论文标题
用于自旋眼镜基态的热带张量网络
Tropical Tensor Network for Ground States of Spin Glasses
论文作者
论文摘要
我们提出了一种统一的精确张量网络方法，以计算基态能量，确定最佳配置并计算自旋玻璃的溶液数量。该方法基于张量网络，其在半度性上定义的热带代数。收缩热带张量网络可提供基态能量；通过张量网络收缩进行区分可提供基态配置；混合热带代数和普通代数可计算基态变性。该方法将图形模型，张量网络，可区分编程和量子电路模拟的概念汇总在一起，并轻松利用图形处理单元（GPU）的计算能力。对于应用，我们计算了高达1024旋转的正方形晶格上的iSing自旋玻璃的确切基态能量，在最高216个旋转的立方晶格上，以及3个常规随机图220旋转，在单个GPU上旋转；我们在不到100秒内获得了（+/-）J ising自旋玻璃的（+/-）J ising自旋玻璃的精确基态能量，并研究了（+/-）J旋转玻璃的残留熵的确切值；最后，我们研究了最高18 x 18的平方晶格玻璃杯的地面能量和熵。我们的方法为自旋玻璃和组合优化问题的精确算法提供了基准和基准，并评估了启发式算法和平均场理论。
We present a unified exact tensor network approach to compute the ground state energy, identify the optimal configuration, and count the number of solutions for spin glasses. The method is based on tensor networks with the Tropical Algebra defined on the semiring. Contracting the tropical tensor network gives the ground state energy; differentiating through the tensor network contraction gives the ground state configuration; mixing the tropical algebra and the ordinary algebra counts the ground state degeneracy. The approach brings together the concepts from graphical models, tensor networks, differentiable programming, and quantum circuit simulation, and easily utilizes the computational power of graphical processing units (GPUs). For applications, we compute the exact ground state energy of Ising spin glasses on square lattice up to 1024 spins, on cubic lattice up to 216 spins, and on 3 regular random graphs up to 220 spins, on a single GPU; We obtain exact ground state energy of (+/-)J Ising spin glass on the chimera graph of D-Wave quantum annealer of 512 qubits in less than 100 seconds and investigate the exact value of the residual entropy of (+/-)J spin glasses on the chimera graph; Finally, we investigate ground-state energy and entropy of 3-state Potts glasses on square lattices up to size 18 x 18. Our approach provides baselines and benchmarks for exact algorithms for spin glasses and combinatorial optimization problems, and for evaluating heuristic algorithms and mean-field theories.
