Repetition code forms a fundamental basis for quantum error correction experiments. To date, it stands as the sole code that has achieved large distances and extremely low error rates. Its applications span the spectrum of evaluating hardware limitations, pinpointing hardware defects, and detecting rare events. However, current methods for decoding repetition codes under circuit level noise are suboptimal, leading to inaccurate error correction thresholds and introducing additional errors in event detection. In this work, we establish that repetition code under circuit level noise has an exact solution, and we propose an optimal maximum likelihood decoding algorithm called planar. The 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 with 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.
Repetition code forms a fundamental basis for quantum error correction experiments. To date, it stands as the sole code that has achieved large distances and extremely low error rates. Its applications span the spectrum of evaluating hardware limitations, pinpointing hardware defects, and detecting rare events. However, current methods for decoding repetition codes under circuit level noise are suboptimal, leading to inaccurate error correction thresholds and introducing additional errors in event detection. In this work, we establish that repetition code under circuit level noise has an exact solution, and we propose an optimal maximum likelihood decoding algorithm called planar. The 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 with 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.
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.
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.
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.
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.
Abstract Quantum error correction is essential for realizing fault-tolerant quantum computing, where both the efficiency and accuracy of the decoding algorithms play critical roles. In this work, we introduce the implementation of the Planar algorithm, a software framework designed for fast and exact decoding of quantum codes with a planar structure. The algorithm first converts the optimal decoding of quantum codes into a partition function computation problem of an Ising spin glass model. Then it utilizes the exact Kac–Ward formula to solve it. In this way, Planar offers the exact maximum likelihood decoding in polynomial complexity for quantum codes with a planar structure, including the surface code with independent code-capacity noise and the quantum repetition code with circuit-level noise. Unlike traditional minimum-weight decoders such as minimum-weight perfect matching (MWPM), Planar achieves theoretically optimal performance while maintaining polynomial-time efficiency. In addition, to demonstrate its capabilities, we exemplify the implementation using the rotated surface code, a commonly used quantum error correction code with a planar structure, and show that Planar achieves a threshold of approximately p uc ≈ 0.109 under the depolarizing error model, with a time complexity scaling of O(N 0.69), where N is the number of spins in the Ising model.
Quantum error correction is essential to achieve fault-tolerant quantum computation in the presence of noisy qubits. Among the most promising approaches to quantum error correction is the surface code, thanks to a scalable two-dimensional architecture, only nearest-neighbor interactions, and a high error threshold. Decoding the surface code, i.e. finding the most likely error chain given a syndrome measurement outcome is a computationally complex task. Traditional decoders rely on classical algorithms, which, especially for larger systems, can be slow and may not always converge to the optimal solution. This thesis presents a novel approach to decoding the surface code using graph neural networks. By mapping the syndrome measurements to a graph and performing graph classification, we find that the graph neural networks can predict the most likely error configuration with high accuracy. Our results show that the GNN-based decoder outperforms the classic minimum weight perfect matching (MWPM) decoder in terms of accuracy. With a phenomenological noise model with depolarizing noise and perfect syndrome measurements, our networks beat MWPM up to code-size 15 across all relevant error rates. Furthermore, the GNN is capable of surpassing MWPM under circuit-level noise up to code size 7. We also show that training the network on repetition code data from a recent experiment [Google Quantum AI, Nature 614, 676 (2023)] produces per-step error rates comparable to those achieved with a matching decoder specifically adapted to the error rates of the physical qubits. This indicates that graph neural network decoders are capable of learning the underlying error distribution on the qubits. Our findings advance the field of quantum error correction and provide a promising new direction for the development of efficient decoding algorithms.