Markov Chain Monte Carlo in Tensor Network Representation

TL;DR

This paper introduces a tensor network-based MCMC method that reduces variance and mitigates the sign problem, significantly improving sampling efficiency in complex physical systems like the 2D Ising model.

Contribution

It presents a novel MCMC formulation using tensor networks with stochastic projectors, eliminating systematic errors and addressing the sign problem.

Findings

Exponential reduction in statistical error with increased bond dimension.

Significant improvements in average signs for systems with negative weights.

Effective sampling in the 2D Ising model demonstrating robustness.

Abstract

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.