Guidable Local Hamiltonian Problems with Implications to Heuristic Ansätze State Preparation and the Quantum PCP Conjecture
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We introduce '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 and that it satisfies certain constraints. 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, which have a short classical description from 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 certain parameter regimes when the guiding state is classically evaluatable. We discuss the implications of these results to heuristic ansätze state preparation and the quantum PCP conjecture. Our completeness results show that, from a complexity-theoretic perspective, classical ansätze prepared by classical heuristics are just as powerful as quantum ansätze prepared by quantum heuristics, so long as one has access to quantum phase estimation. In relation to the quantum PCP conjecture, we (i) define a PCP for 𝖰𝖢𝖬𝖠 and show that it is equal to 𝖭𝖯 under quantum reductions; (ii) show several no-go results for the existence of quantum gap amplification procedures that preserve certain ground state properties; and (iii) 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.
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