The maximum likelihood decoder based on tensor networks has proven highly successful for the 2D surface code, achieving the optimal decoding success rate. However, existing tensor network decoders are typically designed for independent single-qubit error models, and their performance under inter-qubit correlated error models remains unexplored. This is due to two major challenges. The first challenge lies in constructing the tensor network for correlated errors, since the same final Pauli error can arise from many different combinations of independent and correlated errors, preventing a direct factorization of the error probability. The second challenge is that even after a tensor network is constructed, it generally contains huge-dimensional tensors and is therefore not efficiently contractible. In this work, to address the first difficulty, we introduce additional binary indices and two transformations to construct a multi-index tensor network for maximum-likelihood decoding with correlated errors. To address the second difficulty, we use reparametrization, elimination, and index classification to decompose the huge-dimensional tensors into lower-dimensional tensors. This yields an efficiently contractible tensor network for error models satisfying the tractability conditions derived in this work. We perform numerical simulations for a representative correlated error model and show that the maximum-likelihood decoder implemented with our multi-index tensor network construction achieves a higher finite-size threshold than the widely used MWPM decoder.
The maximum likelihood decoder based on tensor networks has proven highly successful for the 2D surface code, achieving the optimal decoding success rate. However, existing tensor network decoders are typically designed for independent single-qubit error models, and their performance under inter-qubit correlated error models remains unexplored. This is due to two major challenges. The first challenge lies in constructing the tensor network for correlated errors, since the same final Pauli error can arise from many different combinations of independent and correlated errors, preventing a direct factorization of the error probability. The second challenge is that even after a tensor network is constructed, it generally contains huge-dimensional tensors and is therefore not efficiently contractible. In this work, to address the first difficulty, we introduce additional binary indices and two transformations to construct a multi-index tensor network for maximum-likelihood decoding with correlated errors. To address the second difficulty, we use reparametrization, elimination, and index classification to decompose the huge-dimensional tensors into lower-dimensional tensors. This yields an efficiently contractible tensor network for error models satisfying the tractability conditions derived in this work. We perform numerical simulations for a representative correlated error model and show that the maximum-likelihood decoder implemented with our multi-index tensor network construction achieves a higher finite-size threshold than the widely used MWPM decoder.
In this work we develop a general tensor network decoder for 2D codes. Specifically, we propose a decoder that approximates maximally likelihood decoding for 2D stabiliser and subsystem codes subject to Pauli noise. For a code consisting of n qubits our decoder has a runtime of O(nlogn+nχ3), where χ is an approximation parameter. We numerically demonstrate the power of this decoder by studying four classes of codes under three noise models, namely regular surface codes, irregular surface codes, subsystem surface codes and colour codes, under bit-flip, phase-flip and depolarising noise. We show that the thresholds yielded by our decoder are state-of-the-art, and numerically consistent with optimal thresholds where available, suggesting that the tensor network decoder well approximates optimal decoding in all these cases. Novel to our decoder is an efficient and effective approximate contraction scheme for arbitrary 2D tensor networks, which may be of independent interest. We have also released an implementation of this algorithm as a stand-alone Julia package: SweepContractor.jl.
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.
Quantum error correction (QEC) is essential for fault-tolerant quantum computation. Often in QEC errors are assumed to be independent and identically distributed and can be discretised to a random Pauli error during the execution of a quantum circuit. In real devices, however, the noise profile is much more complex and contains non-trivial spatiotemporal correlations, such as cross-talk, non-Markovianity, and their mixtures. Here, we examine the performance of two paradigmatic QEC codes in the presence of complex noise by using process tensors to represent spatiotemporal correlations beyond iid errors. This integration is an instance of the recently proposed \textit{strategic code}, which combines QEC with process tensors. In particular, we construct the maximum likelihood (ML) decoder for a quantum error correction code with a process tensor. To understand the computational overhead and implications of this approach, we implement our framework numerically for small code instances and evaluate its performance. We also propose a method to evaluate the performance of strategic codes and construct the ML decoder with an efficient tensor network approximation. Our results highlight the possible detrimental effects of correlated noise and potential pathways for designing decoders that account for such effects.
Decoding algorithms based on approximate tensor-network (TN) contraction have proven tremendously successful in decoding two-dimensional (2D) local quantum codes such as toric or surface codes and color codes, effectively achieving optimal decoding accuracy. In this work, we introduce several techniques to generalize TN decoding to higher dimensions so that it can be applied to three-dimensional (3D) codes as well as 2D codes with noisy syndrome measurements (phenomenological noise or circuit-level noise). The 3D case is significantly more challenging than 2D, as the involved approximate tensor contraction is dramatically less well behaved than its 2D counterpart. Nonetheless, we numerically demonstrate that the decoding accuracy of our approach outperforms state-of-the-art decoders on the 3D surface code, both in the point and loop sectors, as well as for depolarizing noise. Our techniques could prove useful in near-term experimental demonstrations of quantum error correction, when decoding is to be performed off line and accuracy is of the utmost importance. To this end, we show how TN decoding can be applied to circuit-level noise and demonstrate that it outperforms the matching decoder on the rotated surface code. Published by the American Physical Society 2024
Readout errors are a major bottleneck to extracting reliable information from near-term quantum processors, especially when spatial correlations are non-negligible. We present a unified tensor-network framework that models the readout process as a matrix product operator (MPO), enabling efficient characterization and mitigation beyond uncorrelated approximations. The MPO model is trained via likelihood optimization on calibration data and applies to multiple tasks, including nonlocal observable estimation, random circuit sampling, and random-measurement protocols, such as classical shadows and learning-based tomography. Experiments on a superconducting processor and numerical simulations up to 20 qubits show that the MPO model captures correlated readout errors that uncorrelated models miss, with a sample cost that grows only near-linearly with system size. When extended to two-dimensional systems, the framework can also be integrated with tensor-network quantum error-correction decoders by performing joint inference over data and readout errors. These results establish tensor-network readout error mitigation as a scalable and versatile approach for noise-aware quantum data processing.
We describe two implementations of the optimal error correction algorithm known as the maximum likelihood decoder (MLD) for the 2D surface code with a noiseless syndrome extraction. First, we show how to implement MLD exactly in time O(n2), where n is the number of code qubits. Our implementation uses a reduction from MLD to simulation of matchgate quantum circuits. This reduction however requires a special noise model with independent bit-flip and phase-flip errors. Secondly, we show how to implement MLD approximately for more general noise models using matrix product states (MPS). Our implementation has running time O(nχ3) where χ is a parameter that controls the approximation precision. The key step of our algorithm, borrowed from the DMRG method, is a subroutine for contracting a tensor network on the two-dimensional grid. The subroutine uses MPS with a bond dimension χ to approximate the sequence of tensors arising in the course of contraction. We benchmark the MPS-based decoder against the standard minimum weight matching decoder observing a significant reduction of the logical error probability for χ≥4.