The Parameterized Complexity of Quantum Verification

TL;DR

This paper explores the parameterized complexity of QMA problems based on non-Clifford gates, providing algorithms, bounds, and complexity insights for quantum circuit satisfiability and related problems.

Contribution

It introduces a classical algorithm with runtime exponential in non-Clifford gates, reduces the search space for quantum witnesses, and establishes new lower bounds under ETH.

Findings

Classical algorithm with runtime exponential in non-Clifford gates

Reduction of the search space to a stabilizer subspace of size t

New lower bounds on T-count for circuit satisfiability and W-state

Abstract

We initiate the study of parameterized complexity of $\textsf{QMA}$ 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 ($\textsf{ETH}$). Lastly, we explore the parameterized complexity of the quantum non-identity check problem.