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

Towards Neural Variational Monte Carlo That Scales Linearly with System Size

Or Sharir, Garnet Kin‐Lic Chan, Animashree Anandkumar

Quantum many-body problems are some of the most challenging problems in science and are central to demystifying some exotic quantum phenomena, e.g., high-temperature superconductors. The combination of neural networks (NN) for representing quantum states, coupled with the Variational Monte Carlo (VMC) algorithm, has been shown to be a promising method for solving such problems. However, the run-time of this approach scales quadratically with the number of simulated particles, constraining the practically usable NN to - in machine learning terms - minuscule sizes (<10M parameters). Considering the many breakthroughs brought by extreme NN in the +1B parameters scale to other domains, lifting this constraint could significantly expand the set of quantum systems we can accurately simulate on classical computers, both in size and complexity. We propose a NN architecture called Vector-Quantized Neural Quantum States (VQ-NQS) that utilizes vector-quantization techniques to leverage redundancies in the local-energy calculations of the VMC algorithm - the source of the quadratic scaling. In our preliminary experiments, we demonstrate VQ-NQS ability to reproduce the ground state of the 2D Heisenberg model across various system sizes, while reporting a significant reduction of about ${\times}10$ in the number of FLOPs in the local-energy calculation.Quantum many-body problems are some of the most challenging problems in science and are central to demystifying some exotic quantum phenomena, e.g., high-temperature superconductors. The combination of neural networks (NN) for representing quantum states, coupled with the Variational Monte Carlo (VMC) algorithm, has been shown to be a promising method for solving such problems. However, the run-time of this approach scales quadratically with the number of simulated particles, constraining the practically usable NN to - in machine learning terms - minuscule sizes (<10M parameters). Considering the many breakthroughs brought by extreme NN in the +1B parameters scale to other domains, lifting this constraint could significantly expand the set of quantum systems we can accurately simulate on classical computers, both in size and complexity. We propose a NN architecture called Vector-Quantized Neural Quantum States (VQ-NQS) that utilizes vector-quantization techniques to leverage redundancies in the local-energy calculations of the VMC algorithm - the source of the quadratic scaling. In our preliminary experiments, we demonstrate VQ-NQS ability to reproduce the ground state of the 2D Heisenberg model across various system sizes, while reporting a significant reduction of about ${\times}10$ in the number of FLOPs in the local-energy calculation.

arXiv:2212.11296Dec 21, 20220 citations

A new algorithm for variational quantum Monte Carlo method

HX Huang, ZX Cao, 曹泽星

Jan 1, 19980 citations

A New Algorithm for Variational Quantum Monte Carlo Method

Huang Hong

Chemical Research in Chinese UniversitiesJan 1, 19980 citations

Geometry optimization in quantum Monte Carlo with solution mapping: Application to formaldehyde

Charles A. Schuetz, Michael Frenklach, Alexander C. Kollias +1 more

The solution mapping methodology is presented as a method of geometry optimization of molecules in the quantum Monte Carlo method. Applied to formaldehyde as a test system, this approach is found to yield optimized bond lengths and bond angle in the diffusion Monte Carlo method that lie within experimental error. The variational Monte Carlo optimized geometry also lies within experimental error, with the exception of the CH bond length, which is slightly underestimated. Additionally, the resulting quadratic representation of the potential energy surface in the region of the minimum is used to calculate three of the force constants and harmonic frequencies.The solution mapping methodology is presented as a method of geometry optimization of molecules in the quantum Monte Carlo method. Applied to formaldehyde as a test system, this approach is found to yield optimized bond lengths and bond angle in the diffusion Monte Carlo method that lie within experimental error. The variational Monte Carlo optimized geometry also lies within experimental error, with the exception of the CH bond length, which is slightly underestimated. Additionally, the resulting quadratic representation of the potential energy surface in the region of the minimum is used to calculate three of the force constants and harmonic frequencies.

The Journal of Chemical PhysicsOct 31, 20037 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.

{\times}10

in the number of FLOPs in the local-energy calculation.