Symmetries of Codeword Stabilized Quantum Codes

TL;DR

This paper explores symmetry properties of codeword stabilized (CWS) quantum codes, establishing that codes with certain symmetries can be represented by symmetric self-dual additive codes, and introduces a new canonical form for these codes.

Contribution

It introduces a new canonical representation for CWS codes and demonstrates how permutation symmetry can be preserved in the choice of underlying additive codes.

Findings

Symmetry in CWS codes can be preserved through specific representations.

A new canonical form for CWS codes is proposed based on union stabilizer codes.

The symmetry of the graph state in standard form may differ from the code's symmetry, with exceptions like the toric code.

Abstract

Symmetry is at the heart of coding theory. Codes with symmetry, especially cyclic codes, play an essential role in both theory and practical applications of classical error-correcting codes. Here we examine symmetry properties for codeword stabilized (CWS) quantum codes, which is the most general framework for constructing quantum error-correcting codes known to date. A CWS code Q can be represented by a self-dual additive code S and a classical code C, i.,e., Q=(S,C), however this representation is in general not unique. We show that for any CWS code Q with certain permutation symmetry, one can always find a self-dual additive code S with the same permutation symmetry as Q such that Q=(S,C). As many good CWS codes have been found by starting from a chosen S, this ensures that when trying to find CWS codes with certain permutation symmetry, the choice of S with the same symmetry will suffice. A key step for this result is a new canonical representation for CWS codes, which is given in terms of a unique decomposition as union stabilizer codes. For CWS codes, so far mainly the standard form (G,C) has been considered, where G is a graph state. We analyze the symmetry of the corresponding graph of G, which in general cannot possess the same permutation symmetry as Q. We show that it is indeed the case for the toric code on a square lattice with translational symmetry, even if its encoding graph can be chosen to be translational invariant.