One of the central challenges in quantum error correction is determining the performance of a code in the low-error regimes needed to implement utility-scale computations. While performance at these error rates is not amenable to direct Monte Carlo simulation, it can be extrapolated from simulations at higher logical error rates, assuming the logical error rate scales predictably with increasing distance or decreasing physical error rate. However, the expected scaling depends sensitively on the minimum weight of uncorrectable error patterns. In many cases, the minimum weight is unknown since it depends not only on the theoretical code distance, but also on details of the implementation. Markov chain Monte Carlo (MCMC) methods, as adapted to quantum error correction by Bravyi and Vargo, provide a way to estimate logical failure rates in these low-error regimes via simulation. While offering significant gains over Monte Carlo, the described Metropolis algorithm makes small changes to the current logical failure patterns which results in slow convergence. In this paper, we argue that typical failure patterns include a large number of easily correctable errors that coexist alongside a malignant core. This observation motivates two new approaches to better evaluate code performance. First, we describe a pruning algorithm designed to obviate these correctable errors and focus on the problematic low-weight core. Second, we develop a novel family of Metropolis-Hastings algorithms, referred to as subregion MCMC. This technique is parameterized by the fraction of the error pattern that is resampled at each step, effectively interpolating between Monte Carlo and single step MCMC. We show that a judicious choice of this parameter results in far faster convergence than prior work.
One of the central challenges in quantum error correction is determining the performance of a code in the low-error regimes needed to implement utility-scale computations. While performance at these error rates is not amenable to direct Monte Carlo simulation, it can be extrapolated from simulations at higher logical error rates, assuming the logical error rate scales predictably with increasing distance or decreasing physical error rate. However, the expected scaling depends sensitively on the minimum weight of uncorrectable error patterns. In many cases, the minimum weight is unknown since it depends not only on the theoretical code distance, but also on details of the implementation. Markov chain Monte Carlo (MCMC) methods, as adapted to quantum error correction by Bravyi and Vargo, provide a way to estimate logical failure rates in these low-error regimes via simulation. While offering significant gains over Monte Carlo, the described Metropolis algorithm makes small changes to the current logical failure patterns which results in slow convergence. In this paper, we argue that typical failure patterns include a large number of easily correctable errors that coexist alongside a malignant core. This observation motivates two new approaches to better evaluate code performance. First, we describe a pruning algorithm designed to obviate these correctable errors and focus on the problematic low-weight core. Second, we develop a novel family of Metropolis-Hastings algorithms, referred to as subregion MCMC. This technique is parameterized by the fraction of the error pattern that is resampled at each step, effectively interpolating between Monte Carlo and single step MCMC. We show that a judicious choice of this parameter results in far faster convergence than prior work.
The ultimate goal of quantum error correction is to create logical qubits with very low error rates (e.g. 1e-12) and assemble them into large-scale quantum computers capable of performing many (e.g. billions) of logical gates on many (e.g. thousands) of logical qubits. However, it is necessarily difficult to directly assess the performance of such high-quality logical qubits using standard Monte Carlo sampling because logical failure events become very rare. Building on existing approaches to this problem, we develop three complementary techniques to characterize the rare-event regime for general quantum low-density parity-check (qLDPC) codes under circuit noise. (I) We propose a well-motivated, low-parameter ansatz for the failure spectrum (the fraction of fault sets of each size that fail) that empirically fits all the QEC systems we studied and predicts logical error rates at all physical error rates. (II) We find min-weight logical operators of syndrome measurement circuits and exactly compute the number of min-weight failing configurations. (III) We generalize the splitting method to qLDPC codes using multi-seeded Metropolis sampling to improve convergence for systems with many inequivalent logical operators. We apply these tools to distance-6, -12, and -18 bivariate bicycle codes under circuit noise, observing strong low-error-rate performance with the recently proposed Relay decoder but also considerable scope for further improvement.
The splitting method estimates logical error rates of quantum error-correcting codes deep in the rare-event regime, where direct Monte Carlo sampling is prohibitively expensive. It relies on a Markov chain Monte Carlo (MCMC) sampler of faulty configurations, usually driven by one-bit-flip proposals. However, one-bit flips have been reported to be trapped on some codes, which biases the estimate. To address this issue, we propose a new kernel that combines locally balanced one-bit-flip proposals with a cluster-exchange move. We benchmark both kernels and direct Monte Carlo on twelve codes under code-capacity bit-flip noise in two ways, by comparing the mean fault weight with an MCMC-independent reference and by measuring the computation time to reach a fixed accuracy. In the first, our kernel agrees with the reference on every code, including those where one-bit flips are trapped. In the second, it is faster than one-bit flips wherever they are trapped, by orders of magnitude on some codes. Furthermore, at the lowest error rate, it is faster than direct Monte Carlo on every code, by more than four orders of magnitude on some.
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.
We describe a practical approach for accessing the logical failure rates of quantum error-correcting (QEC) circuits under low physical (component) failure rate regimes. Standard Monte Carlo is often the de facto approach for studying the failure rates of quantum circuits. However, in the study of fault-tolerant error-correcting circuits, the ability to extend this approach to low physical failure rates is limited. In particular, the use of Monte Carlo to access circuits that are relatively large or have high correcting power becomes more difficult as we lower the input failure rates of the individual components (gates) in the circuit. For these reasons, many simulations studying the circuit model go no lower than end-to-end logical failure rates in the 10 − 6 regime.In this report, we outline an approach that borrows from earlier work by Bravyi and Vargo \cite{bv13} to the more complex circuit noise model. Earlier works studied both the capacity and phenomenological noise models \cite{bv13,Beverland2019,bonilla2021xzzx}, but the work is insufficient for generating similar simulations in the circuit-noise model. To the best of our knowledge, our team is the first to develop a full prescription of the rare event simulation by splitting technique for the circuit-based noise model. We have also generated promising results that are confirmed by standard Monte Carlo simulation under an accessible regime. This work shows that we can access noise in the circuit-model prescription of quantum error-correcting code to failure rates below 10 − 20 regime.
Quantum error correction (QEC) enables reliable computation on noisy hardware by encoding logical information across many physical qubits and periodically measuring parities to detect errors. A decoder is the classical algorithm that uses these measurements to infer which error most likely occurred, so that the system can correct it. The decoder's accuracy-how rarely it makes the wrong guess-directly determines the scale of quantum computation that can be reliably executed. With a wealth of competing decoding algorithms, a QEC system designer needs reliable methods to evaluate them. Today, the dominant approach is to evaluate decoders using Monte Carlo simulation. However, simulation has several drawbacks such as requiring many samples to produce low variance estimates.
In this work, we develop a new systematic analysis for evaluating decoders. We introduce a novel formal semantics of a core language for QEC programs that captures the de facto standard Stim circuit format, providing a principled theoretical foundation for the emerging space of fault-tolerant quantum systems design. Given a QEC program and a decoder, our verifier can quantify both the decoder accuracy and the decoder robustness to drift in physical error rate. Our approach has two key components: (i) a structured search over the space of possible errors; and (ii) a constrained polynomial optimization kernel. A thorough empirical evaluation of our approach suggests that it can outperform simulation, especially in low error rate regimes, and that it can be deployed to quantify decoder robustness over an interval of physical error rates.
Quantum computers have the potential to solve certain problems exponentially faster than classical computers. However, realizing this potential in practice is a major challenge, as it requires precise control over large-scale quantum systems operating at extremely low error rates. Quantum error correction (QEC) provides a path to achieving such error rates, but its substantial resource overhead poses a significant practical barrier. In addition, identifying which computational problems offer a provable quantum advantage—and which remain fundamentally intractable—remains an open question. This thesis presents progress towards addressing both of these challenges. The first part of this thesis describes advances that substantially reduce the resource overhead of QEC. We begin by presenting realizations of logical circuits in dynamically reconfigurable arrays of neutral atoms. By jointly decoding the logical qubits, we reduce the cost of implementing such transversal Clifford circuits by a factor proportional to the code distance. We then develop new theories of fault tolerance to extend these savings to universal quantum computation with magic state inputs. Finally, we introduce techniques for fast correlated decoding, enabling practical implementations of these improvements in experimental hardware. Collectively, these advances accelerate progress toward large-scale computation and reduce the cost of QEC by over an order of magnitude. The second part of the thesis explores the relative power of quantum and classical computation in combinatorial optimization, a class of problems that is ubiquitous in science and engineering and foundational to the theory of computational complexity. We experimentally implement the optimized quantum adiabatic algorithm (QAA) in neutral atom arrays and observe evidence of a superlinear speedup over classical simulated annealing on certain hard problem instances. To interpret these results, we develop a theoretical framework to compare the performance of QAA with a broad class of classical Markov chain Monte Carlo algorithms. We identify conditions under which a quantum quadratic speedup is achievable and propose modifications to the QAA to reliably realize this advantage. Together, these contributions advance both the practical implementation and theoretical understanding of quantum algorithms.