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.
We initiate the study of parameterized complexity of
QMA problems in terms of the number of non-Clifford gates in the problem description. We show that for the problem of parameterized quantum circuit satisfiability, there exists a classical algorithm solving the problem with a runtime scaling exponentially in the number of non-Clifford gates but only polynomially with the system size. This result follows from our main result, that for any Clifford +
t T-gate quantum circuit satisfiability problem, the search space of optimal witnesses can be reduced to a stabilizer subspace isomorphic to at most
t qubits (independent of the system size). Furthermore, we derive new lower bounds on the
T-count of circuit satisfiability instances and the
T-count of the
W-state assuming the classical exponential time hypothesis (
ETH). Lastly, we explore the parameterized complexity of the quantum non-identity check problem.
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.
We present authorized quantum computation, where only a user with a non-cloneable quantum authorization key can perform a unitary operation created by an authenticated programmer. The security of our authorized quantum computation is based on the quantum computational complexity problem of forging the keys from an obfuscated quantum gate sequence. Under the assumption of the existence of a \textit{sufficiently-random gate shuffling algorithm}, the problem is shown to be in the NQP (Non-deterministic Quantum Polynomial)-hard class by reducing it to a NQP-Complete problem, the exact non-identity check problem. Therefore, our authorized quantum computation can be computationally secure against attacks using quantum computers.
As quantum computing resources remain scarce and error rates high, minimizing the resource consumption of quantum circuits is essential for achieving practical quantum advantage. Here we consider the natural problem of, given a circuit
C, computing a circuit
C′ that behaves equivalently on a desired subspace, and that minimizes a quantum resource type, expressed as the count or depth of (i) arbitrary gates, or (ii) non-Clifford gates, or (iii) superposition gates, or (iv) entanglement gates. We show that, when
C is expressed over any gate-set that can implement the H and TOF gates exactly, a standard property of approximately universal gate-sets, each of the above optimization problems is hard for co-NQP, and hence outside the Polynomial Hierarchy, unless the Polynomial Hierarchy collapses. This complements a recent result of Van de Wetering and Amy (arXiv, 2023) which establishes an NP-hardness lower bound when equivalence is over the full state space, and tightens the gap to the corresponding
NPNQP upper bound known for cases (i)-(iii) over Clifford+T and (i)-(iv) over H+TOF circuits. Our result also generalizes a result by Tanaka (Int. J. Quantum Inf., 2010) to other gate-sets and quantum resources.