© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:1807.04955 · Physical review. B./Physical review. B

Self-learning Monte Carlo method with Behler-Parrinello neural networks

Yuki Nagai, Masahiko Okumura, Akinori Tanaka

Published Mar 9, 2020• 28 citations

Abstract

We propose a general way to construct an effective Hamiltonian in the self-learning Monte Carlo method (SLMC), which speeds up Monte Carlo simulations by training an effective model to propose uncorrelated configurations in the Markov chain. Its applications are, however, limited. This is because it is not obvious to find the explicit form of the effective Hamiltonians. Particularly, it is difficult to make trainable effective Hamiltonians including many-body interactions. In order to overcome this critical difficulty, we introduce the Behler-Parrinello neural networks (BPNNs) as effective Hamiltonian without any prior knowledge, which is used to construct the potential-energy surfaces in interacting many particle systems for molecular dynamics. We combine SLMC with BPNN by focusing on a divisibility of Hamiltonian and propose how to construct the elementwise configurations. We apply it to quantum impurity models. We observed significant improvement of the acceptance ratio from 0.01 (the effective Hamiltonian with the explicit form) to 0.76 (BPNN). This drastic improvement implies that the BPNN effective Hamiltonian includes many-body interaction, which is omitted in the effective Hamiltonian with the explicit forms. The BPNNs make SLMC more promising.

DOI: 10.1103/physrevb.101.115111

Related papers

Self-learning Monte Carlo with deep neural networks

Huitao Shen, Junwei Liu, Liang Fu

The self-learning Monte Carlo (SLMC) method is a general algorithm to speedup MC simulations. Its efficiency has been demonstrated in various systems by introducing an effective model to propose global moves in the configuration space. In this paper, we show that deep neural networks can be naturally incorporated into SLMC, and without any prior knowledge can learn the original model accurately and efficiently. Demonstrated in quantum impurity models, we reduce the complexity for a local update from $O({\ensuremath{\beta}}^{2})$ in Hirsch-Fye algorithm to $O(\ensuremath{\beta}ln\ensuremath{\beta})$, which is a significant speedup especially for systems at low temperatures.

arXiv:1801.01127•Physical review. B./Physical review. B•May 29, 2018•78 citations

Symmetry-enforced self-learning Monte Carlo method applied to the Holstein model

Chuang Chen, Xiao Yan Xu, Junwei Liu +3 more

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.

arXiv:1802.06177•Physical review. B./Physical review. B•Jul 11, 2018•55 citations

Self-learning Monte Carlo Method: A Review

Gaopei Pan, Chuang Chen, Zi Yang Meng

The Self-Learning Monte Carlo (SLMC) method is a Monte Carlo approach that has emerged in recent years by integrating concepts from machine learning with conventional Monte Carlo techniques. Designed to accelerate the numerical study of interacting many-body systems, SLMC significantly improves sampling efficiency by constructing an effective model -- via machine learning methods -- based on configurations generated by conventional Monte Carlo methods and then proposes global updates based on the effective model. This enhancement leads to a substantial reduction in autocorrelation time, especially near the critical region, where traditional methods typically suffer from critical slowing down and increased computation complexity. Moreover, SLMC maintains statistical accuracy by implementing a cumulative update scheme that rigorously satisfies the detailed balance condition. And more recent applications have extended the SLMC to convolutional neural networks with applications not only in condensed matter physics but also high-energy physics, quantum chemistry, and quantum simulations. The generic applicability and high computational efficiency make SLMC a powerful and scalable framework for quantum Monte Carlo simulations of strongly correlated electron systems, extending the reach of numerical investigations beyond the limitations of conventional techniques.

arXiv:2507.12554•Jul 16, 2025•1 citation

Self-learning Monte Carlo method and cumulative update in fermion systems

Junwei Liu, Huitao Shen, Yang Qi +2 more

We develop the self-learning Monte Carlo (SLMC) method, a general-purpose numerical method recently introduced to simulate many-body systems, for studying interacting fermion systems. Our method uses a highly efficient update algorithm, which we design and dub ``cumulative update'', to generate new candidate configurations in the Markov chain based on a self-learned bosonic effective model. From a general analysis and a numerical study of the double exchange model as an example, we find that the SLMC with cumulative update drastically reduces the computational cost of the simulation, while remaining statistically exact. Remarkably, its computational complexity is far less than the conventional algorithm with local updates.

arXiv:1611.09364•Physical review. B./Physical review. B•Jun 7, 2017•82 citations

Self-learning Monte Carlo method

Zi Yang Meng, Junwei Liu, Yang Qi +1 more

Monte Carlo simulation is an unbiased numerical tool for studying classical and quantum many-body systems. One of its bottlenecks is the lack of a general and efficient update algorithm for large size systems close to the phase transition, for which local updates perform badly. In this Rapid Communication, we propose a general-purpose Monte Carlo method, dubbed self-learning Monte Carlo (SLMC), in which an efficient update algorithm is first learned from the training data generated in trial simulations and then used to speed up the actual simulation. We demonstrate the efficiency of SLMC in a spin model at the phase transition point, achieving a 10–20 times speedup.

Physical Review Letters•Jan 1, 2017•1 citation

Self-learning Monte Carlo method

Junwei Liu, Yang Qi, Zi Yang Meng +1 more

Monte Carlo simulation is an unbiased numerical tool for studying classical and quantum many-body systems. One of its bottlenecks is the lack of a general and efficient update algorithm for large size systems close to the phase transition, for which local updates perform badly. In this Rapid Communication, we propose a general-purpose Monte Carlo method, dubbed self-learning Monte Carlo (SLMC), in which an efficient update algorithm is first learned from the training data generated in trial simulations and then used to speed up the actual simulation. We demonstrate the efficiency of SLMC in a spin model at the phase transition point, achieving a 10--20 times speedup.

arXiv:1610.03137•Physical review. B./Physical review. B•Jan 4, 2017•251 citations

We propose a general way to construct an effective Hamiltonian in the self-learning Monte Carlo method (SLMC), which speeds up Monte Carlo simulations by training an effective model to propose uncorrelated configurations in the Markov chain. Its applications are, however, limited. This is because it is not obvious to find the explicit form of the effective Hamiltonians. Particularly, it is difficult to make trainable effective Hamiltonians including many-body interactions. In order to overcome this critical difficulty, we introduce the Behler-Parrinello neural networks (BPNNs) as effective Hamiltonian without any prior knowledge, which is used to construct the potential-energy surfaces in interacting many particle systems for molecular dynamics. We combine SLMC with BPNN by focusing on a divisibility of Hamiltonian and propose how to construct the elementwise configurations. We apply it to quantum impurity models. We observed significant improvement of the acceptance ratio from 0.01 (the effective Hamiltonian with the explicit form) to 0.76 (BPNN). This drastic improvement implies that the BPNN effective Hamiltonian includes many-body interaction, which is omitted in the effective Hamiltonian with the explicit forms. The BPNNs make SLMC more promising.

The self-learning Monte Carlo (SLMC) method is a general algorithm to speedup MC simulations. Its efficiency has been demonstrated in various systems by introducing an effective model to propose global moves in the configuration space. In this paper, we show that deep neural networks can be naturally incorporated into SLMC, and without any prior knowledge can learn the original model accurately and efficiently. Demonstrated in quantum impurity models, we reduce the complexity for a local update from $O({\ensuremath{\beta}}^{2})$ in Hirsch-Fye algorithm to $O(\ensuremath{\beta}ln\ensuremath{\beta})$, which is a significant speedup especially for systems at low temperatures.

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.

The Self-Learning Monte Carlo (SLMC) method is a Monte Carlo approach that has emerged in recent years by integrating concepts from machine learning with conventional Monte Carlo techniques. Designed to accelerate the numerical study of interacting many-body systems, SLMC significantly improves sampling efficiency by constructing an effective model -- via machine learning methods -- based on configurations generated by conventional Monte Carlo methods and then proposes global updates based on the effective model. This enhancement leads to a substantial reduction in autocorrelation time, especially near the critical region, where traditional methods typically suffer from critical slowing down and increased computation complexity. Moreover, SLMC maintains statistical accuracy by implementing a cumulative update scheme that rigorously satisfies the detailed balance condition. And more recent applications have extended the SLMC to convolutional neural networks with applications not only in condensed matter physics but also high-energy physics, quantum chemistry, and quantum simulations. The generic applicability and high computational efficiency make SLMC a powerful and scalable framework for quantum Monte Carlo simulations of strongly correlated electron systems, extending the reach of numerical investigations beyond the limitations of conventional techniques.

We develop the self-learning Monte Carlo (SLMC) method, a general-purpose numerical method recently introduced to simulate many-body systems, for studying interacting fermion systems. Our method uses a highly efficient update algorithm, which we design and dub ``cumulative update'', to generate new candidate configurations in the Markov chain based on a self-learned bosonic effective model. From a general analysis and a numerical study of the double exchange model as an example, we find that the SLMC with cumulative update drastically reduces the computational cost of the simulation, while remaining statistically exact. Remarkably, its computational complexity is far less than the conventional algorithm with local updates.

Monte Carlo simulation is an unbiased numerical tool for studying classical and quantum many-body systems. One of its bottlenecks is the lack of a general and efficient update algorithm for large size systems close to the phase transition, for which local updates perform badly. In this Rapid Communication, we propose a general-purpose Monte Carlo method, dubbed self-learning Monte Carlo (SLMC), in which an efficient update algorithm is first learned from the training data generated in trial simulations and then used to speed up the actual simulation. We demonstrate the efficiency of SLMC in a spin model at the phase transition point, achieving a 10–20 times speedup.

Monte Carlo simulation is an unbiased numerical tool for studying classical and quantum many-body systems. One of its bottlenecks is the lack of a general and efficient update algorithm for large size systems close to the phase transition, for which local updates perform badly. In this Rapid Communication, we propose a general-purpose Monte Carlo method, dubbed self-learning Monte Carlo (SLMC), in which an efficient update algorithm is first learned from the training data generated in trial simulations and then used to speed up the actual simulation. We demonstrate the efficiency of SLMC in a spin model at the phase transition point, achieving a 10--20 times speedup.