We identify regimes where postselection can be used scalably in quantum error correction (QEC) to improve performance. We use statistical mechanical models to analytically quantify the performance and thresholds of postselected QEC, with a focus on the surface code. Based on the nonequilibrium magnetization of these models, we identify a simple heuristic technique for postselection that determines whether to abort without requiring a decoder. Along with performance gains, this heuristic allows us to derive analytic expressions for postselected conditional logical thresholds and abort thresholds of surface codes. We find that such postselected QEC is characterized by four distinct thermodynamic phases and detail the implications of this phase space for practical, scalable quantum computation.
We identify regimes where postselection can be used scalably in quantum error correction (QEC) to improve performance. We use statistical mechanical models to analytically quantify the performance and thresholds of postselected QEC, with a focus on the surface code. Based on the nonequilibrium magnetization of these models, we identify a simple heuristic technique for postselection that determines whether to abort without requiring a decoder. Along with performance gains, this heuristic allows us to derive analytic expressions for postselected conditional logical thresholds and abort thresholds of surface codes. We find that such postselected QEC is characterized by four distinct thermodynamic phases and detail the implications of this phase space for practical, scalable quantum computation.
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 ansatze for the conventional QEC setting of non-post-selected codes whose statistical mechanics mappings involve random-bond disorder. Specifically, we introduce an effective tension model for the below-threshold regime, and a new scaling ansatz for the near-threshold regime, derived from an analysis of the domain wall energy cost distributions. 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.
Quantum error correcting codes can enable large quantum computations provided physical error rates are sufficiently low. We combine post-selection with surface code error correction through the use of exclusive decoders, which abort on decoding instances that are deemed too difficult. For the most discriminating of exclusive decoders, we demonstrate a threshold of 50% under depolarizing noise (or 32(1)% for the fault-tolerant case), and up to a quadratic improvement in logical failure rates below threshold. Furthermore, with a modest exclusion criterion, we identify a regime at low error rates where the exclusion rate decays with code distance, providing a pathway for scalable and time-efficient quantum computing with post-selection. Our exclusive decoder applied to magic state distillation yields a 75% reduction in the number of physical qubits, and a 60% reduction in the total spacetime volume, including accounting for repetitions. Other applications include error mitigation, and in concatenated schemes. Quantum error correction produces an enormous amount of data about the quantum system, including information about whether an uncorrectable error is likely. In this work the authors analyse a new decoder that can abort when decoding is deemed too difficult, yielding improved performance overall.
The large overhead imposed by quantum error correction is a critical challenge to the realization of quantum computers, and motivates searching for alternative error correcting codes and fault-tolerant circuit constructions. Postselection is a powerful tool that builds large programs out of probabilistically generated sub-circuits, and has been shown to increase the threshold of quantum error correction based on fusing fixed-size resource states or concatenated codes. In this work, we present an approach to lower the overhead of quantum computing using scalable postselection, based on directly postselecting sub-circuits with a size extensive in the code distance using decoder soft information. We introduce a metric, the partial gap, that estimates what the logical gap of a resource state will be after it is consumed, and show that postselection based on the partial gap leads to scalable improvements in the logical error rate. In the specific context of implementing logical gates via teleportation through a cluster state, we demonstrate that scalable postselection provides a 4× reduction in the overhead per logical gate, at the same logical error probability.
Decoding stabilizer codes such as the surface and toric codes involves evaluating free-energy differences in a disordered statistical mechanics model, in which the randomness comes from the observed pattern of error syndromes. We study the statistical distribution of logical failure rates across observed syndromes in the toric code, and show that, within the coding phase, logical failures are predominantly caused by exponentially unlikely syndromes. Therefore, postselecting on not seeing these exponentially unlikely syndrome patterns offers a scalable accuracy gain. In general, the logical error rate can be suppressed from pf to pfb, where b≥2 in general; in the specific case of the toric code with perfect syndrome measurements, we find numerically that b=3.1(1). Our arguments apply to general topological stabilizer codes, and can be extended to more general settings as long as the decoding failure probability obeys a large deviation principle.
As quantum computers remain susceptible to noise, QEC is essential for preserving logical information during computations. However, the performance of QEC codes breaks down beyond certain noise thresholds, revealing fundamental limits on their ability to protect quantum information. These limits can be characterized using information-theoretic measures such as the coherent information, which quantifies the maximum rate at which logical information can be reliably transmitted through a noisy quantum channel. In this work, we establish a direct connection between the CI and the binary cross-entropy loss used when training neural network decoders. Specifically, we show that the CI constitutes a sharp lower bound on the achievable loss for decoders that track logical operators across noisy channels. To this end, we develop a transformer-based neural network model based on maximum likelihood decoding. We train this network to estimate the CI and evaluate its performance on the surface code under three noise models: code capacity, phenomenological, and circuit-level noise. Our results demonstrate that the network accurately predicts CI and yields threshold estimates that closely match known theoretical limits. When used as a decoder, the network significantly outperforms the minimum weight perfect matching decoder in terms of logical error rate. We also introduce a novel soft post-selection scheme that independently treats uncertainty in both logical operators and relies on confidence-based filtering of the network's output. We prove that such post-selection strategies, based on the MLD cosets, are optimal, and demonstrate their scalability in terms of both logical error rate and abort probability. These findings establish transformer-based architectures as powerful tools for QEC and provide the first numerical evidence supporting the optimality and scalability of MLD-based post-selection.
Quantum error correction (QEC) is the cornerstone of fault‑tolerant quantum computation, enabling logical qubits to be protected against physical noise by encoding them into highly entangled states. In this paper we propose that the very ability of a QEC code to protect against errors is itself subject to a phase transition. We formulate a theory of the Quantum Error Correction Phase Transition (QECP), in which the system undergoes a sharp change from a regime of successful error suppression to a regime where logical errors proliferate, as a control parameter – such as the physical error rate, the code distance, or the connectivity of the underlying hardware – is varied. Drawing analogies to conventional symmetry‑breaking and topological phase transitions, we construct a statistical mechanical model that captures the competition between error‑correction resources and noise. We analyze the critical behavior, derive scaling relations, and discuss implications for the design of scalable quantum architectures. The existence of a QECP provides a new physical lens through which to view quantum information processing and suggests that fault‑tolerance thresholds may be understood as critical points in an extended parameter space.