A 2021 experiment by Google Quantum AI demonstrated exponential suppression of bit- or phase-flip logical errors with cyclic stabilizer measurements on a superconducting processor, but explicitly left open the milestones needed for scalable fault tolerance: crossing the surface-code threshold, achieving practical suppression factors (Λ≈10\Lambda\approx10 for a 1000:1 physical-to-logical qubit ratio), containing leakage and crosstalk at the 10−310^{-3} level, and mitigating sporadic device-wide high-energy-particle events. Later work has now resolved parts of this question — distance-3/5 surface-code scaling without post-selection (2023), per-cycle leakage removal (2023), and a below-threshold distance-7 surface-code memory with real-time decoding and Λ=2.14\Lambda=2.14 (2025) — while the surviving open core is to push the demonstrated error suppression from Λ≈2\Lambda\approx2 toward the practical Λ≈10\Lambda\approx10 regime and to eliminate the rare correlated error events (approximately once per hour in the most advanced 2025 device) that still set the logical-error floor.

Background

Quantum error correction (QEC) encodes one logical qubit redundantly into many physical qubits so that errors can be detected and corrected. In stabilizer codes, dedicated 'measure' qubits repeatedly measure parities (stabilizers) of groups of 'data' qubits; the pattern of parity flips (detection events) reveals where errors occurred. The distance dd of a code is the minimum number of physical faults needed to cause a logical error, so below a critical physical error rate (the threshold) the logical error rate falls exponentially as dd grows. The surface code arranges data and measure qubits on a two-dimensional checkerboard (X-type and Z-type measure qubits alternating), and is the leading candidate for fault tolerance because its threshold is high and it needs only nearest-neighbour couplings; the one-dimensional repetition code, which protects against bit-flips or phase-flips but not both, is its simpler precursor. A convenient performance metric is the error suppression factor Λ\Lambda: the ratio by which the logical error rate falls when the code distance is increased by two, so that the logical error rate scales as Λ−⌊(d+1)/2⌋\Lambda^{-\lfloor(d+1)/2\rfloor}. Estimates of the physical-to-logical qubit overhead for useful algorithms become practical only around Λ≈10\Lambda \approx 10 (roughly a 1000:1 qubit ratio), whereas near-threshold devices give Λ\Lambda barely above 1. In 2021, Google Quantum AI ('Exponential suppression of bit or phase errors with cyclic error correction', Nature 595, 383–387) ran repetition codes up to d=11d=11 on the 54-qubit Sycamore superconducting processor, suppressing logical error per round by more than 100× from 5 to 21 qubits, stable over 50 rounds of cyclic stabilizer measurements, and validated the results against a simple depolarizing noise model (one in which each faulty operation applies a random Pauli error). Their correlation analysis localized errors to the 3×10−33\times10^{-3} level but exposed non-Pauli mechanisms absent from that model: leakage of population out of the two-level computational subspace, layout-induced crosstalk (unwanted error coupling between nominally independent qubit operations, here between next-nearest neighbours), and sporadic device-wide high-energy events that transiently destroy qubit coherence across the whole chip by generating excess quasiparticles (unpaired electrons) in the superconducting metal. A distance-2 surface-code experiment achieved error detection only with post-selection (discarding experimental runs in which error indicators fired, 27% of runs per round), which is not scalable. The paper's error budget identified controlled-Z (CZ, the two-qubit entangling gate used in the code's circuits) infidelity and data-qubit decoherence during measurement and reset as the dominant contributions, and its conclusion stated that reaching the surface-code threshold and eventually Λ≈10\Lambda\approx10 would require substantial reductions of these error rates together with 'further research into mitigation of error mechanisms such as high-energy particles'. Subsequent milestones define the state of the art: an ETH Zurich team demonstrated repeated QEC cycles in a distance-3 surface code (Nature 605, 669–674, 2022); Google demonstrated distance-3 versus distance-5 surface-code scaling without post-selection (Nature 614, 676–681, 2023), per-cycle leakage removal (Nature Physics 19, 1780–1786, 2023), and, on Google's Willow processor, a 101-qubit distance-7 surface-code memory operating below the surface-code threshold with a real-time decoder (the classical algorithm that converts the streaming syndrome record into correction decisions fast enough to keep up with the measurement cycles), Λ=2.14±0.02\Lambda = 2.14 \pm 0.02, 0.143% logical error per cycle, and beyond-break-even performance (the logical qubit outperforming the best physical qubit on the device) sustained over a million cycles (Nature 638, 920–926, 2025).

Mitigation of these correlated events is itself an open research direction: the events have been traced to quasiparticle bursts from cosmic rays and background radioactivity (Nature Physics 18, 107–111, 2022), and candidate approaches under study — superconducting gap engineering near the Josephson junctions, quasiparticle/phonon trapping, shielding, and radiation-aware watchdog detection-and-repair schemes integrated with decoding — have so far been characterized or simulated rather than demonstrated at scale inside fault-tolerant QEC operation (arXiv:2603.13460; arXiv:2506.16834).

Problem Statement

Take the experimentally established regime of cyclic surface-code quantum error correction on superconducting processors — where a distance-7 code has been operated below the surface-code threshold with real-time decoding, but only at suppression factor Λ=2.14\Lambda=2.14 — and reach the practical performance level targeted by the 2021 experiment that established this program. Specifically: (1) demonstrate suppression of logical errors per cycle at the level of an error suppression factor of order Λ≈10\Lambda\approx10 per two steps of code distance (supporting a roughly 1000:1 physical-to-logical qubit ratio) in a two-dimensional surface-code memory under cyclic stabilizer measurements without post-selection; by the established error budgets, reaching this suppression requires roughly order-of-magnitude reductions of the dominant operational error rates — controlled-Z two-qubit-gate infidelity and data-qubit decoherence during measurement and reset — beyond the 2021 levels, and the demonstration must be accompanied by predictive noise models that include the non-Pauli channels observed in the 2021 correlation analysis (leakage out of the computational subspace and layout-induced crosstalk at the ∼3×10−3\sim3\times10^{-3} level); and (2) mitigate rare device-wide high-energy-particle events so that these correlated events no longer set the logical-error floor (the ∼1.7×10−6\sim1.7\times10^{-6}-per-round floor observed in 2023 and the approximately once-per-hour correlated events limiting the 2025 below-threshold device) in large-scale fault-tolerant operation. Crossing the surface-code threshold and demonstrating distance scaling in 2D without post-selection, which the 2021 work left open, are already achieved and are not part of this problem.

References

  1. Google Quantum AI (Zijun Chen, Kevin J. Satzinger, Juan Atalaya, et al.), 'Exponential suppression of bit or phase errors with cyclic error correction', Nature 595, 383-387 (2021). DOI 10.1038/s41586-021-03588-y; arXiv:2102.06132; https://www.nature.com/articles/s41586-021-03588-y
  2. Rajeev Acharya et al. (Google Quantum AI), 'Suppressing quantum errors by scaling a surface code logical qubit', Nature 614, 676-681 (2023). DOI 10.1038/s41586-022-05434-1; arXiv:2207.06431; https://www.nature.com/articles/s41586-022-05434-1
  3. Rajeev Acharya et al. (Google Quantum AI), 'Quantum error correction below the surface code threshold', Nature 638, 920-926 (2025). DOI 10.1038/s41586-024-08449-y; arXiv:2408.13687; https://www.nature.com/articles/s41586-024-08449-y
  4. Kevin C. Miao, Matt McEwen, Juan Atalaya, et al. (Google Quantum AI), 'Overcoming leakage in quantum error correction', Nature Physics 19, 1780-1786 (2023). DOI 10.1038/s41567-023-02226-w; arXiv:2211.04728; https://www.nature.com/articles/s41567-023-02226-w
  5. Matt McEwen, Lara Faoro, Kunal Arya, et al., 'Resolving catastrophic error bursts from cosmic rays in large arrays of superconducting qubits', Nature Physics 18, 107-111 (2022). DOI 10.1038/s41567-021-01432-8; https://www.nature.com/articles/s41567-021-01432-8
  6. Sebastian Krinner, Nathan Lacroix, Ants Remm, et al., 'Realizing repeated quantum error correction in a distance-three surface code', Nature 605, 669-674 (2022). DOI 10.1038/s41586-022-04566-8; https://www.nature.com/articles/s41586-022-04566-8
  7. Alec Eickbusch, Matt McEwen, Volodymyr Sivak, et al. (Google Quantum AI), 'Demonstrating dynamic surface codes', arXiv:2412.14360 (2024, revised June 2025). https://arxiv.org/abs/2412.14360
  8. H. Douglas Pinckney, Thomas McJunkin, Alan W. Hunt, et al., 'Characterization of Radiation-Induced Errors in Superconducting Qubits Protected with Various Gap-Engineering Strategies', arXiv:2603.13460 (2026). https://arxiv.org/abs/2603.13460
  9. Marzio Vallero, Gioele Casagranda, Flavio Vella, Paolo Rech, 'Radiation-Induced Fault Detection in Superconducting Quantum Devices', arXiv:2506.16834 (2025). https://arxiv.org/abs/2506.16834