Quantum Monte Carlo encompasses a large family of computational methods whose common aim is the study of complexquantum systems. One of the major goals of these approaches is to provide a reliable solution (or an accurate approximation) of the quantummany-body problem. The diverse flavors of quantum Monte Carlo approaches all share the common use of theMonte Carlo method to handle the multi-dimensional integrals that arise in the different formulations of the many-body problem.
Quantum Monte Carlo methods allow for a direct treatment and description of complex many-body effects encoded in thewave function, going beyondmean-field theory. In particular, there exist numerically exact andpolynomially-scalingalgorithms to exactly study static properties ofboson systems withoutgeometrical frustration. Forfermions, 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-bodySchrödinger 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, andrelativistic effects can be neglected. This is true for a wide range of electronic problems incondensed matter physics, inBose–Einstein condensates andsuperfluids such asliquid helium. The ability to solve the Schrödinger equation for a given system allows prediction of its behavior, with important applications ranging frommaterials science to complexbiological systems.
The difficulty is however that solving the Schrödinger equation requires the knowledge of the many-body wave function in the many-bodyHilbert 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 modernparallel computing technology in a reasonable amount of time. Traditionally, approximations for the many-body wave function as anantisymmetric function of one-bodyorbitals[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 theHartree–Fock (HF) approximation, or converging very slowly, as inconfiguration 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 interactingboson systems, while providing an approximate description of interactingfermion systems. Most methods aim at computing theground state wavefunction of the system, with the exception ofpath integral Monte Carlo and finite-temperatureauxiliary-field Monte Carlo, which calculate thedensity 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-evolvedwave function, as done in thetime-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.
