Exact Non-identity check is NQP-complete

TL;DR

This paper proves that the problem of exactly determining whether a quantum circuit is equivalent to the identity operation is NQP-complete, highlighting its computational difficulty and implications for quantum resource optimization.

Contribution

The paper establishes the NQP-completeness of the exact non-identity check problem, differentiating it from related QMA-complete problems and impacting quantum circuit optimization.

Findings

Exact non-identity check is NQP-complete.

Exact equivalence check is also NQP-complete.

Minimizing quantum resources without changing the unitary is computationally hard.

Abstract

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.