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Unitary Property Testing Lower Bounds by Polynomials

Authors Adrian She, Henry Yuen

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Author Details

Adrian She
  • University of Toronto, Canada
Henry Yuen
  • Columbia University, New York, NY, USA

Funding

  • She, Adrian: NSERC Canada Graduate Scholarship.
  • Yuen, Henry: AFOSR award FA9550-21-1-0040, NSF CAREER award CCF-2144219, and the Sloan Foundation.

Acknowledgements

We thank Joshua Grochow for helpful conversations about invariant theory. We thank Shivam Nadimpalli for suggesting the problem about testing k-locality of a Hamiltonian. We thank Kunal Marwaha and Gregory Rosenthal for helpful discussions. We thank the reviewers for their feedback.

Cite AsGet BibTex

Adrian She and Henry Yuen. Unitary Property Testing Lower Bounds by Polynomials. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 96:1-96:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.ITCS.2023.96

Abstract

We study unitary property testing, where a quantum algorithm is given query access to a black-box unitary and has to decide whether it satisfies some property. In addition to containing the standard quantum query complexity model (where the unitary encodes a binary string) as a special case, this model contains "inherently quantum" problems that have no classical analogue. Characterizing the query complexity of these problems requires new algorithmic techniques and lower bound methods. Our main contribution is a generalized polynomial method for unitary property testing problems. By leveraging connections with invariant theory, we apply this method to obtain lower bounds on problems such as determining recurrence times of unitaries, approximating the dimension of a marked subspace, and approximating the entanglement entropy of a marked state. We also present a unitary property testing-based approach towards an oracle separation between QMA and QMA(2), a long standing question in quantum complexity theory.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum complexity theory
  • Theory of computation → Quantum complexity theory
Keywords
  • Quantum query complexity
  • polynomial method
  • unitary property testing
  • quantum proofs
  • invariant theory
  • quantum recurrence time
  • entanglement entropy
  • BQP
  • QMA
  • QMA(2)

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