精确最大似然解码超越树宽：基于秩分解动态规划
Exact Maximum Likelihood Decoding beyond Treewidth via Rank-Decomposition Dynamic Programming
- Centre for Quantum Technologies, National University of Singapore（新加坡国立大学量子技术中心）
- Science, Mathematics and Technology Cluster, Singapore University of Technology and Design（新加坡科技设计大学科学、数学与技术集群）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
提出基于秩分解动态规划的精确最大似然解码算法，利用代数结构降低复杂度，实现多项式时间解码，并验证了运行时间优势。
AI中文摘要：
最大似然解码为随机泡利噪声下的量子纠错提供了最优解码策略。然而，计算逻辑类概率具有挑战性，且领先的精确张量网络收缩方法所需时间随树宽呈指数增长。在本工作中，我们提出了一种基于秩分解动态规划（Rank DP）的新型解码算法。我们将具有独立局部故障因子的解码配分函数表示为幂的二次和，并应用秩分解动态规划。所得到的精确求值器在算术复杂度上对输入规模呈多项式增长，而对故障划分下Tanner图的秩宽呈指数增长，包括分解构建过程。经过高斯消元后，该算法在独立单量子比特泡利噪声下，对打孔量子里德-穆勒码和距离递增的Steane级联码族实现多项式时间解码。相应张量网络的标准收缩需要超多项式时间。非负实现给出了在显式算术假设下的相对浮点误差界。数值实验表明，在选定的码容量和电路级实例上，该算法相对于所测试的张量网络实现具有运行时间优势，并可对多达1,023个量子比特的里德-穆勒码进行完整似然评估。我们进一步利用这些似然值从综合征中学习电路噪声参数，评估稀有后选择概率，并量化解码器最优性差距。我们的工作为在量子解码和噪声表征中利用代数结构开辟了新途径。
英文摘要：
Maximum-likelihood decoding provides an optimal decoding strategy for quantum error correction under stochastic Pauli noise. However, computing logical-class probabilities is challenging, and the leading exact tensor-network contraction requires time exponential in treewidth. In this work, we introduce a new decoding algorithm based on rank-decomposition dynamic programming (Rank DP). We express decoding partition functions with independent local fault factors as quadratic sums of powers and apply Rank DP. The resulting exact evaluator has arithmetic complexity polynomial in the input size and exponential in the Tanner graph's rank-width with respect to the fault partition, including decomposition construction. After Gaussian elimination, it gives polynomial-time decoding for punctured quantum Reed-Muller codes and a Steane-concatenated family with growing distance under independent single-qubit Pauli noise. Standard contraction of the corresponding tensor networks requires superpolynomial time. A nonnegative realization gives relative floating-point error bounds under explicit arithmetic assumptions. Numerical experiments demonstrate runtime advantages over the tested tensor-network implementations on selected code-capacity and circuit-level instances, with full likelihood evaluation for Reed-Muller codes up to $1{,}023$ qubits. We further use these likelihoods to learn circuit noise parameters from syndromes, evaluate rare postselection probabilities, and quantify decoder optimality gaps. Our work opens new avenues for exploiting algebraic structure in quantum decoding and noise characterization.
