  The performance of a quantum information processing protocol is ultimately
judged by distinguishability measures that quantify how distinguishable the
actual result of the protocol is from the ideal case. The most prominent
distinguishability measures are those based on the fidelity and trace distance,
due to their physical interpretations. In this paper, we propose and review
several algorithms for estimating distinguishability measures based on trace
distance and fidelity. The algorithms can be used for distinguishing quantum
states, channels, and strategies (the last also known in the literature as
"quantum combs"). The fidelity-based algorithms offer novel physical
interpretations of these distinguishability measures in terms of the maximum
probability with which a single prover (or competing provers) can convince a
verifier to accept the outcome of an associated computation. We simulate many
of these algorithms by using a variational approach with parameterized quantum
circuits. We find that the simulations converge well in both the noiseless and
noisy scenarios, for all examples considered. Furthermore, the noisy
simulations exhibit a parameter noise resilience. Finally, we establish a
strong relationship between various quantum computational complexity classes
and distance estimation problems.
