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

TL;DR

This work leverages an exact mapping between a fully post-selected toric code under bit-flip noise and the clean two-dimensional Ising model on a torus to derive closed-form logical failure rates across all code distances and physical error rates. It identifies four regimes—path-counting, below-threshold, near-threshold, and above-threshold—each with distinct physical mechanisms described by domain-wall energetics, capillary-wave corrections, and finite-size scaling, and provides quantitative tools that extend to non-post-selected codes through an effective surface-tension framework. The results bridge statistical mechanics and quantum error correction, offering analytic benchmarks, data-collapse-inspired scaling forms, and physically interpretable parameters (e.g., σ_eff(p), δ(p)) for designing and benchmarking topological codes in practical regimes. The framework clarifies how disorder and post-selection shape decoding performance and supplies a principled basis for extrapolating to larger system sizes and more complex noise models, with potential impact on decoder development and resource estimates for fault-tolerant quantum computation.

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.

Figures (15)

• Figure 1: Decoding topological codes: A statistical mechanical perspective. (a) The action of a logical operator in a topological code can often be mapped to the insertion of non-trivial defects or domain walls in an equivalent statistical mechanical model (e.g., $\pm J$ random-bond Ising model for certain surface codes). (b) The performance of optimal decoding exhibits distinct regimes as a function of physical error rate $p$ and code distance $L$, mirroring the regimes of the associated statistical mechanical model: Path-Counting, Ordered, Critical, and Disordered. (c) Summary table providing the physical origin, functional behaviour of logical errors, and consequences for quantum error correction within each identified regime. These features are generic to topological codes whose decoding maps to classical statistical mechanics models.
• Figure 2: Domain wall behaviours across the four regimes of the fully post-selected toric code under bit-flip noise. (a) Path-counting regime ($p\ll p_{c}$): On a clean lattice at very low error rates, logical failure is dominated by the shortest possible domain walls, which remain essentially straight and unperturbed. (b) Below-threshold regime ($p<p_{c}$): Below-threshold, thermal capillary wave fluctuations impart roughening to an otherwise predominantly straight interface, yielding small-amplitude corrections on a straight wall. (c) Near-threshold regime ($p\approx p_{c}$): Near-threshold, scale invariance drives domain walls to develop fractal, self-similar structure, with fluctuations on all length scales reflecting critical behaviour. (d) Above-threshold regime ($p>p_{c}$): Above-threshold, thermal fluctuations dominate, fragmenting the system into a dense tangle of short, disconnected domain wall loops.
• Figure 3: Logical failure rates for a fully post-selected toric code are shown as a function of the bit-flip probability $p$ for various lattice sizes $L$. The solid lines represent the exact analytic curves obtained via a mapping to the 2D Ising model, while the scatter points denote the failure rates computed using a tensor network decoder with a truncation bond-dimension of $\chi=2^{8}$. The curves cross at the threshold error $p_c\simeq 0.29$ and logical failure probability $\mathbb{P}_{\mathrm{fail}}=1/2$ as predicted by the 2D Ising model solution.
• Figure 4: Logical failure rate $\mathbb{P}_{\mathrm{fail}}$ for a fully post-selected toric code under bit-flip noise in the path-counting regime. The failure rate is plotted as a function of the physical error rate $p$, with curves parametrized by the lattice size $L$. The dashed lines represent the path-counting approximation, while the solid lines show the exact analytic values. The shaded grey region indicates parameters where the path-counting regime is not expected to apply. Its boundary is obtained by substituting $L=1/p$ into the path-counting approximation (Eq. \ref{['eq:path_count_postselect']}), consistent with the validity scale $p\ll 1/L$ discussed in the main text.
• Figure 5: Comparison of the capillary wave approximation (dashed lines) and the exact analytic calculation (solid lines) of the logical failure rate for a fully post-selected toric code. Results are shown as a function of the physical error rate $p$ for various lattice sizes $L$, with each colour representing a different $L$. The grey shading marks parameters where the capillary-wave expression (Eq. \ref{['eq:belowthreshold']}) is not expected to be accurate. The boundary is obtained by imposing the near-threshold scale $\lvert p-p_c\rvert L=1$ (i.e., $\xi \sim L$; see Sec. \ref{['subsec:near-threshold-postselect']}) and, on the below-threshold side relevant here, substituting $p=p_c-1/L$ into Eq. \ref{['eq:belowthreshold']}; we shade the region above this boundary.