arXiv 2409.083522024-09-12cond-mat.stat-mechcond-mat.dis-nnquant-ph

利用矩阵乘积态学习广义统计力学

Learning Generalized Statistical Mechanics with Matrix Product States

Pablo Díez-Valle, Fernando Martínez-García, Juan José García-Ripoll, Diego Porras

备注：: 6 pages, 6 figures + Supplementary material (5 pages, 5 figures)

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AI总结：

本研究提出基于矩阵乘积态的变分算法，用Tsallis熵定义的广义自由能训练模型，结合变分退火方案避免局部极小，在Ising自旋玻璃问题中验证有效，为广义统计物理研究和组合优化求解提供新路径。

摘要：

我们提出了一种基于Matrix Product States（矩阵乘积态）的变分算法，该算法通过最小化使用Tsallis熵而非标准吉布斯熵定义的广义自由能进行训练。由此，我们的模型能够生成与广义统计力学相关的概率分布。该模型可被高效训练，因为所得自由能及其梯度可通过张量网络收缩精确计算，而标准方法需要通过采样估计吉布斯熵。我们设计了一种通过逐步提升逆温度实现的变分退火方案，可在训练模型时避免陷入局部极小值。我们在Ising自旋玻璃问题中，通过将该方法与精确数值结果及准精确解析近似进行对比，验证了其有效性。我们的工作为研究广义统计物理以及利用张量网络求解组合优化问题开辟了新的可能。

英文摘要：

We introduce a variational algorithm based on Matrix Product States that is trained by minimizing a generalized free energy defined using Tsallis entropy instead of the standard Gibbs entropy. As a result, our model can generate the probability distributions associated with generalized statistical mechanics. The resulting model can be efficiently trained, since the resulting free energy and its gradient can be calculated exactly through tensor network contractions, as opposed to standard methods which require estimating the Gibbs entropy by sampling. We devise a variational annealing scheme by ramping up the inverse temperature, which allows us to train the model while avoiding getting trapped in local minima. We show the validity of our approach in Ising spin-glass problems by comparing it to exact numerical results and quasi-exact analytical approximations. Our work opens up new possibilities for studying generalized statistical physics and solving combinatorial optimization problems with tensor networks.

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