The Guided Local Hamiltonian Problem for Stoquastic Hamiltonians
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866910194971181056 |
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| author | Waite, Gabriel |
| author_facet | Waite, Gabriel |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_25829 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Guided Local Hamiltonian Problem for Stoquastic Hamiltonians Waite, Gabriel Quantum Physics Computational Complexity 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. |
| title | The Guided Local Hamiltonian Problem for Stoquastic Hamiltonians |
| topic | Quantum Physics Computational Complexity |
| url | https://arxiv.org/abs/2509.25829 |