  This work analyses the performance of quantum circuits and general processes
to transform $k$ uses of an arbitrary unitary operation $U$ into another
unitary 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 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 when
$f$ is 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 task $f(U)=U^{-1}$ and the unitary
transposition task $f(U)=U^T$. 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.
