We present a unified theory of the variational Monte Carlo (VMC) and determinant quantum Monte Carlo (DQMC) methods using a novel density matrix formulation of VMC. We introduce an efficient algorithm for VMC to compute correlation functions and expectation values based on the auxiliary field Hirsch-Hubbard-Stratonovic transformation. We show that this new approach to VMC converges significantly faster than its traditional implementations. Furthermore, we generalize the Trotter-Suzuki decomposition to finite imaginary time steps $τ\sim O(1)$ and develop a variational quantum Monte Carlo (VQMC) method accordingly, which is more accurate than VMC and can incorporate quantum fluctuations more efficiently. The two extreme limits of the VQMC method, namely infinitesimal and infinite imaginary time steps, correspond to the DQMC and VMC techniques, respectively. We demonstrate that our VQMC allows us to access lower temperatures in comparison with the conventional DQMC before the sign problem comes into play. We finally show that our VQMC can also enhance the accuracy of the projector Monte Carlo methods by providing better and less biased candidates for the trial wave functions, requiring shorter projection times for a given accuracy and alleviating the sign problem further.We present a unified theory of the variational Monte Carlo (VMC) and determinant quantum Monte Carlo (DQMC) methods using a novel density matrix formulation of VMC. We introduce an efficient algorithm for VMC to compute correlation functions and expectation values based on the auxiliary field Hirsch-Hubbard-Stratonovic transformation. We show that this new approach to VMC converges significantly faster than its traditional implementations. Furthermore, we generalize the Trotter-Suzuki decomposition to finite imaginary time steps $τ\sim O(1)$ and develop a variational quantum Monte Carlo (VQMC) method accordingly, which is more accurate than VMC and can incorporate quantum fluctuations more efficiently. The two extreme limits of the VQMC method, namely infinitesimal and infinite imaginary time steps, correspond to the DQMC and VMC techniques, respectively. We demonstrate that our VQMC allows us to access lower temperatures in comparison with the conventional DQMC before the sign problem comes into play. We finally show that our VQMC can also enhance the accuracy of the projector Monte Carlo methods by providing better and less biased candidates for the trial wave functions, requiring shorter projection times for a given accuracy and alleviating the sign problem further.

arXiv:1810.00864Oct 1, 20182 citations

Time dependent Variational Monte Carlo

Markus Holzmann

Jul 6, 20200 citations

Removing nodal and support-mismatch pathologies in Variational Monte Carlo via blurred sampling

Zhou-Quan Wan, Roeland Wiersema, Shiwei Zhang

Variational Monte Carlo (VMC) is a powerful and fast-growing method for optimizing and evolving parameterized many-body wave functions, especially with modern neural-network quantum states. In practice, however, the stochastic estimators that form the backbone of the method can become unstable or biased due to the presence of nodes, a ubiquitous feature of quantum wave functions. In the continuum, this results in heavy-tailed estimators with potentially divergent variances, while in discrete Hilbert spaces the sampling distribution can miss parts of the support needed to form unbiased estimators. These statistical pathologies lead to unreliable optimization trajectories in stochastic reconfiguration or incorrect variational dynamics in time-dependent Variational Monte Carlo (t-VMC), and severely limit the power of the numerical simulations. We introduce blurred sampling to address these difficulties. The method has a number of rigorous properties that make it well-behaved, effective and computationally efficient. Additionally, it is a post-processing approach that can be used without modifying the underlying sampler and incurs only minimal overhead. We demonstrate its effectiveness in a broad range of problems -- including ground state optimization in both continuum and discrete settings, as well as in time-dependent variational Monte Carlo, where it resolves previously identified failure modes -- and further apply it to large-scale problems in spin dynamics. This work establishes a broadly applicable framework for robust VMC and t-VMC calculations.Variational Monte Carlo (VMC) is a powerful and fast-growing method for optimizing and evolving parameterized many-body wave functions, especially with modern neural-network quantum states. In practice, however, the stochastic estimators that form the backbone of the method can become unstable or biased due to the presence of nodes, a ubiquitous feature of quantum wave functions. In the continuum, this results in heavy-tailed estimators with potentially divergent variances, while in discrete Hilbert spaces the sampling distribution can miss parts of the support needed to form unbiased estimators. These statistical pathologies lead to unreliable optimization trajectories in stochastic reconfiguration or incorrect variational dynamics in time-dependent Variational Monte Carlo (t-VMC), and severely limit the power of the numerical simulations. We introduce blurred sampling to address these difficulties. The method has a number of rigorous properties that make it well-behaved, effective and computationally efficient. Additionally, it is a post-processing approach that can be used without modifying the underlying sampler and incurs only minimal overhead. We demonstrate its effectiveness in a broad range of problems -- including ground state optimization in both continuum and discrete settings, as well as in time-dependent variational Monte Carlo, where it resolves previously identified failure modes -- and further apply it to large-scale problems in spin dynamics. This work establishes a broadly applicable framework for robust VMC and t-VMC calculations.

cond-mat.str-elCross-listedarXiv:2603.18148Physical Review XMar 20, 20260 citations

Finite-temperature properties of strongly correlated systems via variational Monte Carlo

Jahan Claes, Bryan K. Clark

Variational methods are a common approach for computing properties of ground states but have not yet found analogous success in finite-temperature calculations. In this work, we develop a new variational finite-temperature algorithm (VAFT), which combines ideas from minimally entangled typical thermal states (METTS), variational Monte Carlo (VMC) optimization, and path integral Monte Carlo (PIMC). This allows us to define an implicit variational density matrix to estimate finite-temperature properties in two and three dimensions. We benchmark the algorithm on the bipartite Heisenberg model and compare to exact results.Variational methods are a common approach for computing properties of ground states but have not yet found analogous success in finite-temperature calculations. In this work, we develop a new variational finite-temperature algorithm (VAFT), which combines ideas from minimally entangled typical thermal states (METTS), variational Monte Carlo (VMC) optimization, and path integral Monte Carlo (PIMC). This allows us to define an implicit variational density matrix to estimate finite-temperature properties in two and three dimensions. We benchmark the algorithm on the bipartite Heisenberg model and compare to exact results.

arXiv:1612.05260Physical review. B./Physical review. BMay 8, 201722 citations

τ\sim O(1)

and develop a variational quantum Monte Carlo (VQMC) method accordingly, which is more accurate than VMC and can incorporate quantum fluctuations more efficiently. The two extreme limits of the VQMC method, namely infinitesimal and infinite imaginary time steps, correspond to the DQMC and VMC techniques, respectively. We demonstrate that our VQMC allows us to access lower temperatures in comparison with the conventional DQMC before the sign problem comes into play. We finally show that our VQMC can also enhance the accuracy of the projector Monte Carlo methods by providing better and less biased candidates for the trial wave functions, requiring shorter projection times for a given accuracy and alleviating the sign problem further.