**Quantum Monte Carlo** encompasses a large family of computational methods whose common aim is the study of complex quantum systems. One of the major goals of these approaches is to provide a reliable solution (or an accurate approximation) of the quantum many-body problem. The diverse flavors of quantum Monte Carlo approaches all share the common use of the Monte Carlo method to handle the multi-dimensional integrals that arise in the different formulations of the many-body problem.

- Background
- Quantum Monte Carlo methods
- Zero-temperature (only ground state)
- Finite-temperature (thermodynamic)
- Real-time dynamics (closed quantum systems)
- See also
- Notes
- References
- External links

Quantum Monte Carlo methods allow for a direct treatment and description of complex many-body effects encoded in the wave function, going beyond mean-field theory. In particular, there exist numerically exact and polynomially-scaling algorithms to exactly study static properties of boson systems without geometrical frustration. For fermions, there exist very good approximations to their static properties and numerically exact exponentially scaling quantum Monte Carlo algorithms, but none that are both.

In principle, any physical system can be described by the many-body Schrödinger equation as long as the constituent particles are not moving "too" fast; that is, they are not moving at a speed comparable to that of light, and relativistic effects can be neglected. This is true for a wide range of electronic problems in condensed matter physics, in Bose–Einstein condensates and superfluids such as liquid helium. The ability to solve the Schrödinger equation for a given system allows prediction of its behavior, with important applications ranging from materials science to complex biological systems.

The difficulty is however that solving the Schrödinger equation requires the knowledge of the many-body wave function in the many-body Hilbert space, which typically has an exponentially large size in the number of particles. Its solution for a reasonably large number of particles is therefore typically impossible, even for modern parallel computing technology in a reasonable amount of time. Traditionally, approximations for the many-body wave function as an antisymmetric function of one-body orbitals ^{ [1] } have been used, in order to have a manageable treatment of the Schrödinger equation. However, this kind of formulation has several drawbacks, either limiting the effect of quantum many-body correlations, as in the case of the Hartree–Fock (HF) approximation, or converging very slowly, as in configuration interaction applications in quantum chemistry.

Quantum Monte Carlo is a way to directly study the many-body problem and the many-body wave function beyond these approximations. The most advanced quantum Monte Carlo approaches provide an exact solution to the many-body problem for non-frustrated interacting boson systems, while providing an approximate description of interacting fermion systems. 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. In addition to static properties, the time-dependent Schrödinger equation can also be solved, albeit only approximately, restricting the functional form of the time-evolved wave function, as done in the time-dependent variational Monte Carlo.

From a probabilistic point of view, the computation of the top eigenvalues and the corresponding ground state eigenfunctions associated with the Schrödinger equation relies on the numerical solving of Feynman–Kac path integration problems.^{ [2] }^{ [3] }

There are several quantum Monte Carlo methods, each of which uses Monte Carlo in different ways to solve the many-body problem.

- Variational Monte Carlo: A good place to start; it is commonly used in many sorts of quantum problems.
- Diffusion Monte Carlo: 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.
- Reptation Monte Carlo: 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 Carlo
- Path integral ground state: Mainly used for boson systems; for those it allows calculation of physical observables exactly, i.e. with arbitrary accuracy

- Auxiliary-field Monte Carlo: Usually applied to lattice problems, although there has been recent work on applying it to electrons in chemical systems.
- Continuous-time quantum Monte Carlo
- Determinant quantum Monte Carlo or Hirsch–Fye quantum Monte Carlo
- Hybrid quantum Monte Carlo
- Path integral Monte Carlo: Finite-temperature technique mostly applied to bosons where temperature is very important, especially superfluid helium.
- Stochastic Green function algorithm:
^{ [4] }An algorithm designed for bosons that can simulate any complicated lattice Hamiltonian that does not have a sign problem. - World-line quantum Monte Carlo

- Time-dependent variational Monte Carlo: An extension of the variational Monte Carlo to study the dynamics of pure quantum states.

- ↑ "Functional form of the wave function". Archived from the original on July 18, 2009. Retrieved April 22, 2009.
- ↑ Caffarel, Michel; Claverie, Pierre (1988). "Development of a pure diffusion quantum Monte Carlo method using a full generalized Feynman–Kac formula. I. Formalism".
*The Journal of Chemical Physics*.**88**(2): 1088–1099. Bibcode:1988JChPh..88.1088C. doi:10.1063/1.454227. ISSN 0021-9606. - ↑ Korzeniowski, A.; Fry, J. L.; Orr, D. E.; Fazleev, N. G. (August 10, 1992). "Feynman–Kac path-integral calculation of the ground-state energies of atoms".
*Physical Review Letters*.**69**(6): 893–896. Bibcode:1992PhRvL..69..893K. doi:10.1103/PhysRevLett.69.893. PMID 10047062. - ↑ Rousseau, V. G. (May 20, 2008). "Stochastic Green function algorithm".
*Physical Review E*.**77**(5): 056705. arXiv: 0711.3839 . Bibcode:2008PhRvE..77e6705R. doi:10.1103/physreve.77.056705. PMID 18643193. S2CID 2188292.

**Computational chemistry** is a branch of chemistry that uses computer simulation to assist in solving chemical problems. It uses methods of theoretical chemistry, incorporated into computer programs, to calculate the structures and properties of molecules, groups of molecules, and solids. It is essential because, apart from relatively recent results concerning the hydrogen molecular ion, the quantum many-body problem cannot be solved analytically, much less in closed form. While computational results normally complement the information obtained by chemical experiments, it can in some cases predict hitherto unobserved chemical phenomena. It is widely used in the design of new drugs and materials.

**Quantum chemistry**, also called **molecular quantum mechanics**, is a branch of physical chemistry focused on the application of quantum mechanics to chemical systems, particularly towards the quantum-mechanical calculation of electronic contributions to physical and chemical properties of molecules, materials, and solutions at the atomic level. These calculations include systematically applied approximations intended to make calculations computationally feasible while still capturing as much information about important contributions to the computed wave functions as well as to observable properties such as structures, spectra, and thermodynamic properties. Quantum chemistry is also concerned with the computation of quantum effects on molecular dynamics and chemical kinetics.

**Monte Carlo methods**, or **Monte Carlo experiments**, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be deterministic in principle. They are often used in physical and mathematical problems and are most useful when it is difficult or impossible to use other approaches. Monte Carlo methods are mainly used in three problem classes: optimization, numerical integration, and generating draws from a probability distribution.

**Computational physics** is the study and implementation of numerical analysis to solve problems in physics for which a quantitative theory already exists. Historically, computational physics was the first application of modern computers in science, and is now a subset of computational science. It is sometimes regarded as a subdiscipline of theoretical physics, but others consider it an intermediate branch between theoretical and experimental physics - an area of study which supplements both theory and experiment.

In computational physics and chemistry, the **Hartree–Fock** (**HF**) method is a method of approximation for the determination of the wave function and the energy of a quantum many-body system in a stationary state.

The **Lippmann–Schwinger equation** is one of the most used equations to describe particle collisions – or, more precisely, scattering – in quantum mechanics. It may be used in scattering of molecules, atoms, neutrons, photons or any other particles and is important mainly in atomic, molecular, and optical physics, nuclear physics and particle physics, but also for seismic scattering problems in geophysics. It relates the scattered wave function with the interaction that produces the scattering and therefore allows calculation of the relevant experimental parameters.

** Ab initio quantum chemistry methods** are computational chemistry methods based on quantum chemistry. The term

In computational physics, **variational Monte Carlo (VMC)** is a quantum Monte Carlo method that applies the variational method to approximate the ground state of a quantum system.

**Diffusion Monte Carlo** (DMC) or **diffusion quantum 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. When actually attempting the calculation, one finds that for bosons, the algorithm scales as a polynomial with the system size, but for fermions, DMC scales exponentially with the system size. This makes exact large-scale DMC simulations for fermions impossible; however, DMC employing a clever approximation known as the fixed-node approximation can still yield very accurate results.

**Path integral Monte Carlo** (**PIMC**) is a quantum Monte Carlo method used to solve quantum statistical mechanics problems numerically within the path integral formulation. The application of Monte Carlo methods to path integral simulations of condensed matter systems was first pursued in a key paper by John A. Barker.

**Reptation Monte Carlo** is a quantum Monte Carlo method.

A **quantum master equation** is a generalization of the idea of a master equation. Rather than just a system of differential equations for a set of probabilities, quantum master equations are differential equations for the entire density matrix, including off-diagonal elements. A density matrix with only diagonal elements can be modeled as a classical random process, therefore such an "ordinary" master equation is considered classical. Off-diagonal elements represent quantum coherence which is a physical characteristic that is intrinsically quantum mechanical.

In applied mathematics, the **numerical sign problem** is the problem of numerically evaluating the integral of a highly oscillatory function of a large number of variables. Numerical methods fail because of the near-cancellation of the positive and negative contributions to the integral. Each has to be integrated to very high precision in order for their difference to be obtained with useful accuracy.

**David Matthew Ceperley** is a theoretical physicist in the physics department at the University of Illinois Urbana-Champaign or UIUC. He is a world expert in the area of Quantum Monte Carlo computations, a method of calculation that is generally recognised to provide accurate quantitative results for many-body problems described by quantum mechanics.

**Quantum finance** is an interdisciplinary research field, applying theories and methods developed by quantum physicists and economists in order to solve problems in finance. It is a branch of econophysics.

**Path integral molecular dynamics** (**PIMD**) is a method of incorporating quantum mechanics into molecular dynamics simulations using Feynman path integrals. In PIMD, one uses the Born–Oppenheimer approximation to separate the wavefunction into a nuclear part and an electronic part. The nuclei are treated quantum mechanically by mapping each quantum nucleus onto a classical system of several fictitious particles connected by springs governed by an effective Hamiltonian, which is derived from Feynman's path integral. The resulting classical system, although complex, can be solved relatively quickly. There are now a number of commonly used condensed matter computer simulation techniques that make use of the path integral formulation including **Centroid Molecular Dynamics** (**CMD**), **Ring Polymer Molecular Dynamics** (**RPMD**), and the **Feynman-Kleinert Quasi-Classical Wigner (FK-QCW)** method. The same techniques are also used in path integral Monte Carlo (PIMC).

**Mean-field particle methods** are a broad class of *interacting type* Monte Carlo algorithms for simulating from a sequence of probability distributions satisfying a nonlinear evolution equation. These flows of probability measures can always be interpreted as the distributions of the random states of a Markov process whose transition probabilities depends on the distributions of the current random states. A natural way to simulate these sophisticated nonlinear Markov processes is to sample a large number of copies of the process, replacing in the evolution equation the unknown distributions of the random states by the sampled empirical measures. In contrast with traditional Monte Carlo and Markov chain Monte Carlo methods these mean-field particle techniques rely on **sequential interacting samples**. The terminology mean-field reflects the fact that each of the *samples * interacts with the empirical measures of the process. When the size of the system tends to infinity, these random empirical measures converge to the deterministic distribution of the random states of the nonlinear Markov chain, so that the statistical interaction between particles vanishes. In other words, starting with a chaotic configuration based on independent copies of initial state of the nonlinear Markov chain model, the chaos propagates at any time horizon as the size the system tends to infinity; that is, finite blocks of particles reduces to independent copies of the nonlinear Markov process. This result is called the propagation of chaos property. The terminology "propagation of chaos" originated with the work of Mark Kac in 1976 on a colliding mean-field kinetic gas model.

The **time-dependent variational Monte Carlo (t-VMC)** method is a quantum Monte Carlo approach to study the dynamics of closed, non-relativistic quantum systems in the context of the quantum many-body problem. It is an extension of the variational Monte Carlo method, in which a time-dependent pure quantum state is encoded by some variational wave function, generally parametrized as

In many-body physics, the problem of **analytic continuation** is that of numerically extracting the spectral density of a Green function given its values on the imaginary axis. It is a necessary post-processing step for calculating dynamical properties of physical systems from quantum Monte Carlo simulations, which often compute Green function values only at imaginary-times or Matsubara frequencies.

- Hammond, B.J.; W.A. Lester; P.J. Reynolds (1994).
*Monte Carlo Methods in Ab Initio Quantum Chemistry*. Singapore: World Scientific. ISBN 978-981-02-0321-4. OCLC 29594695. - Nightingale, M.P.; Umrigar, Cyrus J., eds. (1999).
*Quantum Monte Carlo Methods in Physics and Chemistry*. Springer. ISBN 978-0-7923-5552-6. - W. M. C. Foulkes; L. Mitáš; R. J. Needs; G. Rajagopal (January 5, 2001). "Quantum Monte Carlo simulations of solids".
*Rev. Mod. Phys*.**73**(1): 33–83. Bibcode:2001RvMP...73...33F. CiteSeerX 10.1.1.33.8129 . doi:10.1103/RevModPhys.73.33. - Raimundo R. dos Santos (2003). "Introduction to Quantum Monte Carlo simulations for fermionic systems".
*Braz. J. Phys*.**33**: 36–54. arXiv: cond-mat/0303551 . Bibcode:2003cond.mat..3551D. doi:10.1590/S0103-97332003000100003. S2CID 44055350. - M. Dubecký; L. Mitas; P. Jurečka (2016). "Noncovalent Interactions by Quantum Monte Carlo".
*Chem. Rev*.**116**(9): 5188–5215. doi:10.1021/acs.chemrev.5b00577. PMID 27081724. - Becca, Federico; Sandro Sorella (2017).
*Quantum Monte Carlo Approaches for Correlated Systems*. Cambridge University Press. ISBN 978-1107129931.

- QMC in Cambridge and around the world Large amount of general information about QMC with links.
- Quantum Monte Carlo simulator (Qwalk)

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