Deterministic transformations between unitary operations: Exponential advantage with adaptive quantum circuits and the power of indefinite causality

Abstract

This work analyses the performance of quantum circuits and general processes to transformkuses of an arbitrary unitary operationUinto another unitary operationf(U). When the desired functionfa homomorphism, i.e.,f(UV)=f(U)f(V), it is known that optimal average fidelity is attainable by parallel circuits and indefinite causality does not provide any advantage. Here we show that the situation changes dramatically when considering anti-homomorphisms, i.e.,f(UV)=f(V)f(U). In particular, we prove that whenfis an anti-homomorphism, sequential circuits could exponentially outperform parallel ones and processes with indefinite causal order could outperform sequential ones. We presented explicit constructions on how to obtain such advantages for the unitary inversion taskf(U)=U−1and the unitary transposition taskf(U)=UT. We also stablish a one-to-one connection between the problem of unitary estimation and parallel unitary transposition, allowing one to easily translate results from one field to the other. Finally, we apply our results to several concrete problem instances and present a method based on computer-assisted proofs to show optimality.