The Parametrized Complexity of Quantum Verification

Authors Srinivasan Arunachalam, Sergey Bravyi , Chinmay Nirkhe , Bryan O'Gorman

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Srinivasan Arunachalam
  • IBM Quantum, Thomas J Watson Research Center, Yorktown Heights, NY, USA
Sergey Bravyi
  • IBM Quantum, Thomas J Watson Research Center, Yorktown Heights, NY, USA
Chinmay Nirkhe
  • IBM Quantum, Thomas J Watson Research Center, Yorktown Heights, NY, USA
  • Electrical Engineering and Computer Sciences, University of California, Berkeley, CA, USA
  • Challenge Institute for Quantum Computation, University of California, Berkeley, CA, USA
Bryan O'Gorman
  • IBM Quantum, Thomas J Watson Research Center, Yorktown Heights, NY, USA


Part of this work was completed while CN and BO were participants in the Simons Institute for the Theory of Computing Summer Cluster on Quantum Computation. Additionally, we thank Sam Gunn, Zeph Landau, Dimitri Maslov and Kristan Temme for insightful discussions.

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Srinivasan Arunachalam, Sergey Bravyi, Chinmay Nirkhe, and Bryan O'Gorman. The Parametrized Complexity of Quantum Verification. In 17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 232, pp. 3:1-3:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum computation theory
  • parametrized complexity
  • quantum verification
  • QMA


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