logical error rates in a real device. In theory, logical errors should be reduced if physical errors are sufficiently sparse in the quantum processor. In practice, demonstrating reduced logical error requires scaling up a device to support a code that can correct at least two errors, without sacrificing state-of-the-art performance. In this work we report a 72-qubit superconducting device supporting a 49-qubit distance-5 (d = 5) surface code that narrowly outperforms its average subset 17-qubit distance-3 surface code, demonstrating a critical step towards scalable quantum error correction.

3Surface codes– are a family of quantum error-correcting codes that encode a logical qubit into the joint entangled state of a d × d square of physical qubits, referred to as data qubits. The logical qubit states are defined by a pair of anti-commuting logical observables XL and ZL. For the example shown in Fig. 1a, a ZL observable is encoded in the joint Z-basis parity of a line of qubits that traverses the lattice from top to bottom, and likewise an XL observable is encoded in the joint X-basis parity traversing left to right. This non-local encoding of information protects the logical qubit from local physical errors, provided we can detect and correct them.

4To detect errors, we periodically measure X and Z parities of adjacent clusters of data qubits with the aid of d2 − 1 measure qubits interspersed throughout the lattice. As shown in Fig. 1b, each measure qubit interacts with its neighbouring data qubits to map the joint data qubit parity onto the measure qubit state, which is then measured. Each parity measurement, or stabilizer, commutes with the logical observables of the encoded qubit as well as every other stabilizer. Consequently, we can detect errors when parity measurements change unexpectedly, without disturbing the logical qubit state.

5A decoder uses the history of stabilizer measurement outcomes to infer likely configurations of physical errors on the device. We can then determine the overall effect of these inferred errors on the logical qubit, thus preserving the logical state. Most surface code logical gates can be implemented by maintaining logical memory and executing different sequences of measurements on the code boundary–. Thus, we focus on preserving logical memory, the core technical challenge in operating the surface code.

6We implement the surface code on an expanded Sycamore device with 72 transmon qubits and 121 tunable couplers,. Each qubit is coupled to four nearest neighbours except on the boundaries, with mean qubit coherence times T1 = 20 μs and T2,CPMG = 30 μs, in which CPMG represents Carr–Purcell–Meiboom–Gill. As in ref., we implement single-qubit rotations, controlled-Z (CZ) gates, reset and measurement, demonstrating similar or improved simultaneous performance as shown in Fig. 1c.

7The distance-5 surface code logical qubit is encoded on a 49-qubit subset of the device, with 25 data qubits and 24 measure qubits. Each measure qubit corresponds to one stabilizer, classified by its basis (X or Z) and the number of data qubits involved (weight, 2 or 4). Ideally, to assess how logical performance scales with code size, we would compare distance-5 and distance-3 logical qubits under identical noise. Although device inhomogeneity makes this comparison difficult, we can compare the distance-5 logical qubit to the average of four distance-3 logical qubit subgrids, each containing nine data qubits and eight measure qubits. These distance-3 logical qubits cover the four quadrants of the distance-5 code with minimal qubit overlap, capturing the average performance of the full distance-5 grid.

8In a single instance of the experiment, we initialize the logical qubit state, run several cycles of error correction, and then measure the final logical state. We show an example in Fig. 2a. To prepare a ZL eigenstate, we first prepare each data qubit in or, an eigenstate of the Z stabilizers. The first cycle of stabilizer measurements then projects the data qubits into an entangled state that is also an eigenstate of the X stabilizers. Each cycle contains CZ and Hadamard gates sequenced to extract X and Z stabilizers simultaneously, and ends with the measurement and reset of the measure qubits. In the final cycle, we also measure the data qubits in the Z basis, yielding both parity information and a measurement of the logical state. Preparing and measuring XL eigenstates proceeds analogously. The instance succeeds if the corrected logical measurement agrees with the known initial state; otherwise, a logical error has occurred.

9Our stabilizer circuits contain a few modifications to the standard gate sequence described above (see Supplementary Information), including phase corrections to correct for unintended qubit frequency shifts and dynamical decoupling gates during qubit idles. We also remove certain Hadamard gates to implement the ZXXZ variant of the surface code,, which helps symmetrize the X- and Z-basis logical error rates. Finally, during initialization, the data qubits are prepared into randomly selected bitstrings. This ensures that we do not preferentially measure even parities in the first few cycles of the code, which could artificially lower logical error rates owing to bias in measurement error (see Supplementary Information).

10After initialization, parity measurements should produce the same value in each cycle, up to known flips applied by the circuit. If we compare a parity measurement to the corresponding measurement in the preceding cycle and their values are inconsistent, a detection event has occurred, indicating an error. We refer to these comparisons as detectors.

11The detection event probabilities for each detector indicate the distribution of physical errors in space and time while running the surface code. In Fig. 2, we show the detection event probabilities in the distance-5 code (Fig. 2b,c) and the distance-3 codes (Fig. 2d,e) running for 25 cycles, as measured over 50,000 experimental instances. For the weight-4 stabilizers, the average detection probability is 0.185 ± 0.018 (1σ) in the distance-5 code and 0.175 ± 0.017 averaged over the distance-3 codes. The weight-2 stabilizers interact with fewer qubits and hence detect fewer errors. Correspondingly, they yield a lower average detection probability of 0.119 ± 0.012 in the distance-5 code and 0.115 ± 0.008 averaged over the distance-3 codes. The relative consistency between code distances suggests that growing the lattice does not substantially increase the component error rates during error correction.

12The average detection probabilities exhibit a relative rise of 12% for distance-5 and 8% for distance-3 over 25 cycles, with a typical characteristic risetime of roughly 5 cycles (see Supplementary Information). We attribute this rise to data qubits leaking into non-computational excited states and anticipate that the inclusion of leakage-removal techniques on data qubits would help to mitigate this rise,–. We reason that the greater increase in detection probability in the distance-5 code is due to increased stray interactions or leakage from simultaneously operating more gates and measurements.

13We test our understanding of the physical noise in our system by comparing the experimental data to a simulation. We begin with a depolarizing noise simulation based on the component error information in Fig. 1c, and then extend to a Pauli simulation with qubit-specific T1 and T2,CPMG, transitions to leaked states, and stray interactions between qubits during CZ gates (see Supplementary Information). We refer to this simulation as Pauli+. Figure 2f shows that this second simulator accurately predicts the average detection probabilities, finding 0.180 ± 0.013 for the weight-4 stabilizers and 0.116 ± 0.011 for the weight-2 stabilizers, with average detection probabilities increasing 7% over 25 cycles (distance-5).

14We next examine pairwise correlations between detection events, which give us fine-grained information about which types of error are occurring during error correction. Figure 2a illustrates a few examples of pairwise detections that are generated by X or Z errors in the surface code. Measurement and reset errors are detected by the same stabilizer in two consecutive cycles, which we classify as a timelike pair. Data qubits may experience an X (Z) error while idling during measurement that is detected by its neighbouring Z (X) stabilizers in the same cycle, forming a spacelike pair. Errors during CZ gates may cause a variety of pairwise detections to occur, including spacetimelike pairs that are separated in both space and time. More complex clusters of detection events arise when a Y error occurs, which generates detection events for both X and Z errors.

15To estimate the probability for each detection event pair from our data, we compute an appropriately normalized correlation pij between detection events occurring on any two detectors i and j (refs.,; see Supplementary Information). In Fig. 2h, we show the estimated probabilities for experimental and simulated distance-5 data, aggregated and averaged according to the different classes of pairs. In addition to the expected pairs, we also quantify how often detection pairs occur that are unexpected in a local depolarizing circuit model. Overall, the Pauli simulation systematically underpredicts these probabilities compared to experimental data, whereas the Pauli+ simulation is closer and predicts the presence of unexpected pairs, which we surmise are related to leakage and stray interactions. These errors can be especially harmful to the surface code because they can generate multiple detection events distantly separated in space or time, which a decoder might wrongly interpret as multiple independent component errors. We expect that mitigating leakage and stray interactions will become increasingly important as error rates decrease.

16We next examine the logical performance of our surface code qubits. To infer the error-corrected logical measurement, the decoder requires a probability model for physical error events. This information may be expressed as an error hypergraph: detectors are vertices, physical error mechanisms are hyperedges connecting the detectors they trigger, and each hyperedge is assigned its corresponding error mechanism probability. We use a generalization of pij to determine these probabilities,.

17Given the error hypergraph, we implement two different decoders: belief-matching, an efficient combination of belief propagation and minimum-weight perfect matching; and tensor network decoding, a slow but accurate approximate maximum-likelihood decoder. The belief-matching decoder first runs belief propagation on the error hypergraph to update hyperedge error probabilities based on nearby detection events,. The updated error hypergraph is then decomposed into a pair of disjoint error graphs, one each for X and Z errors. These graphs are decoded efficiently using minimum-weight perfect matching to select a single probable set of errors.

18By contrast, a maximum-likelihood decoder considers all possible sets of errors consistent with the detection events, splits them into two groups on the basis of whether they flip the logical measurement, and chooses the group with the greater total likelihood. The two likelihoods are each expressed as a tensor network contraction,, that exhaustively sums the probabilities of all sets of errors within each group. We can contract the network approximately, and verify that the approximation converges. This yields a decoder that is nearly optimal given the hypergraph error priors, but is considerably slower. Further improvements could come from a more accurate prior, or by incorporating more fine-grained measurement information,.

19Figure 3 shows a comparison of the logical error performance of the distance-3 and distance-5 codes using the approximate maximum-likelihood decoder. As the ZXXZ variant of the surface code symmetrizes the X and Z bases, differences between the two bases’ logical error per cycle are small and attributable to spatial variations in physical error rates. Thus, for visual clarity, we report logical error probabilities averaged between the X and Z basis; the full dataset may be found in the Supplementary Information. Note that we do not post-select on leakage or high-energy events to capture the effects of realistic non-idealities on logical performance. Over all 25 cycles of error correction, the distance-5 code realizes lower logical error probabilities pL than the average of the subset distance-3 codes.

20We fit the logical fidelity F = 1 − 2pL to an exponential decay. We start the fit at t = 3 to avoid two phenomena that advantage the larger code: the lower detection probability during the first cycle relative to subsequent cycles (Fig. 2b,d), and the higher effective threshold caused by the confinement of errors to thin time slices in few-cycle experiments. We obtain a logical error per cycle ε5 = (2.914 ± 0.016)% (1σ statistical and fit uncertainty) for the distance-5 code, compared to an average of ε3 = (3.028 ± 0.023)% for the subset distance-3 codes, a relative error reduction of about 4%. When decoding with the faster belief-matching decoder, we fit a logical error per cycle of (3.056 ± 0.015)% for the distance-5 code, compared to an average of (3.118 ± 0.025)% for the distance-3 codes, a relative error reduction of about 2%. We note that the distance-5 logical error per cycle is slightly higher than those of two of the distance-3 codes individually, and that leakage accumulation may cause distance-5 performance to degrade faster than that of distance-3 as logical error probability approaches 50%.

21In principle, the logical performance of a distance-5 code should improve faster than that of a distance-3 code as physical error rates decrease. Over time, we improved our physical error rates, for example by optimizing single- and two-qubit gates, measurement and data qubit idling (see Supplementary Information). In Fig. 3c, we show the corresponding performance progression of distance-5 and distance-3 codes. The larger code improved about twice as fast until finally overtaking the smaller code, validating the benefit of increased-distance protection in practice.

22To understand the contributions of individual components to our logical error performance, we follow ref. and simulate the distance-5 and distance-3 codes while varying the physical error rates of the various circuit components. As the logical-error-suppression factoris approximately inversely proportional to the physical error rate, we can budget how much each physical error mechanism contributes to 1/Λ3/5 (as shown in Fig. 4a) to assess scaling. This error budget shows that CZ error and data qubit decoherence during measurement and reset are dominant contributors.

23Even as known error sources are suppressed in future devices, new dominant error mechanisms may arise as lower logical error rates are realized. To test the behaviour of codes with substantially lower error rates, we use the bit-flip repetition code, a one-dimensional version of the surface code. The bit-flip repetition code does not correct for phase-flip errors and is thus unsuitable for quantum algorithms. However, correcting only bit-flip errors allows it to achieve much lower logical error probabilities.

24Without post-selection, we achieve a logical error per cycle of (1.7 ± 0.3) × 10−6 using a distance-25 repetition code decoded with minimum-weight perfect matching (Fig. 4b). We attribute many of these logical errors in the higher-distance codes to a high-energy impact, which can temporarily impart widespread correlated errors to the system. These events may be identified by spikes in detection event counts, and such error mechanisms must be mitigated for scalable quantum error correction to succeed. In this case, there was one such event; after removing it (0.15% of trials), we observe a logical error per cycle of (1.6 ± 0.8) × 10−7 (see Supplementary Information). The repetition code results demonstrate that low logical error rates are possible in a superconducting system, but finding and mitigating highly correlated errors such as cosmic ray impacts will be an important area of research moving forwards.

25To understand how our surface code results project forwards to future devices, we simulate the logical error performance of surface codes ranging from distance-3 to 25, while also scaling the physical error rates shown in Fig. 1c. For efficiency, the simulation considers only Pauli errors. Figure 4c,d illustrates the contours of this parameter space, which has three distinct regions. When the physical error rate is high (for example, the initial runs of our surface code in Fig. 3c), logical error probability increases with increasing system size (εd+2 > εd). On the other hand, low physical error rates show the desired exponential suppression of logical error (εd+2 < εd). This threshold behaviour can be subtle, and there exists a crossover regime in which, owing to finite-size effects, increasing system size initially suppresses the logical error per cycle before later increasing it. We believe our experiment lies in this regime.

26Although our device is close to threshold, reaching algorithmically relevant logical error rates with manageable resources will require an error-suppression factor Λd/(d+2) ≫ 1. On the basis of the error budget and simulations in Fig. 4, we estimate that component performance must improve by at least 20% to move below threshold, and substantially improve beyond that to achieve practical scaling. However, these projections rely on simplified models and must be validated experimentally, testing larger code sizes with longer durations to eventually realize the desired logical performance. This work demonstrates the first step in that process, suppressing logical errors by scaling a quantum error-correcting code—the foundation of a fault-tolerant quantum computer.

27Any methods, additional references, Nature Portfolio reporting summaries, source data, extended data, supplementary information, acknowledgements, peer review information; details of author contributions and competing interests; and statements of data and code availability are available at 10.1038/s41586-022-05434-1.