标题： 张量网络表示中的马尔可夫链蒙特卡洛方法 显示英文标题

标题： Markov Chain Monte Carlo in Tensor Network Representation

摘要： 马尔可夫链蒙特卡罗（MCMC）是一种从复杂概率分布中抽样的强大工具。尽管具有很强的多功能性，MCMC 经常受到强自相关和负符号问题的影响，导致统计误差的收敛速度变慢。我们提出了一种基于张量网络表示的新 MCMC 形式，以减少群体方差并系统地缓解这些问题。通过在张量网络框架中引入随机投影仪并采用马尔可夫链采样，我们的方法消除了与张量收缩中的低秩近似相关的系统误差，同时保持了张量网络方法的高精度。我们在二维伊辛模型上展示了所提出方法的有效性，随着边界维数截断的增加，统计误差呈指数级减少。此外，我们解决了具有负权重系统的符号问题，在边界维数截断增加时显示平均符号有显著改善。所提出的框架为复杂系统中的准确统计估计提供了一个稳健的解决方案，并为计算物理及其更广泛的应用铺平了道路。 显示英文摘要

摘要： Markov chain Monte Carlo (MCMC) is a powerful tool for sampling from complex probability distributions. Despite its versatility, MCMC often suffers from strong autocorrelation and the negative sign problem, leading to slowing down the convergence of statistical error. We propose a novel MCMC formulation based on tensor network representations to reduce the population variance and mitigate these issues systematically. By introducing stochastic projectors into the tensor network framework and employing Markov chain sampling, our method eliminates the systematic error associated with low-rank approximation in tensor contraction while maintaining the high accuracy of the tensor network method. We demonstrate the effectiveness of the proposed method on the two-dimensional Ising model, achieving an exponential reduction in statistical error with increasing bond dimension cutoff. Furthermore, we address the sign problem in systems with negative weights, showing significant improvements in average signs as bond dimension cutoff increases. The proposed framework provides a robust solution for accurate statistical estimation in complex systems, paving the way for broader applications in computational physics and beyond.