论量子码最小距离问题的难度
On the Hardness of the Minimum Distance Problem of Quantum Codes
AI总结：
本研究通过将经典最小距离问题归约到量子问题，结合无4环图图态距离下界的技术工具，证明精确或近似求解稳定子量子码及CSS码的最小距离均为NP难，填补了量子码距离问题难度研究的空白。
摘要：
我们研究了求解量子纠错码距离问题的难度。经典码的同类问题已知是NP难的，即便在近似形式下也是如此。对于量子码，各类与译码相关的问题已被证明是NP难的，但距离问题的难度此前尚未得到研究。本工作中，我们证明精确或近似求解稳定子量子码的最小距离都是NP难的。该结果通过将经典最小距离问题归约到量子问题得到，归约过程使用了量子码的CWS框架（即利用经典码和一幅图构造量子码的框架）。证明该结果用到的一项核心技术工具是无4环图的所谓图态距离的下界。具体而言，我们证明对于无4环图$G$，其图态距离要么是$δ$，要么是$δ+1$，其中$δ$是$G$的最小顶点度。由于存在从稳定子码到CSS码的知名归约，我们的结果也意味着求解CSS码的最小距离同样是NP难的。
英文摘要：
We study the hardness of the problem of finding the distance of quantum error-correcting codes. The analogous problem for classical codes is known to be NP-hard, even in approximate form. For quantum codes, various problems related to decoding are known to be NP-hard, but the hardness of the distance problem has not been studied before. In this work, we show that finding the minimum distance of stabilizer quantum codes exactly or approximately is NP-hard. This result is obtained by reducing the classical minimum distance problem to the quantum problem, using the CWS framework for quantum codes, which constructs a quantum code using a classical code and a graph. A main technical tool used for our result is a lower bound on the so-called graph state distance of 4-cycle free graphs. In particular, we show that for a 4-cycle free graph $G$, its graph state distance is either $δ$ or $δ+1$, where $δ$ is the minimum vertex degree of $G$. Due to a well-known reduction from stabilizer codes to CSS codes, our results also imply that finding the minimum distance of CSS codes is also NP-hard.
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