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This work analyses the performance of quantum circuits and general processesto transform $k$ uses of an arbitrary unitary operation $U$ into anotherunitary operation $f(U)$. When the desired function $f$ a homomorphism, i.e.,$f(UV)=f(U)f(V)$, it is known that optimal average fidelity is attainable byparallel circuits and indefinite causality does not provide any advantage. Herewe show that the situation changes dramatically when consideringanti-homomorphisms, i.e., $f(UV)=f(V)f(U)$. In particular, we prove that when$f$ is an anti-homomorphism, sequential circuits could exponentially outperformparallel ones and processes with indefinite causal order could outperformsequential ones. We presented explicit constructions on how to obtain suchadvantages for the unitary inversion task $f(U)=U^{-1}$ and the unitarytransposition task $f(U)=U^T$. We also stablish a one-to-one connection betweenthe problem of unitary estimation and parallel unitary transposition, allowingone to easily translate results from one field to the other. Finally, we applyour results to several concrete problem instances and present a method based oncomputer-assisted proofs to show optimality.

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