The Guided Local Hamiltonian Problem for Stoquastic Hamiltonians

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.