Beyond Integral-Domain Stabilizer Codes

TL;DR

This paper develops methods for constructing quantum stabilizer codes in systems with composite local dimensions, expanding the applicability of quantum error correction to more complex quantum systems.

Contribution

It introduces new constructions for stabilizer codes in composite-dimensional systems and discusses logical encodings and code counts, broadening the scope of quantum error correction.

Findings

Provided transformations for known stabilizer codes to composite dimensions

Enhanced understanding of symplectic spaces in composite systems

Enabled full utilization of computational space in certain quantum systems

Abstract

Quantum error-correcting codes aim to protect information in quantum systems to enable fault-tolerant quantum computations. The most prevalent method, stabilizer codes, has been well developed for many varieties of systems, however, largely the case of composite number of levels in the system has been avoided. This relative absence is due to the underlying ring theoretic tools required for analyzing such systems. Here we explore composite local-dimension quantum stabilizer codes, providing a pair of constructions for transforming known stabilizer codes into valid ones for composite local-dimensions. In addition remarks on logical encodings and the counts possible are discussed. This work lays out central methods for working with composite dimensional systems, enabling full use of the computational space of some systems, and expanding the understanding of the symplectic spaces involved.