Phase-state duality in reversible circuit design

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