Geometric Graph-Theoretic Aspects of Quantum Stabilizer Codes

TL;DR

This paper introduces a systematic graph-theoretic method for constructing and analyzing binary quantum stabilizer codes, demonstrated through the Gottesman eight-qubit code, enhancing understanding of their error-correcting capabilities.

Contribution

It presents a new procedure to associate graphs with stabilizer codes via a three-step process, facilitating analysis of quantum error correction.

Findings

Successfully constructed graphs for stabilizer codes

Verified error-correcting capabilities using graph-theoretic methods

Applied method to the Gottesman eight-qubit code

Abstract

We propose a systematic procedure for the construction of graphs associated with binary quantum stabilizer codes. The procedure is characterized by means of the following three step process. First, the stabilizer code is realized as a codeword-stabilized (CWS) quantum code. Second, the canonical form of the CWS code is determined and third, the input vertices are attached to the graphs. In order to verify the effectiveness of the procedure, we implement the Gottesman stabilizer code characterized by multi-qubit encoding operators for the resource-efficient error correction of arbitrary single-qubit errors. Finally, the error-correcting capabilities of the Gottesman eight-qubit quantum stabilizer code is verified in graph-theoretic terms as originally advocated by Schlingemann and Werner.