Identity check is QMA-complete

TL;DR

This paper proves that the problem of determining whether a quantum circuit is close to the identity, called identity check, is QMA-complete, and extends this to a generalized equivalence check problem, establishing their computational complexity.

Contribution

The paper introduces the identity check problem for quantum circuits and proves it is QMA-complete, also showing the generalized equivalence check is similarly QMA-complete.

Findings

Identity check is QMA-complete.

Equivalence check is also QMA-complete.

Both problems involve deciding circuit similarity to the identity or a common invariant subspace.

Abstract

We define the problem identity check: Given a classical description of a quantum circuit, 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 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 equivalence check is also in QMA and hence QMA-complete.