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.
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 introduce methodologies for highly scalable quantum Monte Carlo simulations of electron-phonon models, and we report benchmark results for the Holstein model on the square lattice. The determinant quantum Monte Carlo (DQMC) method is a widely used tool for simulating simple electron-phonon models at finite temperatures, but it incurs a computational cost that scales cubically with system size. Alternatively, near-linear scaling with system size can be achieved with the hybrid Monte Carlo (HMC) method and an integral representation of the Fermion determinant. Here, we introduce a collection of methodologies that make such simulations even faster. To combat "stiffness" arising from the bosonic action, we review how Fourier acceleration can be combined with time-step splitting. To overcome phonon sampling barriers associated with strongly bound bipolaron formation, we design global Monte Carlo updates that approximately respect particle-hole symmetry. To accelerate the iterative linear solver, we introduce a preconditioner that becomes exact in the adiabatic limit of infinite atomic mass. Finally, we demonstrate how stochastic measurements can be accelerated using fast Fourier transforms. These methods are all complementary and, combined, may produce multiple orders of magnitude speedup, depending on model details.
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.
Solving the Schrödinger equation for interacting many-body quantum systems faces computational challenges due to exponential scaling with system size. This complexity limits the study of important phenomena in materials science and physics. We develop an Artificial Neural Network (ANN)-driven algorithm to simulate fermionic systems on lattices. Our method uses Pauli matrices to represent quantum states, incorporates Markov Chain Monte Carlo sampling, and leverages an adaptive momentum optimizer. We demonstrate the algorithm's accuracy by simulating the Heisenberg Hamiltonian on a one-dimensional lattice, achieving results with an error in the order of 10−4 compared to exact diagonalization. Furthermore, we successfully model a magnetic phase transition in a two-dimensional lattice under an applied magnetic field. Importantly, our approach avoids the sign problem common to traditional Fermionic Monte Carlo methods, enabling the investigation of frustrated systems. This work demonstrates the potential of ANN-based algorithms for efficient simulation of complex quantum systems, opening avenues for discoveries in condensed matter physics and materials science.
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.
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.