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arXiv:2107.14454 · Physical review. E

Reducing autocorrelation time in determinant quantum Monte Carlo using the Wang-Landau algorithm: Application to the Holstein model

Meng Yao, Da Wang, Qiang-Hua Wang

Published Aug 16, 2021• 4 citations
arXivHTMLPDF

Abstract

When performing a Monte Carlo calculation, the running time should, in principle, be much longer than the autocorrelation time in order to get reliable results. Among different lattice fermion models, the Holstein model is notorious for its particularly long autocorrelation time. In this paper, we employ the Wang-Landau algorithm in the determinant quantum Monte Carlo to achieve the flat-histogram sampling in the "configuration weight space," which can greatly reduce the autocorrelation time by sacrificing some sampling efficiency. The proposal is checked in the Holstein model on both square and honeycomb lattices. Based on such a Wang-Landau assisted determinant quantum Monte Carlo method, some models with long autocorrelation times can now be simulated possibly.

DOI
10.1103/physreve.104.025305

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When performing a Monte Carlo calculation, the running time should, in principle, be much longer than the autocorrelation time in order to get reliable results. Among different lattice fermion models, the Holstein model is notorious for its particularly long autocorrelation time. In this paper, we employ the Wang-Landau algorithm in the determinant quantum Monte Carlo to achieve the flat-histogram sampling in the "configuration weight space," which can greatly reduce the autocorrelation time by sacrificing some sampling efficiency. The proposal is checked in the Holstein model on both square and honeycomb lattices. Based on such a Wang-Landau assisted determinant quantum Monte Carlo method, some models with long autocorrelation times can now be simulated possibly.
It has been shown that the Metropolis algorithm can be implemented on quantum computers in a way that avoids the sign problem. However, flat histogram techniques are often preferred as they don't suffer from the same limitations that afflict Metropolis for problems of real-world interest and provide a host of other benefits. In particular, the Wang-Landau method is known for its efficiency and accuracy. In this work we design, implement, and validate a quantum algorithm for Wang-Landau sampling, greatly expanding the range of quantum many body problems solvable by Monte Carlo simulation.
Monte Carlo (MC) simulations are essential computational approaches with widespread use throughout all areas of science. We present a method for accelerating lattice MC simulations using fully connected and convolutional artificial neural networks that are trained to perform local and global moves in configuration space, respectively. Both networks take local spacetime MC configurations as input features and can, therefore, be trained using samples generated by conventional MC runs on smaller lattices before being utilized for simulations on larger systems. This approach is benchmarked for the case of determinant quantum Monte Carlo (DQMC) studies of the two-dimensional Holstein model. We find that both artificial neural networks are capable of learning an unspecified effective model that accurately reproduces the MC configuration weights of the original Hamiltonian and achieve an order of magnitude speedup over the conventional DQMC algorithm. Our approach is broadly applicable to many classical and quantum lattice MC algorithms.
We discuss that flat-histogram techniques can be appropriately applied in the sampling of quantum Monte Carlo simulation in order to improve the statistical quality of the results at long imaginary time or low excitation energy. Typical imaginary-time correlation functions calculated in quantum Monte Carlo are subject to exponentially growing errors as the range of imaginary time grows and this smears the information on the low energy excitations. We show that we can extract the low energy physics by modifying the Monte Carlo sampling technique to one in which configurations which contribute to making the histogram of certain quantities flat are promoted. We apply the diagrammatic Monte Carlo (diag-MC) method to the motion of a single hole in the t-J model and we show that the implementation of flat-histogram techniques allows us to calculate the Green's function in a wide range of imaginary-time. In addition, we show that applying the flat-histogram technique alleviates the "sign"-problem associated with the simulation of the single-hole Green's function at long imaginary time.
The self-learning Monte Carlo method (SLMC), using a trained effective model to guide Monte Carlo sampling processes, is a powerful general-purpose numerical method recently introduced to speed up simulations in (quantum) many-body systems. In this Rapid Communication, we further improve the efficiency of SLMC by enforcing physical symmetries on the effective model. We demonstrate its effectiveness in the Holstein Hamiltonian, one of the most fundamental many-body descriptions of electron-phonon coupling. Simulations of the Holstein model are notoriously difficult due to a combination of the typical cubic scaling of fermionic Monte Carlo and the presence of extremely long autocorrelation times. Our method addresses both bottlenecks. This enables simulations on large lattices in the most difficult parameter regions, and an evaluation of the critical point for the charge density wave transition at half filling with high precision. We argue that our work opens a research area of quantum Monte Carlo, providing a general procedure to deal with ergodicity in situations involving Hamiltonians with multiple, distinct low-energy states.
To evaluate the effectiveness of machine learning in systems with competing interactions, we developed a self-learning quantum Monte Carlo (SLQMC) method to simulate the phase transition in the classical Holstein-spin-fermion model. In SLQMC, machine learning techniques are employed to approximate the free energy, thereby bypassing the need for exact diagonalization and significantly reducing computational cost. We assess the performance of SLQMC using both linear regression and neural network models. Our results show that both models are capable of capturing the phase transition from the antiferromagnetic state to the charge-density-wave state. However, the sampling efficiency decreases near the AFM-CDW phase transition, which is attributed to the increased mean-squared-error of the machine learning model. Additionally, the sampling efficiency decreases with increasing lattice size. This suppression is due to the increased root-mean-squared-error as the machine learning model is applied to a large lattice and the finite-size effect, wherein the energy gap between the ground state and low-energy excited states decreases as the lattice grows. Our findings highlight the necessity of highly accurate machine learning models to simulate theoretical models with complex, competing microscopic interactions on a large lattice.
To evaluate the effectiveness of machine learning in systems with competing interactions, we developed a self-learning quantum Monte Carlo (SLQMC) method to simulate the phase transition in the classical Holstein spin-fermion model. In SLQMC, machine-learning techniques are employed to approximate the free energy, thereby bypassing the need for exact diagonalization and significantly reducing computational cost. We assess the performance of SLQMC using both linear regression and neural network models. Our results show that both models are capable of capturing the phase transition from the antiferromagnetic (AFM) state to the charge-density-wave (CDW) state. However, the sampling efficiency decreases near the AFM-CDW phase transition, which is attributed to the increased mean-squared error of the machine-learning model. Additionally, the sampling efficiency decreases with increasing lattice size. This suppression is due to the increased root-mean-squared error as the machine-learning model is applied to a large lattice and the finite-size effect, wherein the energy gap between the ground state and low-energy excited states decreases as the lattice grows. Our findings highlight the necessity of highly accurate machine-learning models to simulate theoretical models with complex, competing microscopic interactions on a large lattice.