Phase-state duality in reversible circuit design
Matthew Amy, Neil J. Ross · Physical Review A · 2021
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 $\mathrm{Clifford}+T$ 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 four-qubit Toffoli gate, and measurement-assisted circuits, including Jones' Toffoli gate and Gidney's temporary logical-and gate. 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 $T$-count efficient constructions of some classical functions in space-constrained regimes, notably multiply controlled Toffoli gates and temporary products.