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