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Identity check is QMA-complete

Dominik Janzing·Pawel Wocjan·Thomas Beth
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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.

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Dominik Janzing, Pawel Wocjan, Thomas Beth. 2003-05-09. Identity check is QMA-complete. https://arxiv.org/abs/quant-ph/0305050

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