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arXiv:1507.00767

Unbiased Monte Carlo for the age of tensor networks

Andrew J. Ferris

Published Jul 2, 2015• 6 citations
arXivHTMLPDF

Abstract

A new unbiased Monte Carlo technique called Tensor Network Monte Carlo (TNMC) is introduced based on sampling all possible renormalizations (or course-grainings) of tensor networks, in this case matrix-product states. Tensor networks are a natural language for expressing a wide range of discrete physical and statistical problems, such as classical and quantum systems on a lattice at thermal equilibrium. By simultaneously sampling multiple degrees of freedom associated with each bond of the tensor network (and its renormalized form), we can achieve unprecedented low levels of statistical fluctuations which simultaneously parallel the impressive accuracy scaling of tensor networks while avoiding completely the variational bias inherent to those techniques, even with small bond dimensions. The resulting technique is essentially an aggressive multi-sampling technique that can account for the great majority of the partition function in a single sample. The method is quite general and can be combined with a variety of tensor renormalization techniques appropriate to different geometries and dimensionalities.

DOI
10.48550/arxiv.1507.00767

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A new unbiased Monte Carlo technique called Tensor Network Monte Carlo (TNMC) is introduced based on sampling all possible renormalizations (or course-grainings) of tensor networks, in this case matrix-product states. Tensor networks are a natural language for expressing a wide range of discrete physical and statistical problems, such as classical and quantum systems on a lattice at thermal equilibrium. By simultaneously sampling multiple degrees of freedom associated with each bond of the tensor network (and its renormalized form), we can achieve unprecedented low levels of statistical fluctuations which simultaneously parallel the impressive accuracy scaling of tensor networks while avoiding completely the variational bias inherent to those techniques, even with small bond dimensions. The resulting technique is essentially an aggressive multi-sampling technique that can account for the great majority of the partition function in a single sample. The method is quite general and can be combined with a variety of tensor renormalization techniques appropriate to different geometries and dimensionalities.
Disordered lattice spin systems are crucial in both theoretical and applied physics. However, understanding their properties poses significant challenges to Monte Carlo simulations. In this work, we investigate the two-dimensional random-bond Ising model using the recently proposed tensor network Monte Carlo (TNMC) method. This method generates biased samples from conditional probabilities computed via tensor network contractions and corrects the bias using the Metropolis scheme. Consequently, the proposals provided by tensor networks function as block updates for Monte Carlo simulations. Through extensive numerical experiments, we demonstrate that TNMC simulations can be performed on lattices as large as $1024\ifmmode\times\else\texttimes\fi{}1024$ spins with moderate computational resources, a substantial increase from the previous maximum size of $64\ifmmode\times\else\texttimes\fi{}64$ in Markov-chain Monte Carlo. Notably, we observe an almost complete absence of critical slowing down, enabling the efficient collection of unbiased samples and averaging over a large number of random realizations of bond disorders. We successfully pinpoint the multicritical point along the Nishimori line with significant precision and accurately determined the bulk and surface critical exponents. Our findings suggest that TNMC is a highly efficient algorithm for exploring disordered and frustrated systems in two dimensions.
Efficient sampling of two-dimensional statistical physics systems remains a central challenge in computational statistical physics. Traditional Markov chain Monte Carlo (MCMC) methods, including cluster algorithms, provide only partial solutions, as their efficiency collapses for large systems in the presence of frustration and quenched disorder. The recently proposed Tensor Network Monte Carlo (TNMC) method offers a promising alternative, yet its original implementation suffers from inefficiencies due to the lack of scalable parallel sampling. In this work, we introduce BatchTNMC, a GPU-optimized and parallelized implementation of TNMC tailored for large-scale simulations of two-dimensional spin glasses. By leveraging batch processing and parallel sampling across multiple disorder realizations, our implementation achieves speedups of up to five orders of magnitude compared with the original serial scheme. Benchmarking on two-dimensional spin glasses demonstrates dramatic gains in efficiency: for instance, on a single GPU, BatchTNMC concurrently produces 1000 uncorrelated and unbiased samples across 1000 disorder realizations on 1024×10241024\times 10241024×1024 lattices in just 3.3 hours, with an acceptance probability of 37%. These results establish BatchTNMC as a scalable and powerful computational framework for the study of two-dimensional disordered spin glass systems.
We propose a tensor-network Monte Carlo (TNMC) approach for unitary evolution following the compression sequence of the time-evolving block decimation (TEBD) algorithm. In the TNMC approach, the obtained results contain evaluable statistical errors rather than truncation errors, unlike ordinary singular-value-decomposition-based methods such as the TEBD algorithm. Consequently, one can estimate unbiased expectation values within statistical errors even with a finite bond dimension. Since the sampling scheme is introduced in the simulations of unitary evolution, the proposed Monte Carlo scheme may suffer from a sign problem. We observe that the sign problem can be mitigated by increasing the bond dimension. We apply the proposed TNMC approach to the Hamiltonian and the Floquet dynamics. Numerical experiments show that the TNMC approach can estimate accurate expectation values of observables even when the TEBD method with the same bond dimension cannot. The proposed approach can be a new direction for improving the classical simulatability of unitary evolution.
Techniques for approximately contracting tensor networks are limited in how efficiently they can make use of parallel computing resources. In this work we demonstrate and characterize a Monte Carlo approach to the tensor network renormalization group method which can be used straightforwardly on modern computing architectures. We demonstrate the efficiency of the technique and show that Monte Carlo tensor network renormalization provides an attractive path to improving the accuracy of a wide class of challenging computations while also providing useful estimates of uncertainty and a statistical guarantee of unbiased results.
When combined with highly expressive ansatz functions such as neural quantum states, variational Monte Carlo (VMC) constitutes a versatile numerical approach to tackle the quantum many-body problem in and out of equilibrium. However, its traditional formulation exhibits a subtle estimation bias leading to inaccuracies, which can be particularly detrimental when addressing real time dynamics. In this work, we investigate two avenues to circumvent said estimation bias. First, we propose an unbiased variant of time-dependent VMC using self-normalized importance sampling with respect to a cutoff-based deformation of the Born distribution. We demonstrate the feasibility and accuracy of the approach in pathological and generic cases of quench dynamics. Furthermore, we explore an alternative sampling strategy based on active learning via the tensor cross interpolation (TCI). While we find that our choice of tensor network architecture lacks the required low rank property, the proposed TCI-based algorithm complements the conventional importance sampling paradigm, providing an alternative perspective that may be further explored in future work.
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.