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 $\delta$ or $\delta+1$, where $\delta$ 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.

翻译：我们研究了量子纠错码距离问题的难度。经典码的类似问题已知是NP难的，即使在近似形式下也是如此。对于量子码，与解码相关的各种问题已知是NP难的，但距离问题的难度此前尚未被研究。在本工作中，我们证明精确或近似地找到稳定子码的最小距离是NP难的。这一结果是通过将经典最小距离问题归约到量子问题而获得的，利用了量子码的CWS框架，该框架使用经典码和一张图构造量子码。我们结果的一个主要技术工具是对所谓的4-环自由图的图态距离的下界。特别地，我们证明对于一张4-环自由图$G$，其图态距离要么是$\delta$，要么是$\delta+1$，其中$\delta$是$G$的最小顶点度数。由于从稳定子码到CSS码的著名归约，我们的结果也意味着找到CSS码的最小距离同样是NP难的。