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Chinese Physics B

P lanar : A software for exact decoding quantum error correction codes with planar structure

Dongyang Feng, Hanyan Cao, Pan Zhang

Published Apr 10, 2025• 1 citation
Publisher

Abstract

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.

DOI
10.1088/1674-1056/adcb26

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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.
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.
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.
Fault tolerance is a prerequisite for scalable quantum computing. Architectures based on 2D topological codes are effective for near-term implementations of fault tolerance. To obtain high performance with these architectures, we require a decoder which can adapt to the wide variety of error models present in experiments. The typical approach to the problem of decoding the surface code is to reduce it to minimum-weight perfect matching in a way that provides a suboptimal threshold error rate, and is specialized to correct a specific error model. Recently, optimal threshold error rates for a variety of error models have been obtained by methods which do not use minimum-weight perfect matching, showing that such thresholds can be achieved in polynomial time. It is an open question whether these results can also be achieved by minimum-weight perfect matching. In this work, we use belief propagation and a novel algorithm for producing edge weights to increase the utility of minimum-weight perfect matching for decoding surface codes. This allows us to correct depolarizing errors using the rotated surface code, obtaining a threshold of 17.76 ± 0.02 % . This is larger than the threshold achieved by previous matching-based decoders ( 14.88 ± 0.02 % ), though still below the known upper bound of ∼ 18.9 % .
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.
Decoding algorithms are essential to fault-tolerant quantum-computing architectures. In this perspective we explore decoding algorithms for the surface code; a prototypical quantum low-density parity-check code that underlies many of the leading efforts to demonstrate scalable quantum computing. Central to our discussion is the minimum-weight perfect-matching decoder. The decoder works by exploiting underlying structure that arises due to materialised symmetries among surface-code stabilizer elements. By concentrating on these symmetries, we begin to address the question of how a minimum-weight perfect-matching decoder might be generalised for other families of codes. We approach this question first by investigating examples of matching decoders for other codes. These include decoding algorithms that have been specialised to correct for noise models that demonstrate a particular structure or bias with respect to certain codes. In addition to this, we propose a systematic way of constructing a minimum-weight perfect-matching decoder for codes with certain characteristic properties. The properties we make use of are common among topological codes. We discuss the broader applicability of the proposal, and we suggest some questions we can address that may show us how to design a generalised matching decoder for arbitrary stabilizer codes.
Surface codes exploit topological protection to increase error resilience in quantum computing devices and can in principle be implemented in existing hardware. They are one of the most promising candidates for active error correction, not least due to a polynomial-time decoding algorithm which admits one of the highest predicted error thresholds. We consider the dependence of this threshold on underlying assumptions including different noise models, and analyze the performance of a minimum weight perfect matching (MWPM) decoding compared to a mathematically optimal maximum likelihood (ML) decoding. Our ML algorithm tracks the success probabilities for all possible corrections over time and accounts for individual gate failure probabilities and error propagation due to the syndrome measurement circuit. We present the very first evidence for the true error threshold of an optimal circuit level decoder, allowing us to draw conclusions about what kind of improvements are possible over standard MWPM.