Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Persistent Population Biases in Branching Random Walk Algorithms

David J. Geroski

Published Jan 1, 2015• 0 citations
OpenAlex

Abstract

We explore the existence of a residual population bias in Diffusion Monte Carlo algorithms. We develop a model problem, which mimics the single-particle, one dimensional simple harmonic oscillator of unitary ground state energy, to study these biases. We then apply an importance sampling algorithms with a population of a single random walker to calculate the ground state energy and wavefunction of our modeled system in cases of both perfect and imperfect importance sampling. In the former case, we perfectly calculate both the ground state energy and corresponding wavefunction of our system, thus validating our calculations and assumed answer. In the latter case, we show the existence of a residual population bias for populations of small numbers of random walkers. We demonstrate the convergence of our small population calculations and show that we cannot remove this bias through a number of single walker methods. Finally, we demonstrate the overall decay of this residual population bias as the population of random walkers expands, allowing our calculations to converge to the true answer. Key Words: Diffusion Monte Carlo, Population Dynamics, Random Walk

Related papers

Finite population-size effects in projection Monte Carlo methods

Nicolas J. Cerf, Olivier Martin

Projection (Green's function and diffusion) MOnte Carlo techniques sample a wave function by a stochastic iterative procedure. These method converge to a stationary distribution which is biased, i.e., differs from the exact ground state wave function. This bias occurs because of the use of population control procedures. We demonstrate that these biased Monte Carlo algorithms lead to a modified effective mass which is equal to the desired mass only in the limit of an infinite population of walkers. In general, the bias for the energy scales as 1/N for a population of walkers of size N and is proportional to the expectation value of the kinetic energy. Finally, we consider various strategies to reduce this bias.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Apr 1, 1995•22 citations

Population size bias in diffusion Monte Carlo

Massimo Boninsegni, Saverio Moroni

The size of the population of random walkers required to obtain converged estimates in diffusion Monte Carlo (DMC) increases dramatically with system size. We illustrate this by comparing ground state energies of small clusters of parahydrogen (up to 48 molecules) computed by DMC and path integral ground state (PIGS) techniques. We contend that the bias associated with a finite population of walkers is the most likely cause of quantitative numerical discrepancies between PIGS and DMC energy estimates reported in the literature, for this few-body Bose system. We discuss the viability of DMC as a general-purpose ground state technique, and argue that PIGS, and even finite temperature methods, enjoy more favorable scaling, and are therefore a superior option for systems of large size.

arXiv:1209.1663•Physical Review E•Nov 28, 2012•55 citations

Grid-based diffusion Monte Carlo for fermions without the fixed-node approximation

Alexander A. Kunitsa, So Hirata

A diffusion Monte Carlo algorithm is introduced that can determine the correct nodal structure of the wave function of a few-fermion system and its ground-state energy without an uncontrolled bias. This is achieved by confining signed random walkers to the points of a uniform infinite spatial grid, allowing them to meet and annihilate one another to establish the nodal structure without the fixed-node approximation. An imaginary-time propagator is derived rigorously from a discretized Hamiltonian, governing a non-Gaussian, sign-flipping, branching, and mutually annihilating random walk of particles. The accuracy of the resulting stochastic representations of a fermion wave function is limited only by the grid and imaginary-time resolutions and can be improved in a controlled manner. The method is tested for a series of model problems including fermions in a harmonic trap as well as the He atom in its singlet or triplet ground state. For the latter case, the energies approach from above with increasing grid resolution and converge within 0.015E_{h} of the exact basis-set-limit value for the grid spacing of 0.08 a.u. with a statistical uncertainty of 10^{-5}E_{h} without an importance sampling or Jastrow factor.

arXiv:1811.03000•Physical review. E•Jan 27, 2020•12 citations

Bilinear diffusion quantum Monte Carlo methods

F. Arias de Saavedra, Malvin H. Kalos

The standard method of quantum Monte Carlo for the solution of the Schrödinger equation in configuration space can be described quite generally as devising a random walk that generates-at least asymptotically-populations of random walkers whose probability density is proportional to the wave function of the system being studied. While, in principle, the energy eigenvalue of the Hamiltonian can be calculated with high accuracy, estimators of operators that do not commute the Hamiltonian cannot. Bilinear quantum Monte Carlo (BQMC) is an alternative in which the square of the wave function is sampled in a somewhat indirect way. More specifically, one uses a pair of walkers at positions x and y and introduces stochastic dynamics to sample phi(i)(x)t(x,y)phi(j)(y), where phi(i)(x) and phi(j)(y) are eigenfunctions of (possibly different) Hamiltonians, and t(x,y) is a kernel that correlates positions x and y. Using different Hamiltonians permits the accurate computation of small energy differences. We review the conceptual basis of BQMC, discuss qualitatively and analytically the problem of the fluctuations in the branching, and present partial solutions to that problem. Finally we exhibit numerical results for some model systems including harmonic oscillators and the hydrogen and helium atoms. Further research will be necessary to make this a practical and generally applicable scheme.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Feb 27, 2003•7 citations

Theory and Application of a Pure-sampling Quantum Monte Carlo Algorithm

Egor Ospadov

The objective of pure-sampling quantum Monte Carlo is to calculate physical properties that are independent of the importance sampling function being employed in the calculation, save for the mismatch of its nodal hypersurface with that of the exact wave function. To achieve this objective, we describe a pure-sampling algorithm that combines features of forward-walking methods of pure-sampling and reptation quantum Monte Carlo. The importance sampling is performed by using a single-determinant basis set composed of Slater-type orbitals. We implement our algorithm by systematically increasing an algorithmic parameter until the properties sampled from the electron distributions converge to statistically equivalent values, extrapolated in the limit of zero time-step. In doing so, we are able to unambiguously determine the values for the ground-state fixed-node energies and one-electron properties of various molecules. These quantities are free from importance sampling bias, population control bias, time-step bias, extrapolation-model bias, and the finite-field approximation. We applied our algorithm to the ground-states of lithium hydride, water and ethylene molecules, and found excellent agreement with the accepted literature values for the energy and a variety of other properties for those systems. Some of our one-electron properties of ethylene had not been calculated before at any level of theory. In a detailed comparison, we found reptation quantum Monte Carlo, our closest competitor, to be less efficient by at least a factor of two. It requires different sets of time-steps to accurately determine the ground-state energy and one-electron properties, whereas our algorithm can achieve the same objective by using a single set of time-step values.

•May 13, 2016•0 citations

Diffusion Monte Carlo methods with a fixed number of walkers

Roland Assaraf, Michel Caffarel, Anatole Khélif

In this paper we discuss various aspects of diffusion Monte Carlo methods using a fixed number of walkers. First, a rigorous proof of the divergence of pure diffusion Monte Carlo (PDMC) methods (DMC without branching in which the weights are carried along trajectories) is given. Second, a bias-free Monte Carlo method combining DMC and PDMC approaches, and based on a minimal stochastic reconfiguration of the population, is discussed. Finally, some illustrative calculations for a system of coupled quantum rotators are presented.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Apr 1, 2000•91 citations
We explore the existence of a residual population bias in Diffusion Monte Carlo algorithms. We develop a model problem, which mimics the single-particle, one dimensional simple harmonic oscillator of unitary ground state energy, to study these biases. We then apply an importance sampling algorithms with a population of a single random walker to calculate the ground state energy and wavefunction of our modeled system in cases of both perfect and imperfect importance sampling. In the former case, we perfectly calculate both the ground state energy and corresponding wavefunction of our system, thus validating our calculations and assumed answer. In the latter case, we show the existence of a residual population bias for populations of small numbers of random walkers. We demonstrate the convergence of our small population calculations and show that we cannot remove this bias through a number of single walker methods. Finally, we demonstrate the overall decay of this residual population bias as the population of random walkers expands, allowing our calculations to converge to the true answer. Key Words: Diffusion Monte Carlo, Population Dynamics, Random Walk
Projection (Green's function and diffusion) MOnte Carlo techniques sample a wave function by a stochastic iterative procedure. These method converge to a stationary distribution which is biased, i.e., differs from the exact ground state wave function. This bias occurs because of the use of population control procedures. We demonstrate that these biased Monte Carlo algorithms lead to a modified effective mass which is equal to the desired mass only in the limit of an infinite population of walkers. In general, the bias for the energy scales as 1/N for a population of walkers of size N and is proportional to the expectation value of the kinetic energy. Finally, we consider various strategies to reduce this bias.
The size of the population of random walkers required to obtain converged estimates in diffusion Monte Carlo (DMC) increases dramatically with system size. We illustrate this by comparing ground state energies of small clusters of parahydrogen (up to 48 molecules) computed by DMC and path integral ground state (PIGS) techniques. We contend that the bias associated with a finite population of walkers is the most likely cause of quantitative numerical discrepancies between PIGS and DMC energy estimates reported in the literature, for this few-body Bose system. We discuss the viability of DMC as a general-purpose ground state technique, and argue that PIGS, and even finite temperature methods, enjoy more favorable scaling, and are therefore a superior option for systems of large size.
A diffusion Monte Carlo algorithm is introduced that can determine the correct nodal structure of the wave function of a few-fermion system and its ground-state energy without an uncontrolled bias. This is achieved by confining signed random walkers to the points of a uniform infinite spatial grid, allowing them to meet and annihilate one another to establish the nodal structure without the fixed-node approximation. An imaginary-time propagator is derived rigorously from a discretized Hamiltonian, governing a non-Gaussian, sign-flipping, branching, and mutually annihilating random walk of particles. The accuracy of the resulting stochastic representations of a fermion wave function is limited only by the grid and imaginary-time resolutions and can be improved in a controlled manner. The method is tested for a series of model problems including fermions in a harmonic trap as well as the He atom in its singlet or triplet ground state. For the latter case, the energies approach from above with increasing grid resolution and converge within 0.015E_{h} of the exact basis-set-limit value for the grid spacing of 0.08 a.u. with a statistical uncertainty of 10^{-5}E_{h} without an importance sampling or Jastrow factor.
The standard method of quantum Monte Carlo for the solution of the Schrödinger equation in configuration space can be described quite generally as devising a random walk that generates-at least asymptotically-populations of random walkers whose probability density is proportional to the wave function of the system being studied. While, in principle, the energy eigenvalue of the Hamiltonian can be calculated with high accuracy, estimators of operators that do not commute the Hamiltonian cannot. Bilinear quantum Monte Carlo (BQMC) is an alternative in which the square of the wave function is sampled in a somewhat indirect way. More specifically, one uses a pair of walkers at positions x and y and introduces stochastic dynamics to sample phi(i)(x)t(x,y)phi(j)(y), where phi(i)(x) and phi(j)(y) are eigenfunctions of (possibly different) Hamiltonians, and t(x,y) is a kernel that correlates positions x and y. Using different Hamiltonians permits the accurate computation of small energy differences. We review the conceptual basis of BQMC, discuss qualitatively and analytically the problem of the fluctuations in the branching, and present partial solutions to that problem. Finally we exhibit numerical results for some model systems including harmonic oscillators and the hydrogen and helium atoms. Further research will be necessary to make this a practical and generally applicable scheme.
The objective of pure-sampling quantum Monte Carlo is to calculate physical properties that are independent of the importance sampling function being employed in the calculation, save for the mismatch of its nodal hypersurface with that of the exact wave function. To achieve this objective, we describe a pure-sampling algorithm that combines features of forward-walking methods of pure-sampling and reptation quantum Monte Carlo. The importance sampling is performed by using a single-determinant basis set composed of Slater-type orbitals. We implement our algorithm by systematically increasing an algorithmic parameter until the properties sampled from the electron distributions converge to statistically equivalent values, extrapolated in the limit of zero time-step. In doing so, we are able to unambiguously determine the values for the ground-state fixed-node energies and one-electron properties of various molecules. These quantities are free from importance sampling bias, population control bias, time-step bias, extrapolation-model bias, and the finite-field approximation. We applied our algorithm to the ground-states of lithium hydride, water and ethylene molecules, and found excellent agreement with the accepted literature values for the energy and a variety of other properties for those systems. Some of our one-electron properties of ethylene had not been calculated before at any level of theory. In a detailed comparison, we found reptation quantum Monte Carlo, our closest competitor, to be less efficient by at least a factor of two. It requires different sets of time-steps to accurately determine the ground-state energy and one-electron properties, whereas our algorithm can achieve the same objective by using a single set of time-step values.
In this paper we discuss various aspects of diffusion Monte Carlo methods using a fixed number of walkers. First, a rigorous proof of the divergence of pure diffusion Monte Carlo (PDMC) methods (DMC without branching in which the weights are carried along trajectories) is given. Second, a bias-free Monte Carlo method combining DMC and PDMC approaches, and based on a minimal stochastic reconfiguration of the population, is discussed. Finally, some illustrative calculations for a system of coupled quantum rotators are presented.