Yoshioka Nobuyuki, Akagi Yutaka, Katsura Hosho
Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan.
Phys Rev E. 2019 Mar;99(3-1):032113. doi: 10.1103/PhysRevE.99.032113.
We find an exact mapping from the generalized Ising models with many-spin interactions to equivalent Boltzmann machines, i.e., the models with only two-spin interactions between physical and auxiliary binary variables accompanied by local external fields. More precisely, the appropriate combination of the algebraic transformations, namely the star-triangle and decoration-iteration transformations, allows one to express the model in terms of fewer-spin interactions at the expense of the degrees of freedom. Furthermore, the benefit of the mapping in Monte Carlo simulations is discussed. In particular, we demonstrate that the application of the method in conjunction with the Swendsen-Wang algorithm drastically reduces the critical slowing down in a model with two- and three-spin interactions on the Kagomé lattice.
我们发现了从具有多自旋相互作用的广义伊辛模型到等效玻尔兹曼机的精确映射，即物理和辅助二元变量之间仅具有两自旋相互作用且伴有局部外场的模型。更确切地说，代数变换（即星 - 三角变换和装饰 - 迭代变换）的适当组合，使得人们能够以自由度为代价，用较少自旋相互作用来表示该模型。此外，还讨论了这种映射在蒙特卡罗模拟中的优势。特别地，我们证明了将该方法与斯文森 - 王算法结合应用，能大幅减少 Kagomé 晶格上具有两自旋和三自旋相互作用的模型中的临界慢化现象。
