Mixed-register Stabilizer Codes: A Coding-theoretic Perspective

TL;DR

This paper introduces mixed-register stabilizer codes for qudit systems, providing theoretical insights and constructing optimal codes that leverage varying local dimensions for improved quantum information protection.

Contribution

It offers a coding-theoretic framework for mixed-register quantum devices and constructs optimal stabilizer codes based on coprime local dimensions, revealing new logical subspace structures.

Findings

Identifies forbidden stabilizer encoded information forms.

Provides general results for mixed-register Pauli operators.

Constructs optimal codes with unique logical subspaces.

Abstract

Protecting information in systems that have more than two basis states (qudits) not only offers a promising route for reducing the number of individual quantum locations that must be protected, while more accurately reflecting the structure of realistic quantum hardware, but also has some possibly enticing foundational strengths. While work in the past has largely focused on protecting information in quantum devices with locations that are some consistent local structure, this work considers coding-theoretic constraints on devices constructed from locations which may vary in their local structures -- these are mixed-register quantum devices. In this work we provide some general results for mixed-register Pauli operators, then identify some stabilizer encoded information forms that are forbidden. Building on these insights, we construct coding-theoretically optimal mixed-register stabilizer codes from sets of codes defined on coprime local-dimensions. The construction of such codes results in codes with logical subspaces that do not directly correspond to any of the constituent local-dimensions.