Standard Form of Qudit Stabilizer Groups

TL;DR

This paper generalizes the structure and standard form of qudit stabilizer codes for arbitrary dimensions, providing an efficient algorithm for canonical form and insights into their duality, which aids in quantum error correction.

Contribution

It extends known prime-dimensional stabilizer results to arbitrary dimensions and introduces an explicit algorithm for standard form transformation.

Findings

Established a relation between code dimension and stabilizer size.

Proved all qudit stabilizers can be transformed into a standard form.

Provided an efficient algorithm for canonical form conversion.

Abstract

We investigate stabilizer codes with carrier qudits of equal dimension $D$, an arbitrary integer greater than 1. We prove that there is a direct relation between the dimension of a qudit stabilizer code and the size of its corresponding stabilizer, and this implies that the code and its stabilizer are dual to each other. We also show that any qudit stabilizer can be put in a standard, or canonical, form using a series of Clifford gates, and we provide an explicit efficient algorithm for doing this. Our work generalizes known results that were valid only for prime dimensional systems and may be useful in constructing efficient encoding/decoding quantum circuits for qudit stabilizer codes and better qudit quantum error correcting codes.