We show that the Guided Local Hamiltonian problem for stoquastic Hamiltonians is (promise) BPP-hard. The Guided Local Hamiltonian problem extends the Local Hamiltonian problem by incorporating an additional input known as a guiding state, which is promised to overlap with the ground state. For a range of local Hamiltonian families, prior work shows this problem is (promise) BQP-hard, though for stoquastic Hamiltonians, the complexity was previously unknown. We obtain our results by first reducing from quantum-inspired BPP circuits to 6-local stoquastic Hamiltonians. We prove particular classes of quantum states, known as semi-classical encoded subset states, can guide the estimation of the ground-state energy. Our analysis shows that this BPP-hardness does not depend on locality, i.e., the result holds for 2-local stoquastic Hamiltonians. Additional arguments extend this BPP-hardness to Hamiltonians restricted to a square lattice. We further show that for stoquastic Hamiltonians with a fixed local constraint on a subset of the system qubits, the Guided Local Hamiltonian problem is BQP-hard. In addition to these hardness results, we present a deterministic classical approximation algorithm for the problem under the conditions of constant promise gap, constant overlap, and constant spectral gap, when the guiding state is preparable in constant depth by a geometrically local circuit.

翻译：我们证明了可停可停哈密顿量的引导局域哈密顿问题是(承诺)BPP-难的。引导局域哈密顿问题扩展了局域哈密顿问题，增加了一个称为引导态的附加输入，该输入被承诺与基态有重叠。对于一系列局域哈密顿族，先前的研究表明该问题是(承诺)BQP-难的，但对于可停可停哈密顿量，其复杂性此前未知。我们通过首先从量子启发的BPP电路归约到6-局域可停可停哈密顿量来得到结果。我们证明了特定类别的量子态（称为半经典编码子集态）可以引导基态能量的估计。我们的分析表明，这种BPP-难性不依赖于局域性，即该结果对2-局域可停可停哈密顿量成立。进一步论证将这种BPP-难性扩展到限制在正方晶格上的哈密顿量。我们还证明，对于在系统量子比特子集上具有固定局域约束的可停可停哈密顿量，引导局域哈密顿问题是BQP-难的。除了这些困难性结果，我们还提出了一种确定性经典近似算法，适用于承诺间隙、重叠和谱间隙均为常数，且引导态可由几何局域电路在常数深度内制备的情况。