Scheme for constructing graphs associated with stabilizer quantum codes

TL;DR

This paper introduces a systematic method to construct graphs from binary stabilizer quantum codes, enabling graph-theoretic analysis of their error-correcting properties and potential generalizations to nonbinary and continuous-variable codes.

Contribution

The paper presents a new scheme for graph construction from stabilizer codes, including canonical form identification and graph attachment, facilitating analysis and generalization.

Findings

Successfully constructed graphs for various stabilizer codes

Verified error-correcting capabilities via graph-theoretic methods

Discussed potential extensions to nonbinary and continuous-variable codes

Abstract

We propose a systematic scheme for the construction of graphs associated with binary stabilizer codes. The scheme is characterized by three main steps: first, the stabilizer code is realized as a codeword-stabilized (CWS) quantum code; second, the canonical form of the CWS code is uncovered; third, the input vertices are attached to the graphs. To check the effectiveness of the scheme, we discuss several graphical constructions of various useful stabilizer codes characterized by single and multi-qubit encoding operators. In particular, the error-correcting capabilities of such quantum codes are verified in graph-theoretic terms as originally advocated by Schlingemann and Werner. Finally, possible generalizations of our scheme for the graphical construction of both (stabilizer and nonadditive) nonbinary and continuous-variable quantum codes are briefly addressed.