Accurate noise estimation is essential for fault-tolerant quantum computing, since decoding per formance depends critically on the fidelity of circuit-level noise parameters. We introduce a differ entiable Maximum Likelihood Estimation (dMLE) framework that learns these parameters directly from experimental syndrome data using likelihood gradients, with no decoder in the training loop. Mapping syndrome likelihoods to statistical-mechanical models enables exact and efficient evalua tion via planar solvers for repetition codes and tensor networks (TNs) for surface codes, extending the exact treatment to code distance d = 5 with a small number of rounds, and, via spatial subsam pling combined with time-translation symmetry, to larger codes and larger round counts. Across four experimental superconducting datasets, dMLE reduces logical error rates by up to 30.6% for repetition codes. On the Sycamore d = 3 surface code, a TN decoder using the dMLE prior attains a logical error rate comparable to that reported for state-of-the-art AlphaQubit, using standard syndromes alone without auxiliary leakage data. On the larger d = 5 surface code the dMLE prior lowers the logical error rate of the TN and Tesseract decoders by 7.18% and 5.23% relative to a prior optimized by reinforcement learning; at d = 7 it lowers that of Tesseract by 4.56% and 4.83% in the X and Z bases.
Accurate noise estimation is essential for fault-tolerant quantum computing, since decoding per formance depends critically on the fidelity of circuit-level noise parameters. We introduce a differ entiable Maximum Likelihood Estimation (dMLE) framework that learns these parameters directly from experimental syndrome data using likelihood gradients, with no decoder in the training loop. Mapping syndrome likelihoods to statistical-mechanical models enables exact and efficient evalua tion via planar solvers for repetition codes and tensor networks (TNs) for surface codes, extending the exact treatment to code distance d = 5 with a small number of rounds, and, via spatial subsam pling combined with time-translation symmetry, to larger codes and larger round counts. Across four experimental superconducting datasets, dMLE reduces logical error rates by up to 30.6% for repetition codes. On the Sycamore d = 3 surface code, a TN decoder using the dMLE prior attains a logical error rate comparable to that reported for state-of-the-art AlphaQubit, using standard syndromes alone without auxiliary leakage data. On the larger d = 5 surface code the dMLE prior lowers the logical error rate of the TN and Tesseract decoders by 7.18% and 5.23% relative to a prior optimized by reinforcement learning; at d = 7 it lowers that of Tesseract by 4.56% and 4.83% in the X and Z bases.
Accurate noise models are essential for high-performance quantum error correction, yet characterizing the noise of a quantum device typically requires dedicated experiments. We present a variational framework that learns the noise model directly from quantum-error-correction syndrome and logical-observable data collected during error-corrected memory experiments. The fault-event probabilities are treated as variational parameters and optimized via gradient descent to minimize the binary cross-entropy between the decoder's predictions and experimental logical-observable outcomes. We prove that this objective is principled rather than ad hoc: a sufficiently expressive noise ansatz attains the information-theoretic minimum logical error rate. We instantiate this framework using a tensor-network decoder, which provides exact maximum-likelihood decoding and analytically differentiable gradients with respect to all noise parameters. Using circuit-level data from Google's Sycamore processor, and starting from an uninformed prior, the optimization recovers noise models whose logical error rates agree to within 2% with those of Google's independently characterized detector error model. The mean squared error between the learned and reference noise parameters shows a clear overall decrease throughout training, confirming that the method recovers physically meaningful noise structure, not merely parameters that happen to decode well. We further demonstrate that the optimization can track device drifts in real time via warm-started updates, maintaining near-optimal decoding performance under synthetically evolving noise without the need for re-characterization. The approach is decoder-agnostic in its formulation and naturally extends to correlated noise models.
Quantum error correction (QEC) is indispensable for realizing fault-tolerant quantum computation, yet its effectiveness hinges critically on the classical decoding algorithm that interprets noisy syndrome measurements. Among all possible decoding strategies, maximum likelihood decoding (MLD) is provably optimal, since it identifies the logical group with largest likelihood by summing over all possible errors within logical class consistent with the observed syndrome. Despite its optimality, MLD is computationally intractable in general (#P-hard), motivating a rich landscape of exact and approximate algorithms. In this topical review, we provide a unified perspective on MLD by surveying recent advances through three complementary lenses: statistical mechanics, tensor networks, and artificial intelligence. From the statistical mechanics viewpoint, the MLD problem maps onto evaluating partition functions of disordered spin models, enabling exact solutions for certain codes and noise models as well as threshold estimation via phase-transition analysis. From the tensor network perspective, approximate contraction of tensor networks on the code's factor graph yields decoders that closely approach MLD accuracy with polynomial computational cost. From the artificial intelligence perspective, neural-network-based decoders, including autoregressive generative models and recurrent transformers, learn to approximate the MLD distribution from data, achieving high accuracy with the parallelism afforded by modern hardware accelerators. We discuss the connections among these three approaches, review their application to both simulated and experimental quantum hardware, and outline open challenges including real-time decoding, scalability to large code distances, and generalization to high-rate quantum low-density parity-check codes.
We study the exact maximum-likelihood decoding of quantum error-correcting codes. Our primary focus is on the repetition code under circuit-level noise, which serves as a fundamental basis for quantum error correction experiments and is the only code that has achieved large distances and extremely low error rates. We uncover that the repetition code under circuit-level noise has an exact solution and propose an optimal maximum-likelihood decoding algorithm called "planar." This algorithm is based on the exact solution of the spin-glass partition function on planar graphs and has polynomial computational complexity. Through extensive numerical experiments, we demonstrate that our algorithm uncovers the exact threshold for depolarizing noise and realistic superconductor SI1000 noise. Furthermore, we apply our method to analyze data from recent quantum memory experiments conducted by Google Quantum AI, revealing that part of the error floor was attributed to the decoding algorithm used by Google. Finally, we implemented the repetition code quantum memory on superconducting systems using a 72-qubit quantum chip lacking reset gates, demonstrating that even with an unknown error model, the proposed algorithm achieves a significantly lower logical error rate than the matching-based algorithm. We also discuss the application of our approach to quantum codes whose maximum-likelihood decoding problem corresponds to a spin-glass problem, including the surface code under independent code-capacity noise.
Accurate and efficient decoding is a crucial component for achieving fault-tolerant quantum computing. Realistic circuit-level noise introduces temporal correlations and degeneracy, making optimal (maximum-likelihood) decoding computationally intractable in general. As a result, practical decoders rely on heuristic approximations, and it is generally difficult to quantify how suboptimal they are, as this strongly depends on the code and noise model considered. In this work, we study the accuracy of practical decoding algorithms under circuit-level noise by comparing them against a maximum likelihood decoding benchmark. Our approach propagates the density matrix through the full memory experiment and computes the optimal decoding decision for each syndrome history. We introduce pruning techniques with rigorous bounds, allowing us to access larger numbers of syndrome-extraction rounds. We apply this framework to small instances of the repetition code and a cellular automaton code, and benchmark minimum-weight perfect matching (MWPM), belief propagation with ordered statistics decoding (BP+OSD), Tesseract, and Planar decoders against optimal decoding. While standard decoders remain close to optimal for the repetition code, we find significant deviations for the cellular automaton code, with BP+OSD deteriorating already in experimentally relevant noise regimes. Moreover, the pruning method developed here highlights that, at low physical error rates, only a narrow fraction of syndrome histories contributes significantly to the logical error rate.
Fault-tolerant quantum computation critically depends on the ability to detect and correct physical qubit errors faster than they accumulate. The surface code (SC) has emerged as the most experimentally promising quantum error correction (QEC) framework due to its high threshold and local stabilizer structure, yet its practical performance is bottlenecked by decoder efficiency. Traditional decoding approaches such as minimum-weight perfect matching (MWPM) achieve near-optimal accuracy under independent noise but struggle with correlated errors, circuit-level noise, and real-time latency constraints. This paper investigates how neural network (NN)-based decoders reduce logical error rates in SC quantum systems by learning complex syndrome-to-correction mappings from data. We propose a hybrid convolutional-recurrent neural architecture that processes stabilizer measurement syndromes across both spatial and temporal dimensions, enabling the decoder to capture correlated error patterns inherent in realistic hardware. Systematic simulations under depolarizing, biased, and circuit-level noise models demonstrate that the proposed neural decoder achieves a logical error rate reduction of up to 41.3% compared to MWPM at a physical error rate of p = 0.008, while maintaining sub-millisecond decoding latency compatible with superconducting qubit cycle times. These results establish that data-driven neural decoders represent a scalable and hardware-compatible pathway toward achieving the logical error rates required for practical fault-tolerant quantum computing.
Fault-tolerant quantum computers use decoders to monitor for errors and find a plausible correction. A decoder may provide a \emph{decoder confidence score} (DCS) to gauge its success. We adopt a \emph{swim distance} DCS, computed from the shortest path between syndrome clusters. By contracting tensor networks, we compare its performance under phenomenological noise to the well-known \emph{complementary gap} and find that both reliably estimate the logical error probability (LEP) in a decoding window. We explore ways to use this to mitigate the LEP in entire logical circuits. For shallow circuits, we just abort if any decoding window produces an exceptionally low DCS: for one window of a distance-13 surface code under circuit-level noise, rejecting a mere 0.1% of possible DCS values improves the window's mean LEP by more than 5 orders of magnitude. For larger algorithms comprising up to billions of windows, DCS-based rejection remains effective for enhancing observable estimation. Moreover, one can use the DCS to assign each circuit's output a unique LEP, and use it as a basis for maximum likelihood estimation. This can reduce the effects of noise by an order of magnitude at no quantum cost; we explore the combination of these methods.