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The Journal of Chemical Physics

Quantum chemistry by random walk: A faster algorithm

James B. Anderson

Published Mar 15, 1985• 20 citations
Publisher

Abstract

Two extensions of the fixed-node random walk method of solving the Schrödinger equation are described. A simple iterative procedure is found to reduce time-step error and the use of two trial wave functions in place of one for importance sampling is found to reduce computation effort. In combination the two modifications reduce computation effort for a fixed accuracy by a factor of about 10.

DOI
10.1063/1.448262

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Two extensions of the fixed-node random walk method of solving the Schrödinger equation are described. A simple iterative procedure is found to reduce time-step error and the use of two trial wave functions in place of one for importance sampling is found to reduce computation effort. In combination the two modifications reduce computation effort for a fixed accuracy by a factor of about 10.
Abstract This little paper describes two procedures for achieving higher speeds in fixed-node diffusion QMC calculations with importance sampling. The first is an iterative procedure for reducing timestep error, thereby allowing larger and fewer time steps to obtain a given accuracy. The movement of a walker is divided into two parts: a diffusion step selected at random from an appropriate distribution and a drift step calculated from the drift velocity at the initial location. These give a temporary step to a new location for which the drift velocity is again determined. The average of the two velocities together with the original diffusion step is used to determine a permanent step. In tests, the procedure was found to reduce time-step error by a factor of 5 to 10. The second procedure makes use of two trial functions for importance sampling, the first a simple function for walker motion and the second a more complex and more accurate function for determining energies. These allow the more computationally expensive function to be used at large intervals for samples having a lower serial correlation.
Abstract This work extended the simple diffusion QMC method of solving the Schrödinger equation for a wavefunction ψ; to that of solving for a correction to a trial wavefunction ψo such that ψ= ψo+ . The method may be applied in the manner of successive corrections to obtain very high accuracies. As described, the method is limited to solutions for systems with fixed nodes. The method was illustrated for the cases of the particle-in-a-box and the hydrogen atom. For the latter, successive corrections improved the accuracy in energy by about a factor of ten with each iteration. For higher dimensional systems, difficulties in the fitting procedure may be expected to limit the applicability of the method.
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.
Abstract In this paper the combination of importance sampling° with diffusion QMC was shown to be remarkably effective in improving the accuracy of electonic structure calculations for molecular systems. The key to success was the use of a simple, fairly accurate trial wavefunction ΨT for which the function itself, its derivatives and the local energy Eioc may be easily computed. This, used in combination with the diffusion-with-drift expression for walkers corresponding to the product f = ΨΨT according to allows the energy to be evaluated as an average for all walkers of every step after reaching a fluctuating steady state. Since there is a time-step error associated with the simulation of diffusion, drift, and multiplication, varying the time-step and an extrapolation to zero step size are required. The result for H3+ in its triangular configuration of side length 1.6500 bohr was -1.3439 ± 0.0002 hartrees, a value well below the lowest variational energy levels at the time and having an error bar straddling the best modern value of -1.343 835 hartrees. The use of importance sampling along with higher computer speeds resulted in an uncertainty about a factor of 60 lower than that of the previous calculation.b
Abstract Klein and Pickett provided a discussion of the problem of determining the nodes for a fixed-node QMC calculation proposed earlier by Andersona They suggested using the nodes of a Hartree-Fock wavefunction and noted that a Jastrow wavefunctionb consisting of a Hartree-Fock function multiplied by a permutationally symmetric and positive function to correct for electron correlation could be quite accurate. Only later was it realized that the node structures given by Hartree-Fock wavefunctions were surprisingly simple even for large many-electron systems. Combinations of determinantal functions with Jastrow functions were found especially useful as trial functions for importance sampling in diffusion and Green’s function QMC calculations.
Quantum chemistry is concerned with solving the Schrodinger equation for chemically relevant systems. This is typically done by making useful and systematic approximations which form the basis for model chemistries. A primary contribution of this dissertation, is taking concrete steps toward establishing a new model chemistry based on quantum computation. Electronic energies of the system can be efficiently obtained to fixed accuracy using quantum algorithms exploiting the ability of quantum computers to efficiently simulate the time evolution of quantum systems. The quantum circuits for simulation of arbitrary electronic Hamiltonians are given using quantum bits associated with single particle basis functions. This model chemistry is applied to hydrogen molecule as a case study where all necessary quantum circuits are clearly laid out. Furthermore, our collaboration to experimentally realize a simplified version of the hydrogen molecule quantum circuit is also included in this thesis. Finally, alternatives to the gate model of quantum computation are pursued by exploring models based on the quantum adiabatic theorem and on the generalization of random walks.