Non-binary Unitary Error Bases and Quantum Codes

TL;DR

This paper introduces non-binary unitary error bases for quantum systems of any dimension, enabling the construction of quantum codes from linear codes over n and supporting fault-tolerant operations.

Contribution

It generalizes error bases and quantum code constructions to non-binary systems, expanding the scope of quantum error correction methods.

Findings

Quantum codes can be derived from linear codes over n.

New error bases facilitate fault-tolerant quantum operations.

The approach applies to systems of any dimension.

Abstract

Error operator bases for systems of any dimension are defined and natural generalizations of the bit/sign flip error basis for qubits are given. These bases allow generalizing the construction of quantum codes based on eigenspaces of Abelian groups. As a consequence, quantum codes can be constructed from linear codes over $\ints_n$ for any $n$. The generalization of the punctured code construction leads to many codes which permit transversal (i.e. fault tolerant) implementations of certain operations compatible with the error basis.