  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.
