Subsystem Code Constructions

TL;DR

This paper explores the construction and properties of quantum subsystem codes, providing methods to derive new codes from existing ones, and establishing the existence of many MDS subsystem code families.

Contribution

It introduces a generic method for deriving subsystem codes, shows all pure MDS subsystem codes originate from MDS stabilizer codes, and develops propagation rules for code construction.

Findings

All pure MDS subsystem codes are derived from MDS stabilizer codes.

Numerous families of MDS subsystem codes are established.

Propagation rules enable the construction of longer, shorter, or combined subsystem codes.

Abstract

Subsystem codes are the most versatile class of quantum error-correcting codes known to date that combine the best features of all known passive and active error-control schemes. The subsystem code is a subspace of the quantum state space that is decomposed into a tensor product of two vector spaces: the subsystem and the co-subsystem. A generic method to derive subsystem codes from existing subsystem codes is given that allows one to trade the dimensions of subsystem and co-subsystem while maintaining or improving the minimum distance. As a consequence, it is shown that all pure MDS subsystem codes are derived from MDS stabilizer codes. The existence of numerous families of MDS subsystem codes is established. Propagation rules are derived that allow one to obtain longer and shorter subsystem codes from given subsystem codes. Furthermore, propagation rules are derived that allow one to construct a new subsystem code by combining two given subsystem codes.