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arXiv:2402.04834 · Physical Review Research

BlockBP decoder for the surface code

Aviad Kaufmann, Itai Arad

Published Aug 27, 2025• 3 citations
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Abstract

We present a new decoder for the surface code, which combines the accuracy of the tensor-network decoders with the efficiency and parallelism of the belief-propagation algorithm. Our main idea is to replace the expensive tensor-network contraction step in the tensor-network decoders with the algorithm—a recent approximate contraction algorithm, based on belief propagation. Our decoder is therefore a belief-propagation decoder that works in the degenerate maximal likelihood decoding framework. Unlike conventional tensor-network decoders, our algorithm can run efficiently in parallel, and may therefore be suitable for real-time decoding. We numerically test our decoder and show that for a large range of lattice sizes and noise levels it delivers a logical error probability that outperforms the Minimal Weight Perfect Matching decoder, sometimes by more than an order of magnitude.

DOI
10.1103/z2qt-smfb

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We present a new decoder for the surface code, which combines the accuracy of the tensor-network decoders with the efficiency and parallelism of the belief-propagation algorithm. Our main idea is to replace the expensive tensor-network contraction step in the tensor-network decoders with the algorithm—a recent approximate contraction algorithm, based on belief propagation. Our decoder is therefore a belief-propagation decoder that works in the degenerate maximal likelihood decoding framework. Unlike conventional tensor-network decoders, our algorithm can run efficiently in parallel, and may therefore be suitable for real-time decoding. We numerically test our decoder and show that for a large range of lattice sizes and noise levels it delivers a logical error probability that outperforms the Minimal Weight Perfect Matching decoder, sometimes by more than an order of magnitude.
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Quantum error correction is crucial for universal fault-tolerant quantum computing. Highly accurate and low-time-complexity decoding algorithms play an indispensable role in ensuring quantum error correction works effectively. Among existing decoding algorithms, belief propagation (BP) is notable for its nearly linear time complexity and general applicability to stabilizer codes. However, BP's decoding accuracy without post-processing is unsatisfactory in most situations. This article focuses on improving the decoding accuracy of BP over GF(4) for surface codes. Inspired by machine learning optimization techniques, we first propose Momentum-BP and AdaGrad-BP to reduce oscillations in message updating, breaking the trapping sets of surface codes. We further propose EWAInit-BP, which adaptively updates initial probabilities and provides a 1 to 3 orders of magnitude improvement over traditional BP for planar surface code, toric code, and XZZX surface code without any post-processing method, showing high decoding accuracy even under parallel scheduling. The theoreticalO(1)O(1)O(1)time complexity under parallel implementation and high accuracy of EWAInit-BP make it a promising candidate for high-precision real-time decoders.
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)O(n^2)O(n2), where nnn 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)O(nχ^3)O(nχ3) where χχχ 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 χχχ 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≥4χ\ge 4χ≥4.
The standard approach to Belief Propagation (BP) decoding passes messages on the Tanner graph of the quantum error-correcting (QEC) code, and is known to lack a threshold for the surface code. We propose novel BP decoders that exchange messages on the decoding graph and obtain thresholds (both for code capacity and circuit-level noise) via standalone BP for the surface code under depolarizing noise. Our approach, similarly to the minimum weight perfect matching (MWPM) decoder, is applicable to any graphlike QEC code. The thresholds observed with our decoders are close to those obtained by MWPM. This result opens the path towards scalable hardware-accelerated implementations of MWPM-compatible decoders.
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
Simulating many-body quantum systems on a classical computer is difficult due to the large number of degrees of freedom, causing the computational complexity to grow exponentially with system size. Tensor Networks (TN) is a framework that breaks down large tensors into a network of smaller tensors, enabling efficient simulation of certain many-body quantum systems. To calculate expectation values of local observables or simulate nearest-neighbor interactions, a contraction of the entire network is needed. This is a known hard problem, which cannot be done exactly for systems with spatial dimension D>1 and is the major bottleneck in all tensor-network based algorithms. Various approximate-contraction algorithms have been suggested, all with their strengths and weaknesses. Nevertheless, contracting a 2D TN remains a major numerical challenge, limiting the use of TN techniques for many interesting systems. Recently, a close connection between TN and Probabilistic Graphical Models (PGM) has been shown. In the PGM framework, marginals of complicated probability distributions can be approximated using iterative message passing algorithms such as Belief Propagation (BP). The BP algorithm can be adapted to the TN framework as an efficient contraction algorithm. While BP is extremely efficient and easy to parallelize, it often yields inaccurate results for highly correlated quantum states or frustrated systems. To overcome this, we suggest the BlockBP algorithm, which coarse-grains the system into blocks and performs BP between them. This thesis focuses on: (i) development and implementation of the BlockBP algorithm for infinite lattices; (ii) using this algorithm to study the anti-ferromagnetic Heisenberg model on the Kagome lattice in the thermodynamic limit - a frustrated 2D model that is difficult to simulate using existing numerical methods.