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

Journal of Physics A Mathematical and General

Random walks of a quantum particle on a circle

N. Fjeldso, John Midtdal, F. Ravndal

Published Apr 7, 1988• 7 citations
Publisher

Abstract

When the quantum planar rotor is put on a lattice its dynamics can be approximated by random walks on a circle. This allows for fast and accurate Monte Carlo simulations to determine the topological charge of different configurations of the system and thereby the theta dependence of the lowest energy levels.
DOI
10.1088/0305-4470/21/7/027

Related papers

THE SELF-ORGANIZED MULTI-LATTICE MONTE CARLO SIMULATION

Denis Horváth, Martin Gmitra

Self-organized Monte Carlo simulations of 2D Ising ferromagnet on the square lattice are performed. The essence of the suggested simulation method is an artificial dynamics consisting of the well-known single-spin-flip Metropolis algorithm supplemented by a random walk on the temperature axis. The walk is biased towards the critical region through a feedback based on instantaneous energy and magnetization cumulants, which are updated at every Monte Carlo step and filtered through a special recursion algorithm. The simulations revealed the invariance of the temperature probability distribution function, once some self-organized critical steady regime is reached, which is called here noncanonical equilibrium. The mean value of this distribution approximates the pseudocritical temperature of canonical equilibrium. In order to suppress finite-size effects, the self-organized approach is extended to multi-lattice systems, where the feedback basis on pairs of instantaneous estimates of the fourth-order magnetization cumulant on two systems of different size. These replica-based simulations resemble, in Monte Carlo lattice systems, some of the invariant statistical distributions of standard self-organized critical systems.

arXiv:physics/0308070•International Journal of Modern Physics C•Nov 1, 2004•4 citations

DYNAMICAL LOCALIZATION OF RANDOM QUANTUM WALKS ON THE LATTICE

Alain Joye

This note describes recent results on the localization properties of Random Quantum Walks on the d-dimensional lattice in a regime analogous to the large disorder regime by means of the Fractional Moments Method adapted to the unitary framework.

arXiv:1209.4190•WORLD SCIENTIFIC eBooks•Oct 1, 2013•11 citations

Projector quantum Monte Carlo

Richard M. Martin, Lucia Reining, David M. Ceperley

… as suggested by Fermi, the time-independent Schrödinger equation … can be interpreted as describing the behavior of a system of particles each of which performs a random walk, i.e., diffuses isotropically and at the same time is subject to multiplication, which is determined by the value of the point function V. N. Metropolis and S. Ulam, 1949 Summary In the projector quantum Monte Carlo method, one uses a function of the hamiltonian to sample a distribution proportional to the exact ground-state wavefunction, and thereby computes exact matrix elements of it. An importance sampling transformation makes the algorithm much more efficient. In this chapter we introduce and develop the diffusion Monte Carlo method, which involves drifting, branching random walks. For any excited state, including any system with more than two electrons, one encounters the sign problem, limiting the direct application of these algorithms for most fermion systems. Instead, by using approximate fixed-node or fixed-phase boundary conditions, one can achieve efficiency similar to variational Monte Carlo. We also discuss the application of the projector method in a basis of Slater determinants. In this chapter, we discuss a different quantum Monte Carlo method, projector Monte Carlo (PMC). This general method was first suggested by Fermi [1049]; see the quote at the start of this chapter by two of the inventors of the Monte Carlo method. An implementation of PMC was tried out in the early days of computing [1050]. Advances in methodology, in particular importance sampling, resulted in a significant large-scale application: the exact calculation of the ground-state properties of 256 hard-sphere bosons by Kalos, Levesque, and Verlet [1051] in 1974. Calculations for electronic systems and the fixednode approximation were introduced by Anderson [1052, 1053]. One of the most important projector MC algorithms, the diffusion Monte Carlo algorithm with importance sampling for fermions, was used to compute the correlation energy of homogeneous electron gas by Ceperley and Alder [109] in 1980; the resulting HEG correlation energy was crucial in the development of density functional calculations. Types and properties of projectors In this method, a many-body projector G ( R , R ) = Ĝ is repeatedly applied to filter out the exact many-body ground state from an initial state; the operation of the projector is carried out with a random walk, hence the name of this class of methods.

Cambridge University Press eBooks•May 31, 2016•0 citations

Many-Body Quantum Spin Dynamics with Monte Carlo Trajectories on a Discrete Phase Space

Johannes Schachenmayer, Alexander Pikovski, A. M. Rey

Following the time evolution of a large and high-dimensional quantum system is often computationally intractable. A new semiclassical method of modeling quantum dynamics yields surprisingly good results for nonequilibrium dynamics.

arXiv:1408.4441•Physical Review X•Feb 25, 2015•205 citations

Numerical path integration with Coulomb potential

Jan Myrheim

A simple and efficient method for quantum Monte Carlo simulation is presented, based on discretization of the action in the path integral, and a Gaussian averaging of the potential, which works well e.g. with the Coulomb potential.

arXiv:physics/0306168•Jun 24, 2003•1 citation

Phase-free quantum Monte Carlo method: random walks using general basis sets

Henry Krakauer

Fermion quantum Monte Carlo (QMC) methods that work in a general basis space may be more effective for some problems than the traditional diffusion QMC method. Projection of the ground state energy is achieved by random walks in the space of Slater determinants whose one‐particle orbitals are expressed in terms of the chosen basis set. A complication is that the determinants will in general acquire complex phases. This a consequence of ground state Monte Carlo projection using the Hubbard‐Stratonovich transformation of the two‐body interaction. To control the resulting “sign” decay, we describe a method we have recently introduced for the propagation of phaseless determinants. The approximation relies on importance sampling with a trial wave function. The approximation has features in common with diffusion MC fixed node and lattice‐model constrained path methods. Using a planewave basis and non‐local pseudopotentials, we apply the method to Si atom, dimer, and 2, 16, 54 atom (216 electrons) bulk supercells. Single Slater determinant wave functions from density functional theory calculations were used as |ΨT〉 with no additional optimization. The calculated binding energy of Si2 and cohesive energy of bulk Si are in excellent agreement with experiments and are comparable to the best existing theoretical results.

AIP conference proceedings•Jan 1, 2003•0 citations
Self-organized Monte Carlo simulations of 2D Ising ferromagnet on the square lattice are performed. The essence of the suggested simulation method is an artificial dynamics consisting of the well-known single-spin-flip Metropolis algorithm supplemented by a random walk on the temperature axis. The walk is biased towards the critical region through a feedback based on instantaneous energy and magnetization cumulants, which are updated at every Monte Carlo step and filtered through a special recursion algorithm. The simulations revealed the invariance of the temperature probability distribution function, once some self-organized critical steady regime is reached, which is called here noncanonical equilibrium. The mean value of this distribution approximates the pseudocritical temperature of canonical equilibrium. In order to suppress finite-size effects, the self-organized approach is extended to multi-lattice systems, where the feedback basis on pairs of instantaneous estimates of the fourth-order magnetization cumulant on two systems of different size. These replica-based simulations resemble, in Monte Carlo lattice systems, some of the invariant statistical distributions of standard self-organized critical systems.
… as suggested by Fermi, the time-independent Schrödinger equation … can be interpreted as describing the behavior of a system of particles each of which performs a random walk, i.e., diffuses isotropically and at the same time is subject to multiplication, which is determined by the value of the point function V. N. Metropolis and S. Ulam, 1949 Summary In the projector quantum Monte Carlo method, one uses a function of the hamiltonian to sample a distribution proportional to the exact ground-state wavefunction, and thereby computes exact matrix elements of it. An importance sampling transformation makes the algorithm much more efficient. In this chapter we introduce and develop the diffusion Monte Carlo method, which involves drifting, branching random walks. For any excited state, including any system with more than two electrons, one encounters the sign problem, limiting the direct application of these algorithms for most fermion systems. Instead, by using approximate fixed-node or fixed-phase boundary conditions, one can achieve efficiency similar to variational Monte Carlo. We also discuss the application of the projector method in a basis of Slater determinants. In this chapter, we discuss a different quantum Monte Carlo method, projector Monte Carlo (PMC). This general method was first suggested by Fermi [1049]; see the quote at the start of this chapter by two of the inventors of the Monte Carlo method. An implementation of PMC was tried out in the early days of computing [1050]. Advances in methodology, in particular importance sampling, resulted in a significant large-scale application: the exact calculation of the ground-state properties of 256 hard-sphere bosons by Kalos, Levesque, and Verlet [1051] in 1974. Calculations for electronic systems and the fixednode approximation were introduced by Anderson [1052, 1053]. One of the most important projector MC algorithms, the diffusion Monte Carlo algorithm with importance sampling for fermions, was used to compute the correlation energy of homogeneous electron gas by Ceperley and Alder [109] in 1980; the resulting HEG correlation energy was crucial in the development of density functional calculations. Types and properties of projectors In this method, a many-body projector G ( R , R ) = Ĝ is repeatedly applied to filter out the exact many-body ground state from an initial state; the operation of the projector is carried out with a random walk, hence the name of this class of methods.
Fermion quantum Monte Carlo (QMC) methods that work in a general basis space may be more effective for some problems than the traditional diffusion QMC method. Projection of the ground state energy is achieved by random walks in the space of Slater determinants whose one‐particle orbitals are expressed in terms of the chosen basis set. A complication is that the determinants will in general acquire complex phases. This a consequence of ground state Monte Carlo projection using the Hubbard‐Stratonovich transformation of the two‐body interaction. To control the resulting “sign” decay, we describe a method we have recently introduced for the propagation of phaseless determinants. The approximation relies on importance sampling with a trial wave function. The approximation has features in common with diffusion MC fixed node and lattice‐model constrained path methods. Using a planewave basis and non‐local pseudopotentials, we apply the method to Si atom, dimer, and 2, 16, 54 atom (216 electrons) bulk supercells. Single Slater determinant wave functions from density functional theory calculations were used as |ΨT〉 with no additional optimization. The calculated binding energy of Si2 and cohesive energy of bulk Si are in excellent agreement with experiments and are comparable to the best existing theoretical results.