EXACT NON-IDENTITY CHECK IS NQP-COMPLETE

To understand quantum gate array complexity, we define a problem named exact non-identity check, which is a decision problem to determine whether a given classical description of a quantum circuit is strictly equivalent to the identity or not. We show that the computational complexity of this problem is non-deterministic quantum polynomial-time (NQP)-complete. As corollaries, it is derived that exact non-equivalence check of two given classical descriptions of quantum circuits is also NQP-complete and that minimizing the number of quantum gates for a given quantum circuit without changing the implemented unitary operation is NQP-hard.

"NON-IDENTITY-CHECK" IS QMA-COMPLETE

We describe a computational problem that is complete for the complexity class QMA, a quantum generalization of NP. It arises as a natural question in quantum computing and quantum physics. "Non-identity-check" is the following decision problem: Given a classical description of a quantum circuit (a sequence of elementary gates), determine whether it is almost equivalent to the identity. Explicitly, the task is to decide whether the corresponding unitary is close to a complex multiple of the identity matrix with respect to the operator norm. We show that this problem is QMA-complete. A generalization of this problem is "non-equivalence check": given two descriptions of quantum circuits and a description of a common invariant subspace, decide whether the restrictions of the circuits to this subspace almost coincide. We show that non-equivalence check is also in QMA and hence QMA-complete.