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arXiv:2512.10399 · PRX Quantum

Ising on the donut: Regimes of topological quantum error correction from statistical mechanics

Lucas H. English, Sam Roberts, Stephen D. Bartlett, Andrew C. Doherty, Dominic J. Williamson

quant-phSubmitted Dec 11, 2025• Announced Dec 12, 2025• 0 citations
v1 · Dec 11, 2025v2 · Sep 15, 2026
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Abstract

Utility-scale quantum computers require quantum error correcting codes with large numbers of physical qubits to achieve sufficiently low logical error rates. The performance of quantum error correction (QEC) is generally predicted through large-scale numerical simulations, used to estimate thresholds, finite-size scaling, and exponential suppression of logical errors below threshold. The connection of QEC to models from statistical mechanics provides an alternative tool for analysing QEC performance. However, predicting the behaviour of these models also requires large-scale numerical simulations, as analytic solutions are not generally known. Here we exploit an exact mapping, from a toric code under bit-flip noise that is post-selected on being syndrome free to the exactly-solvable two-dimensional Ising model on a torus, to derive an analytic solution for the logical failure rate across its full domain of physical error rates. In particular, this mapping provides closed-form expressions for the logical failure rate in four distinct regimes: the path-counting, below-threshold (ordered), near-threshold (critical), and above-threshold (disordered) regimes. Our framework places a number of familiar and long-standing numerical observations on firm theoretical ground. It also motivates explicit ansätze for the conventional QEC setting of non-post-selected codes whose statistical mechanics mappings involve random-bond disorder. Specifically, we introduce an effective surface tension model for the below-threshold regime, and a new scaling ansatz for the near-threshold regime, derived from an analysis of the ground state energy cost distribution. By bridging statistical mechanics theory and quantum error correction practice, our results offer a new toolkit for designing, benchmarking, and understanding topological codes beyond current computational limits.

Journal reference
PRX Quantum 7, 043004 (2026)
DOI
10.1103/4wvz-1p8m
Comments
29 pages, comments welcome

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Utility-scale quantum computers require quantum error correcting codes with large numbers of physical qubits to achieve sufficiently low logical error rates. The performance of quantum error correction (QEC) is generally predicted through large-scale numerical simulations, used to estimate thresholds, finite-size scaling, and exponential suppression of logical errors below threshold. The connection of QEC to models from statistical mechanics provides an alternative tool for analysing QEC performance. However, predicting the behaviour of these models also requires large-scale numerical simulations, as analytic solutions are not generally known. Here we exploit an exact mapping, from a toric code under bit-flip noise that is post-selected on being syndrome free to the exactly-solvable two-dimensional Ising model on a torus, to derive an analytic solution for the logical failure rate across its full domain of physical error rates. In particular, this mapping provides closed-form expressions for the logical failure rate in four distinct regimes: the path-counting, below-threshold (ordered), near-threshold (critical), and above-threshold (disordered) regimes. Our framework places a number of familiar and long-standing numerical observations on firm theoretical ground. It also motivates explicit ansätze for the conventional QEC setting of non-post-selected codes whose statistical mechanics mappings involve random-bond disorder. Specifically, we introduce an effective surface tension model for the below-threshold regime, and a new scaling ansatz for the near-threshold regime, derived from an analysis of the ground state energy cost distribution. By bridging statistical mechanics theory and quantum error correction practice, our results offer a new toolkit for designing, benchmarking, and understanding topological codes beyond current computational limits.
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The ability of quantum error correction (QEC) to protect logical qubits against physical noise is traditionally quantified by a noise threshold: a critical noise level below which the logical error rate decreases exponentially with code distance. This paper proposes that QEC itself can undergo a phase transition analogous to a physical phase change. We develop a theoretical framework that treats the logical error rate as an order parameter and maps the behavior of large stabilizer codes onto a statistical mechanics model. By analyzing the scaling of logical error probabilities with code distance and noise strength, we identify a continuous phase transition at the threshold, characterized by critical exponents that depend on the code family and noise model. The transition separates a "fault‑tolerant" phase where logical errors are exponentially suppressed from a "fault‑unreliable" phase where logical errors proliferate. We discuss implications for the design of large‑scale quantum computers, potential experimental signatures in current quantum hardware, and the prospect of discovering new quantum information physics through this phase‑transition perspective.
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Realizing the potential of quantum computing will require achieving sufficiently low logical error rates. Many applications call for error rates in the 10−1510^{-15}10−15 regime, but state-of-the-art quantum platforms typically have physical error rates near 10−310^{-3}10−3. Quantum error correction (QEC) promises to bridge this divide by distributing quantum logical information across many physical qubits so that errors can be detected and corrected. Logical errors are then exponentially suppressed as the number of physical qubits grows, provided that the physical error rates are below a certain threshold. QEC also requires that the errors are local and that performance is maintained over many rounds of error correction, two major outstanding experimental challenges. Here, we implement 1D repetition codes embedded in a 2D grid of superconducting qubits which demonstrate exponential suppression of bit or phase-flip errors, reducing logical error per round by more than 100×100\times100× when increasing the number of qubits from 5 to 21. Crucially, this error suppression is stable over 50 rounds of error correction. We also introduce a method for analyzing error correlations with high precision, and characterize the locality of errors in a device performing QEC for the first time. Finally, we perform error detection using a small 2D surface code logical qubit on the same device, and show that the results from both 1D and 2D codes agree with numerical simulations using a simple depolarizing error model. These findings demonstrate that superconducting qubits are on a viable path towards fault tolerant quantum computing.
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