A Complete and Natural Rule Set for Multi-Qutrit Clifford Circuits

TL;DR

This paper introduces a complete set of rewrite rules for n-qutrit Clifford circuits, providing a foundational framework for understanding quantum circuits in odd prime dimensions.

Contribution

It generalizes the normal form for n-qubit Clifford circuits to qutrits and establishes a complete rewrite system for qutrit Clifford unitaries.

Findings

First complete rewrite system for qutrit Clifford circuits

Simplified rules for reducing circuits to normal form

Explicit presentation of qutrit Clifford group via generators and relations

Abstract

We present a complete set of rewrite rules for n-qutrit Clifford circuits where n is any non-negative integer. This is the first completeness result for any fragment of quantum circuits in odd prime dimensions. We first generalize Selinger's normal form for n-qubit Clifford circuits to the qutrit setting. Then, we present a rewrite system by which any Clifford circuit can be reduced to this normal form. We then simplify the rewrite rules in this procedure to a small natural set of rules, giving a clean presentation of the group of qutrit Clifford unitaries in terms of generators and relations.