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 manybody problem. The diverse flavor of quantum Monte Carlo approaches all share the common use of the Monte Carlo method to handle the multidimensional integrals that arise in the different formulations of the manybody problem. The quantum Monte Carlo methods allow for a direct treatment and description of complex manybody effects encoded in the wave function, going beyond mean field theory and offering an exact solution of the manybody problem in some circumstances. In particular, there exist numerically exact and polynomiallyscaling 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.
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Transcription
Contents
Background
In principle, any physical system can be described by the manybody 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 manybody wave function in the manybody 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 manybody wave function as an antisymmetric function of onebody orbitals^{[1]} have been used, in order to have a manageable treatment of the Schrödinger equation. This kind of formulation has however several drawbacks, either limiting the effect of quantum manybody 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 manybody problem and the manybody wave function beyond these approximations. The most advanced quantum Monte Carlo approaches provide an exact solution to the manybody problem for nonfrustrated interacting boson systems, while providing an approximate, yet typically very accurate, 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 finitetemperature auxiliary field Monte Carlo, which calculate the density matrix. In addition to static properties, the timedependent Schrödinger equation can also be solved, albeit only approximately, restricting the functional form of the timeevolved wave function, as done in the timedependent variational Monte Carlo. From the probabilistic point of view, the computation of the top eigenvalues and the corresponding ground states eigenfunctions associated with the Schrödinger equation relies on the numerical solving of Feynman–Kac path integration problems.^{[2]}^{[3]} The mathematical foundations of Feynman–Kac particle absorption models and their Sequential Monte Carlo and mean field interpretations are developed in.^{[4]}^{[5]}^{[6]}^{[7]}^{[8]}
There are several quantum Monte Carlo methods, each of which uses Monte Carlo in different ways to solve the manybody problem:
Quantum Monte Carlo methods
Zerotemperature (only ground state)
 Variational Monte Carlo: A good place to start; it is commonly used in many sorts of quantum problems.
 Diffusion Monte Carlo: The most common highaccuracy method for electrons (that is, chemical problems), since it comes quite close to the exact groundstate energy fairly efficiently. Also used for simulating the quantum behavior of atoms, etc.
 Reptation Monte Carlo: Recent zerotemperature 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
Finitetemperature (thermodynamic)
 Auxiliary field Monte Carlo: Usually applied to lattice problems, although there has been recent work on applying it to electrons in chemical systems.
 Continuoustime quantum Monte Carlo
 Determinant quantum Monte Carlo or Hirsch–Fye quantum Monte Carlo
 Hybrid quantum Monte Carlo
 Path integral Monte Carlo: Finitetemperature technique mostly applied to bosons where temperature is very important, especially superfluid helium.
 Stochastic Green function algorithm:^{[9]} An algorithm designed for bosons that can simulate any complicated lattice Hamiltonian that does not have a sign problem.
 Worldline quantum Monte Carlo
Realtime dynamics (closed quantum systems)
 Timedependent variational Monte Carlo: An extension of the variational Monte Carlo to study the dynamics of pure quantum states.
See also
 Monte Carlo method
 QMC@Home
 Quantum chemistry
 Density matrix renormalization group
 Timeevolving block decimation
 Metropolis algorithm
 Wavefunction optimization
 Monte Carlo molecular modeling
 Quantum chemistry computer programs
Implementations
Notes
 ^ Functional form of the wave function
 ^ 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 00219606.
 ^ Korzeniowski, A.; Fry, J. L.; Orr, D. E.; Fazleev, N. G. (August 10, 1992). "Feynman–Kac pathintegral calculation of the groundstate energies of atoms". Physical Review Letters. 69 (6): 893–896. Bibcode:1992PhRvL..69..893K. doi:10.1103/PhysRevLett.69.893. PMID 10047062.
 ^ "EUDML  Particle approximations of Lyapunov exponents connected to Schrödinger operators and Feynman–Kac semigroups – P. Del Moral, L. Miclo". eudml.org. Retrieved 20150611.
 ^ Del Moral, Pierre; Doucet, Arnaud (January 1, 2004). "Particle Motions in Absorbing Medium with Hard and Soft Obstacles". Stochastic Analysis and Applications. 22 (5): 1175–1207. doi:10.1081/SAP200026444. ISSN 07362994.
 ^ Del Moral, Pierre (2013). Mean field simulation for Monte Carlo integration. Chapman & Hall/CRC Press. p. 626.
Monographs on Statistics & Applied Probability
 ^ Del Moral, Pierre (2004). Feynman–Kac formulae. Genealogical and interacting particle approximations. Probability and its Applications. Springer. p. 575. ISBN 9780387202686.
Series: Probability and Applications
 ^ Del Moral, Pierre; Miclo, Laurent (2000). Branching and Interacting Particle Systems Approximations of Feynman–Kac Formulae with Applications to NonLinear Filtering (PDF). Lecture Notes in Mathematics. 1729. pp. 1–145. doi:10.1007/bfb0103798. ISBN 9783540673149.
 ^ Rousseau, V. G. (20 May 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.
References
 V. G. Rousseau (May 2008). "Stochastic Green Function (SGF) algorithm". Phys. Rev. E. 77 (5): 056705. arXiv:0711.3839. Bibcode:2008PhRvE..77e6705R. doi:10.1103/PhysRevE.77.056705.
 Hammond, B.J.; W.A. Lester; P.J. Reynolds (1994). Monte Carlo Methods in Ab Initio Quantum Chemistry. Singapore: World Scientific. ISBN 9789810203214. OCLC 29594695.
 Nightingale, M.P.; Umrigar, Cyrus J., eds. (1999). Quantum Monte Carlo Methods in Physics and Chemistry. Springer. ISBN 9780792355526.
 W. M. C. Foulkes; L. Mitáš; R. J. Needs; G. Rajagopal (5 January 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:condmat/0303551. Bibcode:2003cond.mat..3551D. doi:10.1590/S010397332003000100003.
 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.
External links
 QMC in Cambridge and around the world Large amount of general information about QMC with links.
 Semiclassical methods of deformation quantisation in transport theory
 Joint DEMOCRITOSICTP 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 in the Apuan Alps IX – international QMC workshop, Vallico Sotto, Tuscany, Italy, 26 July – 2 August 2014 – Announcement, Poster
 Quantum Monte Carlo and the CASINO program IX – international QMC summer school, Vallico Sotto, Tuscany, Italy, 3–10 August 2014 – Announcement, Poster
 Quantum Monte Carlo simulator (Qwalk)