Tensor-Network Markov-Chain Monte Carlo: Efficient Sampling of Three-Dimensional Spin Glasses.

Sampling equilibrium configurations of three-dimensional (3D) spin glasses with quenched disorder remains a fundamental challenge in statistical physics. The rugged energy landscape, pronounced critical slowing down, and intrinsic ergodicity breaking render standard Monte Carlo methods severely inefficient, particularly for large systems at low temperatures. In this Letter, we introduce the tensor-network Markov-chain Monte Carlo (TNMCMC) approach to address the issue. By formulating large-scale collective updates via tensor-network contractions on 2D lattice slices, TNMCMC yields unbiased samples of the Boltzmann distribution while substantially reducing autocorrelation times relative to conventional MCMC approaches. Numerical benchmarks on 3D spin glasses demonstrate that the method systematically mitigates critical slowing down. Furthermore, application to three- and four-state 3D Potts models shows that TNMCMC can traverse the exponential free-energy barriers of first-order phase transitions that typically hinder local MC updates. These results establish TNMCMC as a promising numerical framework capable of addressing important 3D problems in statistical physics and beyond, such as the precise computation of low-temperature thermodynamic properties of three-dimensional spin glasses and the decoding of quantum error-correcting codes with a three-dimensional topology.