The projective quantum Monte Carlo (PQMC) algorithms are among the most powerful computational techniques to simulate the ground-state properties of quantum many-body systems. However, they are efficient only if a sufficiently accurate trial wave function is used to guide the simulation. In the standard approach, this guiding wave function is obtained in a separate simulation that performs a variational minimization. Here we show how to perform PQMC simulations guided by an adaptive wave function based on a restricted Boltzmann machine. This adaptive wave function is optimized along the PQMC simulation via unsupervised machine learning, avoiding the need of a separate variational optimization. As a byproduct, this technique provides an accurate ansatz for the ground-state wave function, which is obtained by minimizing the Kullback-Leibler divergence with respect to the PQMC samples, rather than by minimizing the energy expectation value as in standard variational optimizations. The high accuracy of this self-learning PQMC technique is demonstrated for a paradigmatic sign-problem-free model, namely, the ferromagnetic quantum Ising chain, showing very precise agreement with the predictions of the Jordan-Wigner theory and of loop quantum Monte Carlo simulations performed in the low-temperature limit.
The projective quantum Monte Carlo (PQMC) algorithms are among the most powerful computational techniques to simulate the ground-state properties of quantum many-body systems. However, they are efficient only if a sufficiently accurate trial wave function is used to guide the simulation. In the standard approach, this guiding wave function is obtained in a separate simulation that performs a variational minimization. Here we show how to perform PQMC simulations guided by an adaptive wave function based on a restricted Boltzmann machine. This adaptive wave function is optimized along the PQMC simulation via unsupervised machine learning, avoiding the need of a separate variational optimization. As a byproduct, this technique provides an accurate ansatz for the ground-state wave function, which is obtained by minimizing the Kullback-Leibler divergence with respect to the PQMC samples, rather than by minimizing the energy expectation value as in standard variational optimizations. The high accuracy of this self-learning PQMC technique is demonstrated for a paradigmatic sign-problem-free model, namely, the ferromagnetic quantum Ising chain, showing very precise agreement with the predictions of the Jordan-Wigner theory and of loop quantum Monte Carlo simulations performed in the low-temperature limit.
We investigate the use of variational wave functions that mimic stochastic recurrent neural networks, specifically, unrestricted Boltzmann machines, as guiding functions in projective quantum Monte Carlo (PQMC) simulations of quantum spin models. As a preliminary step, we investigate the accuracy of such unrestricted neural network states as variational Ans\"atze for the ground state of the ferromagnetic quantum Ising chain. We find that by optimizing just three variational parameters, independently on the system size, accurate ground-state energies are obtained, comparable to those previously obtained using restricted Boltzmann machines with few variational parameters per spin. Chiefly, we show that if one uses optimized unrestricted neural network states as guiding functions for importance sampling, the efficiency of the PQMC algorithms is greatly enhanced, drastically reducing the most relevant systematic bias, namely, the one due to the finite random-walker population. The scaling of the computational cost with the system size changes from the exponential scaling characteristic of PQMC simulations performed without importance sampling, to a polynomial scaling, apparently even at the ferromagnetic quantum critical point. The important role of the protocol chosen to sample hidden-spin configurations, in particular at the critical point, is analyzed. We discuss the implications of these findings for what concerns the problem of simulating adiabatic quantum optimization using stochastic algorithms on classical computers.
Accurately determining the ground-state properties of quantum many-body systems remains a central challenge. In this work, we introduce an autoregressive projective quantum Monte Carlo (PQMC) framework that leverages recurrent neural networks (RNNs) to guide the stochastic dynamics. By incorporating autoregressive sampling into PQMC, we demonstrate substantial improvements in accuracy compared to standard unguided PQMC, while retaining polynomial computational cost. We benchmark our approach against conventional variational RNN ansätze and find that the autoregressive PQMC consistently achieves lower energies and higher fidelity, regardless of system size or whether the Hamiltonian is Hermitian or non-Hermitian. Our results highlight the versatility and power of neural-guided PQMC methods, paving the way for promising scalable simulations of low-energy states in complex quantum many-body systems.
In recent years, generative artificial neural networks based on restricted Boltzmann machines (RBMs) have been successfully employed as accurate and flexible variational wave functions for clean quantum many-body systems. In this article, we explore their use in simulations of disordered quantum Ising chains. The standard dense RBM with all-to-all interlayer connectivity is not particularly appropriate for large disordered systems, since in such systems one cannot exploit translational invariance to reduce the amount of parameters to be optimized. To circumvent this problem, we implement sparse RBMs, whereby the visible spins are connected only to a subset of local hidden neurons, thus reducing the amount of parameters. We assess the performance of sparse RBMs as a function of the range of the allowed connections, and we compare it with that of dense RBMs. Benchmark results are provided for two sign-problem-free Hamiltonians, namely pure and random quantum Ising chains. The RBM Ansätzes are trained using the unsupervised learning scheme based on projective quantum Monte Carlo (PQMC) algorithms. We find that the sparse connectivity facilitates the training process and allows sparse RBMs to outperform their dense counterparts. Furthermore, the use of sparse RBMs as guiding functions for PQMC simulations allows us to perform PQMC simulations at a reduced computational cost, avoiding possible biases due to finite random-walker populations. We obtain unbiased predictions for the ground-state energies and the magnetization profiles with fixed boundary conditions, at the ferromagnetic quantum critical point. The magnetization profiles agree with the Fisher-de Gennes scaling relation for conformally invariant systems, including the scaling dimension predicted by the renormalization-group analysis.
We develop a machine learning method to construct accurate ground-state wave functions of strongly interacting and entangled quantum spin as well as fermionic models on lattices. A restricted Boltzmann machine algorithm in the form of an artificial neural network is combined with a conventional variational Monte Carlo method with pair product (geminal) wave functions and quantum number projections. The combination allows an application of the machine learning scheme to interacting fermionic systems. The combined method substantially improves the accuracy beyond that ever achieved by each method separately, in the Heisenberg as well as Hubbard models on square lattices, thus proving its power as a highly accurate quantum many-body solver.
Restricted Boltzmann machine (RBM) has been proposed as a powerful variational ansatz to represent the ground state of a given quantum many-body system. On the other hand, as a shallow neural network, it is found that the RBM is still hardly able to capture the characteristics of systems with large sizes or complicated interactions. In order to find a way out of the dilemma, here, we propose to adopt the Green’s function Monte Carlo (GFMC) method for which the RBM is used as a guiding wave function. To demonstrate the implementation and effectiveness of the proposal, we have applied the proposal to study the frustrated J 1–J 2 Heisenberg model on a square lattice, which is considered as a typical model with sign problem for quantum Monte Carlo simulations. The calculation results demonstrate that the GFMC method can significantly further reduce the relative error of the ground-state energy on the basis of the RBM variational results. This encourages to combine the GFMC method with other neural networks like convolutional neural networks for dealing with more models with sign problem in the future.
Solving the quantum many-body Schrödinger equation is a fundamental and challenging problem in the fields of quantum physics, quantum chemistry, and material sciences. One of the common computational approaches to this problem is Quantum Variational Monte Carlo (QVMC), in which ground-state solutions are obtained by minimizing the energy of the system within a restricted family of parameterized wave functions. Deep learning methods partially address the limitations of traditional QVMC by representing a rich family of wave functions in terms of neural networks. However, the optimization objective in QVMC remains notoriously hard to minimize and requires second-order optimization methods such as natural gradient. In this paper, we first reformulate energy functional minimization in the space of Born distributions corresponding to particle-permutation (anti-)symmetric wave functions, rather than the space of wave functions. We then interpret QVMC as the Fisher-Rao gradient flow in this distributional space, followed by a projection step onto the variational manifold. This perspective provides us with a principled framework to derive new QMC algorithms, by endowing the distributional space with better metrics, and following the projected gradient flow induced by those metrics. More specifically, we propose "Wasserstein Quantum Monte Carlo" (WQMC), which uses the gradient flow induced by the Wasserstein metric, rather than Fisher-Rao metric, and corresponds to transporting the probability mass, rather than teleporting it. We demonstrate empirically that the dynamics of WQMC results in faster convergence to the ground state of molecular systems.