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Guidable Local Hamiltonian Problems with Implications to Heuristic Ansatz State Preparation and the Quantum PCP Conjecture

Authors Jordi Weggemans , Marten Folkertsma, Chris Cade

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Jordi Weggemans
  • CWI & QuSoft, Amsterdam, The Netherlands
  • Fermioniq, Amsterdam, The Netherlands
Marten Folkertsma
  • CWI & QuSoft, Amsterdam, The Netherlands
Chris Cade
  • Fermioniq, Amsterdam, The Netherlands
  • QuSoft & University of Amsterdam (UvA), The Netherlands

Funding

CC was supported by Fermioniq B.V., and thanks QuSoft and CWI for their accommodation whilst this work was completed. MF and JW were supported by the Dutch Ministry of Economic Affairs and Climate Policy (EZK), as part of the Quantum Delta NL programme.

Acknowledgements

The authors thank Jonas Helsen and Harry Buhrman for useful discussions and Sevag Gharibian and François Le Gall for their comments on an earlier version of the manuscript. We also thank Ronald de Wolf for providing feedback on the introduction. We thank the anonymous reviewers for their helpful comments, and in particular an anonymous STOC reviewer who provided an annotated version of an earlier version of this manuscript with very detailed and helpful comments.

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Jordi Weggemans, Marten Folkertsma, and Chris Cade. Guidable Local Hamiltonian Problems with Implications to Heuristic Ansatz State Preparation and the Quantum PCP Conjecture. In 19th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 310, pp. 10:1-10:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.TQC.2024.10

Abstract

We study "Merlinized" versions of the recently defined Guided Local Hamiltonian problem, which we call "Guidable Local Hamiltonian" problems. Unlike their guided counterparts, these problems do not have a guiding state provided as a part of the input, but merely come with the promise that one exists. We consider in particular two classes of guiding states: those that can be prepared efficiently by a quantum circuit; and those belonging to a class of quantum states we call classically evaluatable, for which it is possible to efficiently compute expectation values of local observables classically. We show that guidable local Hamiltonian problems for both classes of guiding states are QCMA-complete in the inverse-polynomial precision setting, but lie within NP (or NqP) in the constant precision regime when the guiding state is classically evaluatable. Our completeness results show that, from a complexity-theoretic perspective, classical Ansätze selected by classical heuristics are just as powerful as quantum Ansätze prepared by quantum heuristics, as long as one has access to quantum phase estimation. In relation to the quantum PCP conjecture, we (i) define a complexity class capturing quantum-classical probabilistically checkable proof systems and show that it is contained in BQP^NP[1] for constant proof queries; (ii) give a no-go result on "dequantizing" the known quantum reduction which maps a QPCP-verification circuit to a local Hamiltonian with constant promise gap; (iii) give several no-go results for the existence of quantum gap amplification procedures that preserve certain ground state properties; and (iv) propose two conjectures that can be viewed as stronger versions of the NLTS theorem. Finally, we show that many of our results can be directly modified to obtain similar results for the class MA.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum complexity theory
Keywords
  • Quantum complexity theory
  • local Hamiltonian problem
  • quantum state ansatzes
  • QCMA
  • quantum PCP conjecture

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