Synthesizing Toffoli-optimal quantum circuits for arbitrary multi-qubit unitaries

TL;DR

This paper develops methods to synthesize multi-qubit unitaries with minimal Toffoli gates within the Clifford+Toffoli set, providing bounds and algorithms for optimal quantum circuit implementation.

Contribution

It introduces a generating set for Clifford+Toffoli unitaries, derives bounds on Toffoli-count, and develops optimal synthesis algorithms for both approximate and exact implementations.

Findings

Bound on Toffoli-count for multi-qubit unitaries

Development of Toffoli-count optimal synthesis algorithms

Equivalence of implementable unitaries between Clifford+Toffoli and Clifford+CS sets

Abstract

In this paper we study the Clifford+Toffoli universal fault-tolerant gate set. We introduce a generating set in order to represent any unitary implementable by this gate set and with this we derive a bound on the Toffoli-count of arbitrary multi-qubit unitaries. We analyse the channel representation of the generating set elements, with the help of which we infer $|\mathcal{J}_n^{Tof}|<|\mathcal{J}_n^T|$, where $\mathcal{J}_n^{Tof}$ and $\mathcal{J}_n^T$ are the set of unitaries exactly implementable by the Clifford+Toffoli and Clifford+T gate set, respectively. We develop Toffoli-count optimal synthesis algorithms for both approximately and exactly implementable multi-qubit unitaries. With the help of these we prove $|\mathcal{J}_n^{Tof}|=|\mathcal{J}_n^{CS}|$, where $\mathcal{J}_n^{CS}$ is the set of unitaries exactly implementable by the Clifford+CS gate set.