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.
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.
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.
In 2021, Broadbent and Kazmi developed a gate-teleportation-based protocol for computational indistinguishability obfuscation of quantum circuits. This protocol is efficient for Clifford+T circuits with logarithmically many T-gates, where the limiting factor in the efficiency of the protocol is the difficulty, on input a quantum circuit
C, of the classical task of producing a description of the unitary obtained by conjugating a Pauli
P (corresponding to a Bell-measurement outcome) by
C, where this description only depends on the input-output functionality of
CPC†. The task above, in turn, is at least as hard as the problem of determining whether two
n-qubit quantum circuits are perfectly equivalent up to global phase. In 2009, Tanaka defined the corresponding decision problem Exact Non-Identity Check (ENIC) and showed that ENIC is NQP-complete and thus NP-hard as well. Motivated by the 2021 protocol, we consider in this work what happens when we pass from low T-count to low T-depth. We show that, for Clifford+T-circuits of T-depth
O(log(n)), deciding ENIC remains NP-hard. In particular, we show this by relating certain decision problems on Pauli coefficients of Clifford+T unitaries to well-known hardness results for codeword weights in binary linear codes. This effectively rules out the possibility, for Clifford+T-circuits of logarithmic T-depth, of either efficient ENIC or efficient gate-teleportation based computational indistinguishability obfuscation, unless P=NP. It also provides a new, independent proof of the hardness of ENIC, even at low T-depth. Along the way, we prove the NP-hardness of some related circuit problems.
QMA and QCMA are possible quantum analogues of the complexity class NP. In QMA the proof is a quantum state and the verification is a quantum circuit. In contrast, in QCMA the proof is restricted to be a classical state. It is not known whether QMA strictly contains QCMA. Here we show that two known QMA-complete problems can be modified to QCMA-complete problems in a natural way: (1) Deciding whether a 3-local Hamiltonian has low energy states (with energy smaller than a given value) that can be prepared with at most k elementary gates is QCMA-complete, whereas it is QMA-complete when the restriction on the complexity of preparation is dropped. (2) Deciding whether a (classically described) quantum circuit does not act as the identity on all basis states is QCMA-complete. It is QMA-complete to decide whether it does not act on all states as the identity.
Distinguishing logarithmic depth quantum circuits on mixed states is shown to be complete for
QIP, the class of problems having quantum interactive proof systems. Circuits in this model can represent arbitrary quantum processes, and thus this result has implications for the verification of implementations of quantum algorithms. The distinguishability problem is also complete for
QIP on constant depth circuits containing the unbounded fan-out gate. These results are shown by reducing a
QIP-complete problem to a logarithmic depth version of itself using a parallelization technique.