Gibbs state preparation for commuting Hamiltonian: Mapping to classical Gibbs sampling

• DOI: 10.48550/arxiv.2410.04909
• Publication Date: 2024-10-07
• Authors: Yeongwoo Hwang; Jiaqing Jiang
• Keywords: Gibbs sampling; Hamiltonian (control theory); Gibbs state

Gibbs state preparation, or sampling, is a key computational technique extensively used in physics, statistics, and other scientific fields. Recent efforts for designing fast mixing samplers quantum Hamiltonians have largely focused on commuting local (CLHs), non-trivial subclass of which include highly entangled systems such as the Toric code double model. Most previous relied simulating Davies generator, Lindbladian associated with thermalization process nature. Instead using we design different sampler various CLHs by giving reduction to classical Hamiltonians, sense that one can efficiently prepare some CLH $H$ computer long do sampling corresponding Hamiltonian $H^{(c)}$. We demonstrate our able replicate state-of-the-art results well regimes were previously unknown, low temperature region, there exists Hamiltonians. Our reductions are follows. - If 2-local qudit CLH, then $H^{(c)}$ Hamiltonian. 4-local qubit 2D lattice no qubits, planar graph. As an example, algorithm (defected) at any non-zero $\mathcal O(n^2)$ time. assuming terms uniformly correctable, constant-local