Quantum Monte Carlo
Encyclopedia
Quantum Monte Carlo is a large class of computer algorithms that simulate quantum systems with the idea of solving the quantum many-body problem
. They use, in one way or another, the Monte Carlo method
to handle the many-dimensional integrals that arise. Quantum Monte Carlo allows a direct representation of many-body effects in the wave function, at the cost of statistical uncertainty that can be reduced with more simulation time. For boson
s, there exist numerically exact and polynomial
-scaling algorithm
s. For fermion
s, there exist very good approximations and numerically exact exponentially scaling quantum Monte Carlo algorithms, but none that are both.
as long as the constituent particles are not moving "too" fast; that is, they are not moving near the speed of light. This covers a wide range of electronic problems in condensed matter physics
, so if we could solve the Schrödinger equation for a given system, we could predict its behavior, which has important applications in fields from computers to biology. This also includes the nuclei
in Bose–Einstein condensate
and superfluid
s such as liquid helium
. The difficulty is that the Schrödinger equation involves a function of a number of coordinates that is three times the number of particles, and is therefore difficult, if not impossible, to solve even using parallel computing
technology in a reasonable amount of time. Traditionally, theorists have approximated the many-body wave function as an antisymmetric function of one-body orbital
s. This kind of formulation either limits the possible wave functions, as in the case of the Hartree-Fock
(HF) approximation, or converges very slowly, as in configuration interaction
. One of the reasons for the difficulty with an HF initial estimate (ground state seed, also known as Slater determinant
) is that it is very difficult to model the electronic
and nuclear cusps in the wavefunction. However, one does not generally model at this point of the approximation. As two particles approach each other, the wavefunction has exactly known derivative
s.
Quantum Monte Carlo is a way around these problems because it allows us to model a many-body wavefunction of our choice directly. Specifically, we can use a Hartree-Fock approximation as our starting point but then multiplying it by any symmetric function, of which Jastrow functions are typical, designed to enforce the cusp conditions. Most methods aim at computing the ground state
wavefunction of the system, with the exception of path integral Monte Carlo
and finite-temperature auxiliary field Monte Carlo
, which calculate the density matrix
.
There are several quantum Monte Carlo methods, each of which uses Monte Carlo in different ways to solve the many-body problem:
Many-body problem
The many-body problem is a general name for a vast category of physical problems pertaining to the properties of microscopic systems made of a large number of interacting particles. Microscopic here implies that quantum mechanics has to be used to provide an accurate description of the system...
. They use, in one way or another, the Monte Carlo method
Monte Carlo method
Monte Carlo methods are a class of computational algorithms that rely on repeated random sampling to compute their results. Monte Carlo methods are often used in computer simulations of physical and mathematical systems...
to handle the many-dimensional integrals that arise. Quantum Monte Carlo allows a direct representation of many-body effects in the wave function, at the cost of statistical uncertainty that can be reduced with more simulation time. For boson
Boson
In particle physics, bosons are subatomic particles that obey Bose–Einstein statistics. Several bosons can occupy the same quantum state. The word boson derives from the name of Satyendra Nath Bose....
s, there exist numerically exact and polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
-scaling algorithm
Algorithm
In mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning...
s. For fermion
Fermion
In particle physics, a fermion is any particle which obeys the Fermi–Dirac statistics . Fermions contrast with bosons which obey Bose–Einstein statistics....
s, there exist very good approximations and numerically exact exponentially scaling quantum Monte Carlo algorithms, but none that are both.
Background
In principle, any physical system can be described by the many-body Schrödinger equationSchrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....
as long as the constituent particles are not moving "too" fast; that is, they are not moving near the speed of light. This covers a wide range of electronic problems in condensed matter physics
Condensed matter physics
Condensed matter physics deals with the physical properties of condensed phases of matter. These properties appear when a number of atoms at the supramolecular and macromolecular scale interact strongly and adhere to each other or are otherwise highly concentrated in a system. The most familiar...
, so if we could solve the Schrödinger equation for a given system, we could predict its behavior, which has important applications in fields from computers to biology. This also includes the nuclei
Atomic nucleus
The nucleus is the very dense region consisting of protons and neutrons at the center of an atom. It was discovered in 1911, as a result of Ernest Rutherford's interpretation of the famous 1909 Rutherford experiment performed by Hans Geiger and Ernest Marsden, under the direction of Rutherford. The...
in Bose–Einstein condensate
Bose–Einstein condensate
A Bose–Einstein condensate is a state of matter of a dilute gas of weakly interacting bosons confined in an external potential and cooled to temperatures very near absolute zero . Under such conditions, a large fraction of the bosons occupy the lowest quantum state of the external potential, at...
and superfluid
Superfluid
Superfluidity is a state of matter in which the matter behaves like a fluid without viscosity and with extremely high thermal conductivity. The substance, which appears to be a normal liquid, will flow without friction past any surface, which allows it to continue to circulate over obstructions and...
s such as liquid helium
Liquid helium
Helium exists in liquid form only at extremely low temperatures. The boiling point and critical point depend on the isotope of the helium; see the table below for values. The density of liquid helium-4 at its boiling point and 1 atmosphere is approximately 0.125 g/mL Helium-4 was first liquefied...
. The difficulty is that the Schrödinger equation involves a function of a number of coordinates that is three times the number of particles, and is therefore difficult, if not impossible, to solve even using parallel computing
Parallel computing
Parallel computing is a form of computation in which many calculations are carried out simultaneously, operating on the principle that large problems can often be divided into smaller ones, which are then solved concurrently . There are several different forms of parallel computing: bit-level,...
technology in a reasonable amount of time. Traditionally, theorists have approximated the many-body wave function as an antisymmetric function of one-body orbital
Molecular orbital
In chemistry, a molecular orbital is a mathematical function describing the wave-like behavior of an electron in a molecule. This function can be used to calculate chemical and physical properties such as the probability of finding an electron in any specific region. The term "orbital" was first...
s. This kind of formulation either limits the possible wave functions, as in the case of the Hartree-Fock
Hartree-Fock
In computational physics and chemistry, the Hartree–Fock method is an approximate method for the determination of the ground-state wave function and ground-state energy of a quantum many-body system....
(HF) approximation, or converges very slowly, as in configuration interaction
Configuration interaction
Configuration interaction is a post-Hartree–Fock linear variational method for solving the nonrelativistic Schrödinger equation within the Born–Oppenheimer approximation for a quantum chemical multi-electron system. Mathematically, configuration simply describes the linear combination...
. One of the reasons for the difficulty with an HF initial estimate (ground state seed, also known as Slater determinant
Slater determinant
In quantum mechanics, a Slater determinant is an expression that describes the wavefunction of a multi-fermionic system that satisfies anti-symmetry requirements and consequently the Pauli exclusion principle by changing sign upon exchange of fermions . It is named for its discoverer, John C...
) is that it is very difficult to model the electronic
Electronic density
In quantum mechanics, and in particular quantum chemistry, the electronic density is a measure of the probability of an electron occupying an infinitesimal element of space surrounding any given point. It is a scalar quantity depending upon three spatial variables and is typically denoted as either...
and nuclear cusps in the wavefunction. However, one does not generally model at this point of the approximation. As two particles approach each other, the wavefunction has exactly known derivative
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
s.
Quantum Monte Carlo is a way around these problems because it allows us to model a many-body wavefunction of our choice directly. Specifically, we can use a Hartree-Fock approximation as our starting point but then multiplying it by any symmetric function, of which Jastrow functions are typical, designed to enforce the cusp conditions. Most methods aim at computing the ground state
Ground state
The ground state of a quantum mechanical system is its lowest-energy state; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state...
wavefunction of the system, with the exception of path integral Monte Carlo
Path integral Monte Carlo
Path integral Monte Carlo is a quantum Monte Carlo method in the path integral formulation of quantum mechanics.The equations often are applied assuming that quantum exchange does not matter...
and finite-temperature auxiliary field Monte Carlo
Auxiliary field Monte Carlo
Auxiliary field Monte Carlo is a method that allows the calculation, by use of Monte Carlo techniques, of averages of operators in many-body quantum mechanical or classical problems .-Reweighting procedure and numerical sign problem:The distinctive ingredient of "auxiliary field Monte Carlo" is...
, which calculate the density matrix
Density matrix
In quantum mechanics, a density matrix is a self-adjoint positive-semidefinite matrix of trace one, that describes the statistical state of a quantum system...
.
There are several quantum Monte Carlo methods, each of which uses Monte Carlo in different ways to solve the many-body problem:
Quantum Monte Carlo methods
- Stochastic Green function (SGF) algorithm : An algorithm designed for bosons that can simulate any complicated lattice HamiltonianHamiltonian (quantum mechanics)In quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...
that does not have a sign problem. Used in combination with a directed update scheme, this is a powerful tool. - Variational Monte CarloVariational Monte CarloIn mathematical physics, variational Monte Carlo is a quantum Monte Carlo method that applies the variational method to approximate the ground state of the system.The expectation value necessary can be written in the x representation as...
: A good place to start; it is commonly used in many sorts of quantum problems. - Diffusion Monte CarloDiffusion Monte CarloDiffusion Monte Carlo is a quantum Monte Carlo method that uses a Green's function to solve the Schrödinger equation. DMC is potentially numerically exact, meaning that it can find the exact ground state energy within a given error for any quantum system...
: The most common high-accuracy method for electrons (that is, chemical problems), since it comes quite close to the exact ground-state energy fairly efficiently. Also used for simulating the quantum behavior of atoms, etc. - Path integral Monte CarloPath integral Monte CarloPath integral Monte Carlo is a quantum Monte Carlo method in the path integral formulation of quantum mechanics.The equations often are applied assuming that quantum exchange does not matter...
: Finite-temperature technique mostly applied to bosons where temperature is very important, especially superfluid helium. - Auxiliary field Monte CarloAuxiliary field Monte CarloAuxiliary field Monte Carlo is a method that allows the calculation, by use of Monte Carlo techniques, of averages of operators in many-body quantum mechanical or classical problems .-Reweighting procedure and numerical sign problem:The distinctive ingredient of "auxiliary field Monte Carlo" is...
: Usually applied to latticeLattice model (physics)In physics, a lattice model is a physical model that is defined on a lattice, as opposed to the continuum of space or spacetime. Lattice models originally occurred in the context of condensed matter physics, where the atoms of a crystal automatically form a lattice. Currently, lattice models are...
problems, although there has been recent work on applying it to electrons in chemical systems. - Reptation Monte CarloReptation Monte Carlo- References :**...
: Recent zero-temperature method related to path integral Monte Carlo, with applications similar to diffusion Monte Carlo but with some different tradeoffs. - Gaussian quantum Monte CarloGaussian quantum Monte CarloGaussian Quantum Monte Carlo is a quantum Monte Carlo method that shows a potential solution to the fermion sign problem without the deficiencies of alternative approaches. Instead of the Hilbert space, this method works in the space of density matrices that can be spanned by an over-complete basis...
See also
- Stochastic Green Function (SGF) algorithm
- Monte Carlo methodMonte Carlo methodMonte Carlo methods are a class of computational algorithms that rely on repeated random sampling to compute their results. Monte Carlo methods are often used in computer simulations of physical and mathematical systems...
- QMC@HomeQMC@HomeQMC@Home is a distributed computing project for the BOINC client aimed at further developing and testing Quantum Monte Carlo for use in quantum chemistry. It is hosted by the University of Münster with participation by the Cavendish Laboratory...
- Quantum chemistryQuantum chemistryQuantum chemistry is a branch of chemistry whose primary focus is the application of quantum mechanics in physical models and experiments of chemical systems...
- Density matrix renormalization groupDensity matrix renormalization groupThe density matrix renormalization group is a numerical variational technique devised to obtain the low energy physics of quantum many-body systems with high accuracy. It was invented in 1992 by Steven R...
- Time-evolving block decimationTime-evolving block decimationThe time-evolving block decimation algorithm is a numerical scheme used to simulate one-dimensional quantum many-body systems, characterized by at most nearest-neighbour interactions....
- Metropolis algorithm
- Wavefunction optimization
- Monte Carlo molecular modelingMonte Carlo molecular modelingMonte Carlo molecular modeling is the application of Monte Carlo methods to molecular problems. These problems can also be modeled by the molecular dynamics method. The difference is that this approach relies on statistical mechanics rather than molecular dynamics. Instead of trying to reproduce...
- Quantum chemistry computer programsQuantum chemistry computer programsQuantum chemistry computer programs are used in computational chemistry to implement the methods of quantum chemistry. Most include the Hartree–Fock and some post-Hartree–Fock methods. They may also include density functional theory , molecular mechanics or semi-empirical quantum...
Implementations
External links
- QMCWIKI
- Joint DEMOCRITOS-ICTP School on Continuum Quantum Monte Carlo Methods
- FreeScience Library - Quantum Monte Carlo
- UIUC 2007 Summer School on Computational Materials Science: Quantum Monte Carlo from Minerals and Materials to Molecules
- Quantum Monte Carlo and the CASINO program VI - summer school, Vallico Sotto, Tuscany, 30 July - 6 August 2011 - Announcement, Poster
- Quantum Monte Carlo simulator (Qwalk)
