The phase/state duality in reversible circuit design

TL;DR

This paper explores the duality of phase and state in reversible circuit design, introducing new methods for implementing classical functions with reduced T-count in quantum computing.

Contribution

It generalizes classes of reversible circuits using phase and measurement-assisted techniques, providing novel efficient constructions for classical functions in quantum circuits.

Findings

Developed methods for implementing classical functions up to phase.

Introduced measurement-assisted termination techniques for temporary values.

Achieved T-count efficient constructions for multiply-controlled Toffoli gates.

Abstract

The reversible implementation of classical functions accounts for the bulk of most known quantum algorithms. As a result, a number of reversible circuit constructions over the Clifford+$T$ gate set have been developed in recent years which use both the state and phase spaces, or $X$ and $Z$ bases, to reduce circuit costs beyond what is possible at the strictly classical level. We study and generalize two particular classes of these constructions: relative phase circuits, including Giles and Selinger's multiply-controlled $iX$ gates and Maslov's $4$ qubit Toffoli gate, and measurement-assisted circuits, including Jones' Toffoli gate and Gidney's temporary logical-AND. In doing so, we introduce general methods for implementing classical functions up to phase and for measurement-assisted termination of temporary values. We then apply these techniques to find novel $T$-count efficient constructions of some classical functions in space-constrained regimes, notably multiply-controlled Toffoli gates and temporary products.