Exact Non-identity check is NQP-complete

We define a problem "exact non-identity check": Given a classical description of a quantum circuit with an ancilla system, determine whether it is strictly equivalent to the identity or not. We show that this problem is NQP-complete. In a sense of the strict equivalence condition, this problem is different from a QMA-complete problem, non-identity check defined by D. Janzing etc. As corollaries, it is derived that exact equivalence check is also NQP-complete and that it is hard to minimize quantum resources of a given quantum gate array without changing an implemented unitary operation.