On the Hardness of the Minimum Distance Problem of Quantum Codes

TL;DR

This paper proves that determining the minimum distance of quantum error-correcting codes is NP-hard, extending classical code hardness results to the quantum domain using graph-based code constructions.

Contribution

It establishes the NP-hardness of the quantum minimum distance problem by reducing from classical codes and analyzing graph state distances, a novel connection in quantum coding theory.

Findings

Minimum distance of stabilizer quantum codes is NP-hard to compute.

The hardness extends to CSS codes due to their relation to stabilizer codes.

Graph state distance for 4-cycle free graphs is either δ or δ+1, where δ is the minimum vertex degree.

Abstract

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.