Graphical Quantum Error-Correcting Codes

TL;DR

This paper introduces a graph-theoretical framework for constructing quantum error-correcting codes, enabling the creation of both stabilizer and nonadditive codes with improved efficiency and explicit examples.

Contribution

It presents the concept of coding cliques for quantum codes, unifies the construction of stabilizer and nonadditive codes, and provides explicit optimal and high-rate codes.

Findings

Constructed the optimal ((10,24,3)) code.

Developed a family of high-rate nonadditive codes.

Classified all extremal stabilizer codes up to 8 qubits.

Abstract

We introduce a purely graph-theoretical object, namely the coding clique, to construct quantum errorcorrecting codes. Almost all quantum codes constructed so far are stabilizer (additive) codes and the construction of nonadditive codes, which are potentially more efficient, is not as well understood as that of stabilizer codes. Our graphical approach provides a unified and classical way to construct both stabilizer and nonadditive codes. In particular we have explicitly constructed the optimal ((10,24,3)) code and a family of 1-error detecting nonadditive codes with the highest encoding rate so far. In the case of stabilizer codes a thorough search becomes tangible and we have classified all the extremal stabilizer codes up to 8 qubits.