二维拓扑量子码最小重量译码的近似难度
Hardness of approximation for minimum-weight decoding of two-dimensional topological quantum codes
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中文总结 AI 辅助
研究二维拓扑量子码最小重量译码的计算复杂性，在P≠NP假设下证明环面码等的不可近似性间隙，基于MAX-3SAT的近似难度理论完成归约。
中文摘要 AI 辅助
高效译码对于容错量子计算机的实际实现至关重要。我们研究拓扑量子码最小重量译码的计算复杂性。对于去极化信道下的表面码，我们考虑最小重量译码，其目标是找到与X型和Z型校正子都一致的最小重量Pauli错误。对于独立X错误和Z错误模型下的颜色码，我们考虑分离最小重量译码。假设P≠NP，我们为这些问题建立多项式加性不可近似性间隙。具体而言，对于环面上的环面码和4.8.8颜色码，不存在多项式时间算法总能生成其重量与最优值相差在Ω(N^(1/14))以内的解，其中N是量子比特数。对于平面表面码，我们得到Ω(N^(1/18))的间隙。我们的不可近似性结果基于Håstad的MAX-3SAT近似难度理论，我们的归约开发了一种通用模块化框架，用于将逻辑约束嵌入到格上耦合的原始-对偶连接问题中，关键要素是定位论证，用于控制构造不同部分之间的意外相互作用。
英文摘要
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, there exists a constant $c>0$ such that no polynomial-time algorithm can always produce a solution whose weight is within $cN^{1/14}$ of the optimum, where $N$ is the number of qubits, unless $P=NP$. 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.
