Abstract We compare the accuracy and precision of the wave function correction method to that of traditional diffusion quantum Monte Carlo (traditional DQMC) simulations. Direct calculation of corrections to trial wave functions reduces time step error for Be and Ne electronic ground state energies when compared to traditional DQMC results. The fixed‐node method is utilized for these fermionic systems, so error due to incorrect node locations is not addressed. This correction DQMC method achieves more accurate energies at larger time steps, promising decreased computational cost to reach a given accuracy. For an optimized Be trial wave function, the statistical uncertainty per unit of computational effort is also reduced relative to traditional DQMC. Correction DQMC utilizes walker feed integrals and growth factors, as reported previously. We illustrate the method in a simple harmonic oscillator example, and we report high accuracy energies for Be and Ne at several time steps.
Abstract We compare the accuracy and precision of the wave function correction method to that of traditional diffusion quantum Monte Carlo (traditional DQMC) simulations. Direct calculation of corrections to trial wave functions reduces time step error for Be and Ne electronic ground state energies when compared to traditional DQMC results. The fixed‐node method is utilized for these fermionic systems, so error due to incorrect node locations is not addressed. This correction DQMC method achieves more accurate energies at larger time steps, promising decreased computational cost to reach a given accuracy. For an optimized Be trial wave function, the statistical uncertainty per unit of computational effort is also reduced relative to traditional DQMC. Correction DQMC utilizes walker feed integrals and growth factors, as reported previously. We illustrate the method in a simple harmonic oscillator example, and we report high accuracy energies for Be and Ne at several time steps.
Quantum Monte Carlo (QMC) methods represent a powerful family of computational techniques for tackling complex quantum many-body problems and performing calculations of stationary state properties. QMC is among the most accurate and powerful approaches to the study of electronic structure, but its application is often hindered by a steep learning curve; hence it is rarely addressed in undergraduate and postgraduate classes. This tutorial is a step toward filling this gap. We offer an introduction to the diffusion Monte Carlo (DMC) method, which aims to solve the imaginary time Schrödinger equation through stochastic sampling of the configuration space. Starting from the theoretical foundations, the discussion leads naturally to the formulation of a step-by-step algorithm. To illustrate how the method works in simplified scenarios, examples such as the harmonic oscillator and the hydrogen atom are provided. The discussion extends to the fixed-node approximation, a crucial approach for addressing the fermionic sign problem in multi-electron systems. In particular, we examine the influence of trial wave function nodal surfaces on the accuracy of DMC energy by evaluating results from a non-interacting two-fermion system. Extending the method to excited states is feasible in principle, but some additional considerations are needed, supported by practical insights. By addressing the fundamental concepts from a hands-on perspective, we hope this tutorial will serve as a valuable guide for researchers and students approaching DMC for the first time.
We develop a formalism and present an algorithm for optimization of the trial wave function used in fixed-node diffusion quantum Monte Carlo (DMC) methods. The formalism is based on the DMC mixed estimator of the ground-state probability density. We take advantage of a basic property of the walker configuration distribution generated in a DMC calculation, to (i) project out a multideterminant expansion of the fixed-node ground-state wave function and (ii) to define a cost function that relates the fixed-node ground-state and the noninteracting trial wave functions. We show that (a) locally smoothing out the kink of the fixed-node ground-state wave function at the node generates a new trial wave function with better nodal structure and (b) we argue that the noise in the fixed-node wave function resulting from finite sampling plays a beneficial role, allowing the nodes to adjust toward the ones of the exact many-body ground state in a simulated annealing-like process. Based on these principles, we propose a method to improve both single determinant and multideterminant expansions of the trial wave function. The method can be generalized to other wave-function forms such as pfaffians. We test the method in a model system where benchmark configuration-interaction calculations can be performed and most components of the Hamiltonian are evaluated analytically. Comparing the DMC calculations with the exact solutions, we find that the trial wave function is systematically improved. The overlap of the optimized trial wave function and the exact ground state converges to 100% even starting from wave functions orthogonal to the exact ground state. Similarly, the DMC total energy and density converges to the exact solutions for the model. In the optimization process we find an optimal noninteracting nodal potential of density-functional-like form whose existence was predicted in a previous publication [Phys. Rev. B 77, 245110 (2008)]. Tests of the method are extended to a model system with a conventional Coulomb interaction where we show we can obtain the exact Kohn-Sham effective potential from the DMC data.
Abstract In an earlier papera a method for the direct calculation of the difference between a true wavefunction and a trial wavefunction was described for simple diffusion calculations. This paper reports further increases in accuracy obtained by a combination of this difference method with importance sampling. The combination allows the calculation of corrections to the product ΨΨT of true and trial wavefunctions, rather than to the wavefunction Ψ itself. Using a new function g = ΨΨT- Ψ2T, the original diffusion equation with drift and local energies is altered by replacement of ΨΨT with g and by addition of a distributed source term. Since walkers fed by sampling the source term may be positive or negative, cancellation is required, but both sets of walkers proceed to the same distribution and age-based cancellation may be used. Node locations are not changed and remain at their original locations as specified by the trial function. The corrected wavefunction and the energy obtained are the same as for a fixed-node diffusion calculation using the same trial function for importance sampling. The method is useful in reducing statistical error, and because the statistical error occurs only in the difference terms, it is most effective when the trial wavefunction is already reasonably accurate. In test calculations for the hydrogen atom and the helium atom, both nodeless, the statistical error in the QMC calculations was greatly reduced. Thus, for hydrogen a trial function with an expectation value of -0.4998 hartrees led to a QMC energy of -0.499995(4) hartrees.
High Resolution Image Download MS PowerPoint Slide Quantum Monte Carlo (QMC) methods are powerful approaches for solving electronic structure problems. Although they often provide high-accuracy solutions, the precision of most QMC methods is ultimately limited by the trial wave function that must be used. Recently, an approach has been demonstrated to allow the use of trial wave functions prepared on a quantum computer [Huggins et al., Unbiasing fermionic quantum Monte Carlo with a quantum computer. Nature 2022, 603, 416] in the auxiliary-field QMC (AFQMC) method using classical shadows to estimate the required overlaps. However, this approach has an exponential post-processing step to construct these overlaps when performing classical shadows obtained using random Clifford circuits. Here, we study an approach to avoid this exponential scaling step by using a fixed-node Monte Carlo method based on full configuration interaction quantum Monte Carlo. This method is applied to the local unitary cluster Jastrow ansatz. We consider H 4, ferrocene, and benzene molecules using up to 12 qubits as examples. Circuits are compiled to native gates for typical near-term architectures, and we assess the impact of circuit-level depolarizing noise on the method. We also provide a comparison of AFQMC and fixed-node approaches, demonstrating that AFQMC is more robust to errors, although extrapolations of the fixed-node energy reduce this discrepancy. Although the method can be used to reach chemical accuracy, the sampling cost to achieve this is high even for small active spaces, suggesting caution about the prospect of outperforming conventional QMC approaches.
Abstract In this paper, a novel exact fixed‐node quantum Monte Carlo (EFNQMC) algorithm was proposed, which is a self‐optimizing and self‐improving procedure. In contrast to the previous EFNQMC method, the importance function of this method is optimized synchronistically in the diffusion procedure, but not before beginning the EFNQMC computation. In order to optimize the importance function, the improved steepest descent technique is used, in which the step size is automatically adjustable. The procedure is quasi‐Newton type and converges super linearly. The present method also uses a novel trial function, which has correct electron‐electron and electron‐nucleus cusp conditions. The novel EFNQMC algorithm and the novel trial function are employed to calculate the energies of 1 1A1 state of CH2, 1A1 state of C8 and the ground‐states of H2, LiH, Li2 and H2O.
We propose an algorithm for optimizing quantum Monte Carlo wave functions. An improved steepest-descent technique is used, with step size automatically adjustable to obtain a procedure that converges superlinearly. We also propose a novel trial function, which has both correct electron–electron cusp conditions and correct electron–nucleus cusp conditions. To test the optimization procedure and the optimized trial function, the ground states for CH4 and H2O molecules were investigated using variational Monte Carlo (VMC) and fixed-node quantum Monte Carlo (FNQMC) calculations. For CH4 and H2O, the VMC recovered 73.3% and 57.9% of the correlation energy, respectively, and the FNQMC recovered 99.3% and 92.8%, respectively. The optimization procedure is three to five times faster than conventional usual steepest-descent procedures. The trial functions optimized are more accurate than prior trial functions of similar complexity.