Fault-tolerant thresholds for quantum error correction with the surface code
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The surface code is a promising candidate for fault-tolerant quantum computation, achieving a high threshold error rate with nearest-neighbor gates in two spatial dimensions. Here, through a series of numerical simulations, we investigate how the precise value of the threshold depends on the noise model, measurement circuits, and decoding algorithm. We observe thresholds between 0.502(1)% and 1.140(1)% per gate, values which are generally lower than previous estimates.
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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.
“This threshold behaviour can be subtle58, 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.”
From Suppressing quantum errors by scaling a surface code logical qubit (2023) · cites this paper as [58]
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DNA Assembly of Dye Aggregates—A Possible Path to Quantum Computing
Yurke, B. (2023) · Natural computing series · doi:10.1007/978-981-19-9891-1_9 · Cited by 19 · see citing works
Suzuki, Y., Fujii, K., & Koashi, M. (2017) · Physical Review Letters · doi:10.1103/physrevlett.119.190503 · Cited by 18 · see citing works
Scruby, T. R., Browne, D. E., Webster, P., & Vasmer, M. (2022) · Quantum · doi:10.22331/q-2022-05-24-721 · Cited by 18 · see citing works
“In addition, our value is only one order of magnitude smaller than the 2D surface code threshold [6, 7, 21, 48].”
BackgroundA Silicon Surface Code Architecture Resilient Against Leakage Errors
Cai, Z., Fogarty, M. A., Schaal, S., Patomäki, S., Benjamin, S. C., & Morton, J. J. L. (2019) · Quantum · doi:10.22331/q-2019-12-09-212 · Cited by 18 · see citing works
“5% – 1% [82].”
MethodCoherent Parity Check Construction for Quantum Error CorrectionPreprint
Chancellor, N., Kissinger, A., Zohren, S., & Horsman, D. C. (2016) · arXiv (Cornell University) · Cited by 17 · see citing works
Comparing two-qubit and multiqubit gates within the toric code
Schwerdt, D., Shapira, Y., Manovitz, T., & Ozeri, R. (2022) · Physical Review A · doi:10.1103/physreva.105.022612 · Cited by 16 · see citing works
Concatenation of the Gottesman-Kitaev-Preskill code with the XZZX surface code
Zhang, J., Wu, Y.-C., & Guo, G. (2023) · Physical Review A · doi:10.1103/physreva.107.062408 · Cited by 16 · see citing works
Fault-tolerant computing with single-qudit encoding in a molecular spin
Mezzadri, M., Chiesa, A., Lepori, L., & Carretta, S. (2024) · Materials Horizons · doi:10.1039/d4mh00454j · Cited by 16 · see citing works
“Yet, a comprehensive Fault-Tolerant implementation accommodating realistic noise models, which encompass noisy ancillae, encodings, measurements and especially noise during gates, is still missing [23, 36].”
BackgroundLow-overhead quantum computing with the color code
Thomsen, F., Kesselring, M. S., Bartlett, S. D., & Brown, B. J. (2024) · Physical Review Research · doi:10.1103/physrevresearch.6.043125 · Cited by 15 · see citing works
Classical Control of Large-Scale Quantum Computers
Devitt, S. J. (2014) · Lecture notes in computer science · doi:10.1007/978-3-319-08494-7_3 · Cited by 14 · see citing works
“This can reduce the amount of information as the probability that a unit cell of the lattice will experience a parity flip is of the order of the error rate of each qubit, p , which will be approximately 0.1% [48].”
BackgroundInfluentialComparative study of quantum error correction strategies for the heavy-hexagonal lattice
Benito, C., López, E., Peropadre, B., & Bermúdez, A. (2025) · Quantum · doi:10.22331/q-2025-02-06-1623 · Cited by 16 · see citing works
“These codes display a much larger error threshold approaching p th ≈ 1% [63–65] with respect to that of concatenated QEC codes [20, 66, 67], making them very promising.”
Compare the pair: Rotated versus unrotated surface codes at equal logical error rates
O'Rourke, A. R., & Devitt, S. J. (2025) · Physical Review Research · doi:10.1103/physrevresearch.7.033074 · Cited by 14 · see citing works
Fundamental thresholds of realistic quantum error correction circuits from classical spin models
Vodola, D., Rispler, M., Kim, S., & Müller, M. (2022) · Quantum · doi:10.22331/q-2022-01-05-618 · Cited by 14 · see citing works
“This is particularly pressing and valuable for QEC codes because even (near-)optimal decoding is computationally hard [34], or optimal decoders are unknown/strongly dependent on the chosen QEC circuitry and details of the circuit noise models [35].”
BackgroundBenchmarking quantum logic operations relative to thresholds for fault tolerance
Hashim, A., Seritan, S., Proctor, T. J., Rudinger, K., Goss, N., Naik, R., Kreikebaum, J. M., Santiago, D., & Siddiqi, I. A. (2023) · npj Quantum Information · doi:10.1038/s41534-023-00764-y · Cited by 16 · see citing works
Single-step parity check gate set for quantum error correction
Üstün, G., Morello, A., & Devitt, S. J. (2024) · Quantum Science and Technology · doi:10.1088/2058-9565/ad473c · Cited by 13 · see citing works
“Making assumptions about how error syndromes are measured in the system, commonly referred to as code capacity noise model, phenomenological noise model, and circuit-based noise model [10], can lead to significantly different fault-tolerant threshold values.”
Facilitating practical fault-tolerant quantum computing based on color codes
Zhang, J., Wu, Y.-C., & Guo, G.-P. (2024) · Physical Review Research · doi:10.1103/physrevresearch.6.033086 · Cited by 12 · see citing works
“9% using an efficient decoder [15, 16].”
BackgroundTopological dynamical decoupling
Zhang, J., Yu, X.-D., Long, G.-L., & Xue, Q.-K. (2019) · Science China Physics Mechanics and Astronomy · doi:10.1007/s11433-019-9447-5 · Cited by 12 · see citing works
Trellis Decoding for Qudit Stabilizer Codes and Its Application to Qubit Topological Codes
Sabo, E., Aloshious, A. B., & Brown, K. R. (2024) · IEEE Transactions on Quantum Engineering · doi:10.1109/tqe.2024.3401857 · Cited by 12 · see citing works
“Extended codes (with weight one stabilizers) and codes with k = 0 or d ≤ 2 were discarded, and code-capacity (memory model) numerical simulations were performed on the remaining four codes: [[20 , 3 , 6]], [[20 , 4 , 6]], [[20 , 10 , 4]], and [[20 , 13 , 3]].”
MethodInfluentialCoherent control of an NV−center with one adjacent13C
Scharfenberger, B., Munro, W. J., & Nemoto, K. (2014) · New Journal of Physics · doi:10.1088/1367-2630/16/9/093043 · Cited by 11 · see citing works
Low-Overhead Magic State Distillation with Color Codes
Lee, S.-H., Thomsen, F., Fazio, N., Brown, B. J., & Bartlett, S. D. (2025) · PRX Quantum · doi:10.1103/ch5r-cnfq · Cited by 14 · see citing works
Hexagonal matching codes with two-body measurements
Wootton, J. R. (2022) · Journal of Physics A Mathematical and Theoretical · doi:10.1088/1751-8121/ac7a75 · Cited by 10 · see citing works
Convolutional-Neural-Network-Based Hexagonal Quantum Error Correction Decoder
Li, A.-Q., Li, F., Qi-Di, G. A. N., & Ma, H. (2023) · Applied Sciences · doi:10.3390/app13179689 · Cited by 10 · see citing works
“A surface code is a type of stabilizer code whose layout is created by arranging and combining four-dimensional square matrices defined on a two-dimensional plane [23].”
BackgroundImproving Threshold for Fault-Tolerant Color-Code Quantum Computing by Flagged Weight Optimization
Takada, Y., & Fujii, K. (2024) · PRX Quantum · doi:10.1103/prxquantum.5.030352 · Cited by 10 · see citing works
“The thresholds of surface codes under circuit-level noise are estimated to be around 0.5%-1.1% [8–13], depending on each noise assumption and way of performing QEC.”
BackgroundInfluentialGavriel, J., Herr, D., Shaw, A., Bremner, M. J., Paler, A., & Devitt, S. J. (2023) · Physical Review Research · doi:10.1103/physrevresearch.5.033019 · Cited by 12 · see citing works
“[22] Multiple studies have numerically derived the precise scaling, including constant factors for a required logical error rate in the planar surface code as a function of additional hardware constraints, more precise physical error models, and overall physical error rate [21].”
BackgroundNon–Pauli Errors Can Be Efficiently Sampled in Qudit Surface Codes
Ma, Y., Hanks, M., & Kim, M. S. (2023) · Physical Review Letters · doi:10.1103/physrevlett.131.200602 · Cited by 9 · see citing works
Error-rate-agnostic decoding of topological stabilizer codes
Hammar, K., Orekhov, A., Hybelius, P. W., Wisakanto, A. K., Srivastava, B., Kockum, A. F., & Granath, M. (2022) · Physical Review A · doi:10.1103/physreva.105.042616 · Cited by 9 · see citing works
“As these ancillas are also noisy and the measurement circuit involves several two-qubit gates, there will be errors in the syndrome readout, which complicates the decoding significantly [29, 73].”
BackgroundQuantum computer error structure probed by quantum error correction syndrome measurements
Gicev, S., Hollenberg, L. C. L., & Usman, M. (2024) · Physical Review Research · doi:10.1103/physrevresearch.6.043249 · Cited by 8 · see citing works
“Results always used initial layouts corresponding to: • qubits 2, 10, 17, 5, 13, 21, 9, 16, 24 as data qubits, • qubits 1, 7, 3, 12, 18, 8, 14, 23, 19, 25 as flag qubits, • qubits 4, 15, 11, 22 as measure qubits, as shown in Figure 8.”
Das, P., Locharla, A., & Jones, C. (2021) · arXiv (Cornell University) · doi:10.48550/arxiv.2108.06569 · Cited by 8 · see citing works
“While m = ( d − 1 ) rounds are needed to detect as many errors as the code is capable of correcting [54], we observe that LILLIPUT performance saturates at m = 2 syndrome rounds for d = 4.”
BackgroundOld, J., Rispler, M., & Müller, M. (2024) · Quantum Science and Technology · doi:10.1088/2058-9565/ad5eb6 · Cited by 7 · see citing works
“Additionally, surface codes have some of the highest known thresholds for realistic circuit level noise models [9, 10].”
BackgroundOpen-system quantum error correction
Len, Y. L., & Ng, H. K. (2018) · Physical Review A · doi:10.1103/physreva.98.022307 · Cited by 7 · see citing works
Wang, H., Cao, Q., Xue, Y.-J., Ding, L. J., Liu, H.-Y., Dong, Y., & Ma, H. (2022) · Frontiers in Physics · doi:10.3389/fphy.2022.981225 · Cited by 6 · see citing works
“[46], with the d increase, the perror of continuous curve intersection will decrease, and this definition will lead to an overestimation of the threshold.”
BackgroundRobust gates with spin-locked superconducting qubits
Zuk, I., Cohen, D., Gorshkov, A. V., & Retzker, A. (2024) · Physical Review Research · doi:10.1103/physrevresearch.6.013217 · Cited by 6 · see citing works
“[83], we use d = 7 , 9 , 11 so as not to have small-distance effects.”
MethodEDA and Quantum Computing: a symbiotic relationship?
Raghunathan, S., & Stok, L. (2020) · IEEE Design and Test · doi:10.1109/mdat.2020.3016586 · Cited by 6 · see citing works
“Also, error correction and fault-tolerant techniques are only possible when we achieve an error bound of 10 −3 or less [20], and we are orders of magnitude away from that bound.”
BackgroundHardness of Braided Quantum Circuit Optimization in the Surface Code
Wasa, K., Nishio, S., Suetsugu, K., Hanks, M., Stephens, A. M., Yokoi, Y., & Nemoto, K. (2023) · IEEE Transactions on Quantum Engineering · doi:10.1109/tqe.2023.3251358 · Cited by 6 · see citing works
“Topological stabilizer codes, such as surface codes [8], are good candidates for implementing FTQC due to their high thresholds and low-weight local stabilizers [9], [10].”
BackgroundSelf-citationGeneralized Quantum Data-Syndrome Codes and Belief Propagation Decoding for Phenomenological Noise
Kuo, K.-Y., & Lai, C. (2025) · IEEE Transactions on Information Theory · doi:10.1109/tit.2025.3529773 · Cited by 5 · see citing works
“This model, referred to as the phenomenological noise model [15], [36], [37], goes beyond traditional quantum coding theory.”
Quantum error correction with an Ising machine under circuit-level noise
Fujisaki, J., Maruyama, K., Oshima, H., Sato, S., Sakashita, T., Takeuchi, Y., & Fujii, K. (2023) · Physical Review Research · doi:10.1103/physrevresearch.5.043261 · Cited by 5 · see citing works
“For example, a two-qubit gate is implemented in a control-Z gate [37–39] or in a CNOT gate [40, 41].”
BackgroundMethodDecoding topological XYZ 2 codes with reinforcement learning based on attention mechanisms
Chen, Q.-H., Ji, Y., Wang, K.-H., Ma, H., & Ji, N. (2024) · Chinese Physics B · doi:10.1088/1674-1056/ad342b · Cited by 5 · see citing works
Cruikshank, B., & Jacobs, K. A. (2017) · Physical Review Letters · doi:10.1103/physrevlett.119.030503 · Cited by 6 · see citing works
A multilayered Pauli tracking architecture for lattice surgery‐based logical qubits
On, J., Kim, C., Oh, S., Lee, S., & Cha, G. (2022) · ETRI Journal · doi:10.4218/etrij.2022-0037 · Cited by 5 · see citing works
The role of quantum measurements in physical processes and protocols
Cruikshank, B., & Jacobs, K. (2017) · Quantum Science and Technology · doi:10.1088/2058-9565/aa6d3e · Cited by 5 · see citing works
Matrix Product Operator Simulations of Quantum Algorithms
Woolfe, K. J. (2015) · Minerva Access (University of Melbourne) · Cited by 6 · see citing works
Li, A.-Q., 田, C.-W. 策. T., Xu, X., Ma, H., & Liang, J. (2025) · Chinese Physics B · doi:10.1088/1674-1056/adab63 · Cited by 5 · see citing works
Surface code fidelity at finite temperatures
Novais, E., Stanforth, A. J., & Mucciolo, E. R. (2017) · Physical Review A · doi:10.1103/physreva.95.042339 · Cited by 4 · see citing works
Ji, Y., Chen, Q., Wang, R., Ji, N., & Ma, H. (2024) · Quantum Information Processing · doi:10.1007/s11128-024-04377-y · Cited by 4 · see citing works
“The threshold is the value of the physical error rate at the intersection of the physical error rate and the logical error rate for different code distances [47], where the physical error rate is the probability of error at each step in the heavy hexagonal code cycle.”
Effect of quantum error correction on detection-induced coherent errors
Yang, Q., & Liu, D. E. (2022) · Physical Review A · doi:10.1103/physreva.105.022434 · Cited by 5 · see citing works
Resource overheads and attainable rates for trapped-ion lattice surgery
Leone, H., Thinh, L. P., Srikara, S., & Devitt, S. J. (2025) · Physical Review Research · doi:10.1103/physrevresearch.7.023088 · Cited by 3 · see citing works
“The surface code [7] is a popular choice of stabilizer code because of its high error tolerance (between 0.1% and 1% for unbiased noise [8] and up to 43.7% for biased noise [9]) and because it can be implemented using only nearest-neighbor qubit interactions.”
InfluentialSurface code continuous quantum error correction using feedback
Nguyen, T., Hill, C. D., Hollenberg, L. C. L., & James, M. R. (2015) · IEEE Conference on Decision and Control (CDC) · doi:10.1109/cdc.2015.7403339 · Cited by 3 · see citing works
“In terms of the topological error correction, discrete error correction schemes were investigated by Fowler et al. [17] and Stephens [18], where an accuracy threshold was derived for the surface code.”
BackgroundRemarkable Prospect for Quantum-Dot-Coupled Tin Qubits in Silicon
Witzel, W. M., Lutz, J. J., & Luhman, D. R. (2022) · PRX Quantum · doi:10.1103/prxquantum.3.040320 · Cited by 3 · see citing works
“The error probability is below 10−6 for all of our HFI strengths beyond a modest 15 mT Bfield, well below known quantum-error-correction thresholds [70].”
BackgroundFinite-rate sparse quantum codes aplenty
Tremblay, M., Duclos-Cianci, G., & Kourtis, S. (2023) · Quantum · doi:10.22331/q-2023-04-20-985 · Cited by 3 · see citing works
“While codes like surface codes [10, 18, 30] and color codes [25, 26] are sufficient to encode and protect a small constant number of qubits, their performance degrades rapidly when performing computations with more qubits [12, 15].”
BackgroundNoise management to achieve superiority in quantum information systems
Nemoto, K., Devitt, S. J., & Munro, W. J. (2017) · Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences · doi:10.1098/rsta.2016.0236 · Cited by 3 · see citing works
(2018) · WORLD SCIENTIFIC eBooks · doi:10.1142/9789813238411_0015 · Cited by 3 · see citing works
Superconducting-qubit gates robust to parameter fluctuations
Wright, E. M., Damme, L. V., Glaser, N. J., Devra, A., Roy, F., Englhardt, J., Bruckmoser, N., Koch, L., Marx, A., Schirk, J., et al. (2026) · Physical Review Applied · doi:10.1103/tf39-n6s3 · Cited by 2 · see citing works
Channel polarization of two-dimensional-input quantum symmetric channels
Yi, Z., Liang, Z., & Wang, X. (2023) · Quantum Information Processing · doi:10.1007/s11128-023-03949-8 · Cited by 2 · see citing works
Ballistic Frenkel-exciton gate with a high on-off ratio
Sup, A., Yurke, B., & Elliott, R. (2024) · Physical Review A · doi:10.1103/physreva.110.022616 · Cited by 2 · see citing works
- Authors
- Stephens, A. M.
- Affiliations
- National Institute of Informatics (JP)
- Corresponding author
- Stephens, Ashley M.
- ORCID
- Stephens, Ashley M.
- Venue
- Physical Review A
- Publisher
- American Physical Society (APS)
- Published
- 2014-02-14
- Volume
- 89
- Issue
- 2
- ISSN
- 0556-2791 (ISSN-L)
- 1050-2947
- 1094-1622
- OpenAlex
- W2044503188
- Cited by
- 133
- References
- 52
- Citations by year
- 2026: 13 · 2025: 11 · 2024: 15 · 2023: 17 · 2022: 17 · 2021: 8 · 2020: 7 · 2019: 4 · 2018: 11 · 2017: 12 · 2016: 4 · 2015: 6 · 2014: 8
- FWCI
- 9.88
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Physical Review A (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Information and Cryptography
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum many-body systems
- Keywords
- surface code, fault-tolerant quantum computing, error threshold, quantum error correction, noise models, measurement circuit, decoding algorithms
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