Diagonal distance of quantum codes and hardness of the minimum distance problem

The diagonal distance or graph distance is an important parameter of a quantum error-correcting code that characterizes whether the code is degenerate or not. Degeneracy is a property unique to quantum codes, which allows quantum codes, unlike their classical counterparts, to correct more errors than they can uniquely identify. In the CWS framework introduced by Cross, Smith, Smolin and Zeng (2009), a quantum code is constructed using a classical code and a graph. It is known that the diagonal distance of such a code is upper bounded by $\delta+1$, where $\delta$ is the minimum degree of the associated graph. In this paper, we give sufficient conditions on a graph such that a CWS code constructed from it has diagonal distance at least $\delta$, and in fact most of the graphs in our sufficient class achieve the upper bound of $\delta+1$. Using this result, first we give necessary conditions for a CWS code to be degenerate. Secondly, we prove hardness results for the problem of finding the distance of a CWS code. We construct a CWS code from a given classical code, with the distance of the CWS code being equal to the distance of the classical code. This allows us to translate well-known hardness results for computing the minimum distance in classical codes to quantum codes. Specifically, we show that exactly computing the distance of a CWS code is NP-complete, and multiplicatively or additively approximating it is NP-hard under polynomial-time randomized reductions. Our reduction from the classical problems to the quantum problems results in a non-degenerate quantum code, hence our result implies that the quantum problems remain NP-hard even with the promise that the code is non-degenerate. Moreover, using a mapping from stabilizer codes to CSS codes due to Bravyi, Terhal and Leemhuis (2010), we are able to show that the hardness results hold for CSS codes as well.