Multi-qutrit exact synthesis

TL;DR

This paper introduces an exact synthesis algorithm for multi-qutrit unitaries over specific algebraic rings using Clifford+T gates with minimal ancillas, extending previous results and providing new conversion techniques.

Contribution

It develops an exact synthesis algorithm for multi-qutrit unitaries over specialized rings with minimal ancillas, and introduces conversion and catalytic embedding methods.

Findings

Successfully synthesizes multi-qutrit unitaries with at most two ancillas.

Extends known results to larger algebraic rings for exact synthesis.

Provides algorithms for converting 3-level unitaries into multiply-controlled gates.

Abstract

We present an exact synthesis algorithm for qutrit unitaries in $\mathcal{U}_{3^n}(\mathbb{Z}[1/3,e^{2\pi i/3}])$ over the Clifford$+T$ gate set with at most one ancilla. This extends the already known result of qutrit metaplectic gates being a subset of Clifford$+T$ gate set with one ancilla. As an intermediary step, we construct an algorithm to convert 3-level unitaries into multiply-controlled gates, analogous to Gray codes converting 2-level unitaries into multiply-controlled gates. Finally, using catalytic embeddings, we present an algorithm to exactly synthesize unitaries $\mathcal{U}_{3^n}(\mathbb{Z}[1/3,e^{2\pi i/9}])$ over the Clifford$+T$ gate set with at most 2 ancillas. This, in particular, gives an exact synthesis algorithm of single-qutrit Clifford$+\mathcal{D}$ over the multi-qutrit Clifford$+T$ gate set with at most two ancillas.