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 $\mathsf{QCMA}$-complete in the inverse-polynomial precision setting, but lie within $\mathsf{NP}$ (or $\mathsf{NqP}$) in the constant precision regime when the guiding state is classically evaluatable. Our completeness results show that, from a complexity-theoretic perspective, classical Ans\"atze selected by classical heuristics are just as powerful as quantum Ans\"atze 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 $\mathsf{BQP}^{\mathsf{NP}[1]}$ for constant proof queries; (ii) give a no-go result on 'dequantizing' the known quantum reduction which maps a $\mathsf{QPCP}$-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 $\mathsf{MA}$.

翻译：我们研究了近期提出的“引导局部哈密顿问题”的“梅林化”版本，称之为“可引导局部哈密顿问题”。与引导版本不同，这类问题并不将引导态作为输入的一部分提供，仅作出其存在的承诺。我们特别考虑了两类引导态：可通过量子电路高效制备的态；以及属于一类称为经典可评估量子态的态——对此类态，可在经典计算机上高效计算局部观测量的期望值。我们证明，对于逆多项式精度设定，两类引导态对应的可引导局部哈密顿问题均为$\mathsf{QCMA}$-完全问题；而在恒定精度设定下，当引导态为经典可评估时，该问题属于$\mathsf{NP}$（或$\mathsf{NqP}$）。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，由经典启发式算法选择的经典Ansätze与由量子启发式算法制备的量子Ansätze具有同等能力。关于量子PCP猜想，我们：(i) 定义了一个捕获量子-经典概率可检验证明系统的复杂性类，并证明在恒定证明查询次数下该类包含于$\mathsf{BQP}^{\mathsf{NP}[1]}$；(ii) 给出一个关于已知量子归约“去量子化”的不可行结果——该归约将$\mathsf{QPCP}$验证电路映射至具有恒定承诺间隙的局部哈密顿量；(iii) 给出多个关于保持特定基态性质的量子间隙放大过程存在的不可行结果；(iv) 提出两个可视为NLTS定理增强版本的猜想。最后，我们证明许多结果可直接修改以得到关于复杂性类$\mathsf{MA}$的类似结论。