Codeword stabilized quantum codes: algorithm and structure

TL;DR

This paper introduces an algorithm for constructing codeword stabilized quantum codes by reducing the problem to finding maximum cliques in graphs, and provides structural theorems that narrow the search space for optimal codes.

Contribution

It presents a novel graph-based algorithm for CWS code construction and proves structural theorems that limit the search for certain optimal quantum codes.

Findings

No ((7,3,3)) CWS code exists according to the structural theorems.

The algorithm reduces the search space for CWS codes significantly.

Comparison with previous methods shows advantages in certain cases.

Abstract

The codeword stabilized ("CWS") quantum codes formalism presents a unifying approach to both additive and nonadditive quantum error-correcting codes (arXiv:0708.1021). This formalism reduces the problem of constructing such quantum codes to finding a binary classical code correcting an error pattern induced by a graph state. Finding such a classical code can be very difficult. Here, we consider an algorithm which maps the search for CWS codes to a problem of identifying maximum cliques in a graph. While solving this problem is in general very hard, we prove three structure theorems which reduce the search space, specifying certain admissible and optimal ((n,K,d)) additive codes. In particular, we find there does not exist any ((7,3,3)) CWS code though the linear programming bound does not rule it out. The complexity of the CWS search algorithm is compared with the contrasting method introduced by Aggarwal and Calderbank (arXiv:cs/0610159).