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

翻译：吉布斯态制备，或称吉布斯采样，是物理学、统计学及其他科学领域中广泛使用的一项关键计算技术。近期，针对量子哈密顿量设计快速混合吉布斯采样器的研究主要集中于对易局域哈密顿量（CLHs），这是哈密顿量中一个非平凡的子类，包含了如环面码和量子双模型等高纠缠系统。以往大多数吉布斯采样器依赖于模拟戴维斯生成元，这是一种与自然界热化过程相关的林德布拉德算子。我们并未使用戴维斯生成元，而是通过将其约化为经典哈密顿量，为多种对易局域哈密顿量设计了一种不同的吉布斯采样器。其核心在于：只要能够对相应的经典哈密顿量 $H^{(c)}$ 进行高效的经典吉布斯采样，便能在量子计算机上高效地制备某些对易局域哈密顿量 $H$ 的吉布斯态。我们证明，只要存在针对相应经典哈密顿量的快速混合吉布斯采样器，我们的吉布斯采样器不仅能够复现最先进的结果，还能在以往未知的区域内（如低温区）制备吉布斯态。我们的约化方案如下：- 若 $H$ 是一个 2-局域量子比特对易哈密顿量，则 $H^{(c)}$ 是一个 2-局域量子比特经典哈密顿量。- 若 $H$ 是二维晶格上的 4-局域量子比特对易哈密顿量且不存在经典量子比特，则 $H^{(c)}$ 是平面图上的 2-局域量子比特经典哈密顿量。例如，我们的算法能够在 $\mathcal O(n^2)$ 时间内为（有缺陷的）环面码在任何非零温度下制备吉布斯态。- 若 $H$ 是二维晶格上的 4-局域量子比特对易哈密顿量且存在经典量子比特，并假设量子项是一致可校正的，则 $H^{(c)}$ 是一个常数局域的经典哈密顿量。