Distinguishing logarithmic depth quantum circuits on mixed states is shown to be complete for $QIP$, the class of problems having quantum interactive proof systems. Circuits in this model can represent arbitrary quantum processes, and thus this result has implications for the verification of implementations of quantum algorithms. The distinguishability problem is also complete for $QIP$ on constant depth circuits containing the unbounded fan-out gate. These results are shown by reducing a $QIP$-complete problem to a logarithmic depth version of itself using a parallelization technique.

This paper considers the following problem. Two mixed-state quantum circuits Q/sub 0/ and Q/sub 1/ are given, and the goal is to determine which of two possibilities holds: (i) Q/sub 0/ and Q/sub 1/ act nearly identically on all possible quantum state inputs, or (ii) there exists some input state /spl rho/ that Q/sub 0/ and Q/sub 1/ transform into almost perfectly distinguishable outputs. This may be viewed as an abstraction of the problem that asks, given two discrete quantum mechanical processes described by sequences of local interactions, are the processes effectively the same or are they different? We prove that this promise problem is complete for the class QIP of problems having quantum interactive proof systems, and is therefore PSPACE-hard. This is in contrast to the fact that the analogous problem for classical (probabilistic) circuits is in AM, and for unitary quantum circuits is in QMA.