The Quantum Complexity of Markov Chain Monte Carlo

David Layden, Guglielmo Mazzola, Ryan V. Mishmash +4 more

Kefan Cao, Zidong Cui, Lei Wang +1 more

Detailed balance underlies conventional Markov-chain Monte Carlo (MCMC) algorithms. Yet in classical systems, breaking detailed balance generates irreversible probability currents and can accelerate sampling. Whether irreversibility can similarly enhance quantum MCMC remains an intriguing question. Here we show that irreversibility provides a new route to improving the recent quantum-enhanced MCMC (QEMC), which combines quantum proposals with classical acceptance. By introducing state-dependent proposals that break detailed balance while preserving the target stationary distribution, we develop an irreversible quantum-enhanced Monte Carlo (IQEMC). Guided by Landau-Zener transitions, IQEMC promotes large energy descents from high-energy states while maintaining stable transitions near low-energy states. On spin-glass benchmarks, IQEMC outperforms QEMC without increasing computational complexity and, unlike the annealing baseline, exhibits a spectral gap that increases with system size and annealing speed. These results establish irreversibility as a physically grounded mechanism for enhancing quantum MCMC.

Shunta Arai, Tadashi Kadowaki

In this study, we propose quantum annealing-enhanced Markov Chain Monte Carlo (QAEMCMC), where QA is integrated into the MCMC subroutine. QA efficiently explores low-energy configurations and overcomes local minima, enabling the generation of proposal states with a high acceptance probability. We benchmark QAEMCMC for the Sherrington-Kirkpatrick model and demonstrate its superior performance over the classical MCMC method. Our results reveal larger spectral gaps, faster convergence of energy observables, and reduced total variation distance between the empirical and target distributions. QAEMCMC accelerates MCMC and provides an efficient method for complex systems, paving the way for scalable quantum-assisted sampling strategies.

안민정

Tao Chen, Jing Liu, Youjin Deng +1 more

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.

Luigi Accardi, Dmitri Koroliouk