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

Priyanka Mukhopadhyay

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arXiv · 2401.08950

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

Priyanka Mukhopadhyay

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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.

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Priyanka Mukhopadhyay. 2024-01-17. Synthesizing Toffoli-optimal quantum circuits for arbitrary multi-qubit unitaries. https://arxiv.org/abs/2401.08950

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