# Codeword stabilized quantum codes: Algorithm and structure

## Abstract

The codeword stabilized (CWS) quantum code formalism presents a unifying approach to both additive and nonadditive quantum error-correcting codes [IEEE Trans. Inf. Theory 55, 433 (2009)]. 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 provide three structure theorems which reduce the search space, specifying certain admissible and optimal ((n,K,d)) additive codes. In particular, we find that the re 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 [IEEE Trans. Inf. Theory 54, 1700 (2008)]. © 2009 American Institute of Physics.