On the hardness of distinguishing mixed-state quantum computations

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.