Hardness of approximation for minimum-weight decoding of two-dimensional topological quantum codes

2026-08-17

AI Summary

本文探讨了二维拓扑量子码最小权重解码的计算复杂性问题，通过假设P不等于NP，使用Håstad对MAX-3SAT的近似难度结果，证明了多项式时间算法无法达到最优解一定范围内的解。

Abstract

Efficient decoding is essential for the practical realization of fault-tolerant quantum computers. We study the computational complexity of minimum-weight decoding for topological quantum codes. For surface codes under the depolarizing channel, we consider Minimum-Weight decoding, which seeks a minimum-weight Pauli error consistent with both the $X$- and $Z$-syndromes. For color codes under independent $X$- and $Z$-error models, we consider Separate Minimum-Weight decoding. Assuming $P\neq NP$, we establish polynomial additive inapproximability gaps for these problems. Specifically, for the toric code and the $4.8.8$ color code on the torus, no polynomial-time algorithm can always produce a solution whose weight is within $Ω(N^{1/14})$ of the optimum, where $N$ is the number of qubits. For the planar surface code, we obtain an $Ω(N^{1/18})$ gap. Our inapproximability results use Håstad's hardness of approximation for MAX-3SAT. Our reduction develops a general, modular framework for embedding logical constraints into coupled primal--dual join problems on a lattice. A key ingredient is a localization argument that controls unintended interactions between different parts of the construction.

Problem

• Minimum-Weight Decoding
• Topological Quantum Codes
• Computational Complexity

Innovation

• minimum-weight decoding
• topological quantum codes
• inapproximability gap
• polynomial-time algorithm
• localization argument