Optimal Universal Quantum Circuits for Unitary Complex Conjugation

Let
$U_{d}$
be a unitary operator representing an arbitrary
$d$
-dimensional unitary quantum operation. This work presents optimal quantum circuits for transforming a number
$k$
of calls of
$U_{d}$
into its complex conjugate
$\overline {U_{d}}$
. Our circuits admit a parallel implementation and are proven to be optimal for any
$k$
and
$d$
with an average fidelity of
$\left \langle{ {F}}\right \rangle =\frac {k+1}{d(d-k)}$
. Optimality is shown for average fidelity, robustness to noise, and other standard figures of merit. This extends previous works which considered the scenario of a single call (
$k=1$
) of the operation
$U_{d}$
, and the special case of
$k=d-1$
calls. We then show that our results encompass optimal transformations from
$k$
calls of
$U_{d}$
to
$f(U_{d})$
for any arbitrary homomorphism
$f$
from the group of
$d$
-dimensional unitary operators to itself, since complex conjugation is the only non-trivial automorphism on the group of unitary operators. Finally, we apply our optimal complex conjugation implementation to design a probabilistic circuit for reversing arbitrary quantum evolutions.