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 $\mathsf{QCMA}$-complete in the inverse-polynomial precision setting, but lie within $\mathsf{NP}$ (or $\mathsf{NqP}$) in certain parameter regimes when the guiding state is classically evaluatable. We discuss the implications of these results to heuristic ans\"atze state preparation and the quantum PCP conjecture. Our completeness results show that, from a complexity-theoretic perspective, classical ans\"atze prepared by classical heuristics are just as powerful as quantum ans\"atze 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 $\mathsf{QCMA}$ and show that it is equal to $\mathsf{NP}$ 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 $\mathsf{MA}$.

翻译：我们引入了近期定义的“引导式局部哈密顿问题”的“梅林化”版本，称之为“可指导的局部哈密顿”问题。与引导版本不同，这些问题不要求输入中提供引导态，仅保证存在一个满足特定约束的引导态。我们特别考虑了两类引导态：一类可由量子电路高效制备；另一类属于我们称为经典可评估的量子态，这类态具有简短的经典描述，可据此经典高效计算局部可观测量的期望值。我们证明，对于这两类引导态，可指导的局部哈密顿问题在逆多项式精度设定下是$\mathsf{QCMA}$-完全的，但在引导态属于经典可评估态的特定参数范围内，则属于$\mathsf{NP}$（或$\mathsf{NqP}$）。我们讨论了这些结果对启发式变分量子态制备和量子PCP猜想的启示。我们的完全性结果表明，从复杂性理论视角看，只要具备量子相位估计能力，由经典启发式制备的经典变分态与由量子启发式制备的量子变分态具有同等能力。关于量子PCP猜想，我们：（i）定义了$\mathsf{QCMA}$的PCP版本，证明其在量子归约下等价于$\mathsf{NP}$；（ii）证明了若干关于保持特定基态性质的量子能隙放大过程不可存在性的否定结果；（iii）提出了两个可视为NLTS定理强化版本的猜想。最后，我们展示了多数结果可直接修改以获得关于$\mathsf{MA}$类的类似结论。