Codeword stabilized quantum codes on subsystems

TL;DR

This paper extends codeword stabilized quantum codes to include subsystem encoding, providing a unified framework for constructing both additive and non-additive quantum error-correcting codes.

Contribution

It introduces operator quantum codes within the codeword stabilized framework and derives error correction conditions for these codes.

Findings

Constructed examples of additive and non-additive codes

Derived error correction conditions for operator codeword stabilized codes

Unified approach to quantum error correction encoding

Abstract

Codeword stabilized quantum codes provide a unified approach to constructing quantum error-correcting codes, including both additive and non-additive quantum codes. Standard codeword stabilized quantum codes encode quantum information into subspaces. The more general notion of encoding quantum information into a subsystem is known as an operator (or subsystem) quantum error correcting code. Most operator codes studied to date are based in the usual stabilizer formalism. We introduce operator quantum codes based on the codeword stabilized quantum code framework. Based on the necessary and sufficient conditions for operator quantum error correction, we derive a error correction condition for operator codeword stabilized quantum codes. Based on this condition, the word operators of a operator codeword stabilized quantum code are constructed from a set of classical binary errors induced by generators of the gauge group. We use this scheme to construct examples of both additive and non-additive codes that encode quantum information into a subsystem.