Quantum Monte Carlo encompasses a large family of computational methods whose common aim is the study of complex

quantum system Quantum mechanics is a fundamental Scientific theory, theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including qua ...

s. 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 many-body problem is a general name for a vast category of physical problems pertaining to the properties of microscopic systems made of many interacting particles. ''Microscopic'' here implies that quantum mechanics has to be used to provid ...

. The diverse flavors of quantum Monte Carlo approaches all share the common use of the

Monte Carlo method 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 deter ...

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 the

wave function A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements m ...

, going beyond mean-field theory. In particular, there exist numerically exact and

polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An ex ...

ly-scaling

algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...

s to exactly study static properties of

boson In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer s ...

systems without

geometrical frustration In condensed matter physics, the term geometrical frustration (or in short: frustration) refers to a phenomenon where atoms tend to stick to non-trivial positions or where, on a regular crystal lattice, conflicting inter-atomic forces (each one fav ...

. For

fermion In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...

s, there exist very good approximations to their static properties 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 equation The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...

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 Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid phases which arise from electromagnetic forces between atoms. More generally, the s ...

, in

Bose–Einstein condensate In condensed matter physics, a Bose–Einstein condensate (BEC) is a state of matter that is typically formed when a gas of bosons at very low densities is cooled to temperatures very close to absolute zero (−273.15 °C or −459.67&n ...

s and

superfluid Superfluidity is the characteristic property of a fluid with zero viscosity which therefore flows without any loss of kinetic energy. When stirred, a superfluid forms vortices that continue to rotate indefinitely. Superfluidity occurs in two ...

s such as

liquid helium Liquid helium is a physical state of helium at very low temperatures at standard atmospheric pressures. Liquid helium may show superfluidity. At standard pressure, the chemical element helium exists in a liquid form only at the extremely low t ...

. 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 system A biological system is a complex network which connects several biologically relevant entities. Biological organization spans several scales and are determined based different structures depending on what the system is. Examples of biological sys ...

s. The difficulty is however that solving the Schrödinger equation requires the knowledge of the many-body wave function in the many-body

Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natu ...

, 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

orbital Orbital may refer to: Sciences Chemistry and physics * Atomic orbital * Molecular orbital * Hybrid orbital Astronomy and space flight * Orbit ** Earth orbit Medicine and physiology * Orbit (anatomy), also known as the ''orbital bone'' * Orbito ...

s 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 Configuration interaction (CI) 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. Mathemati ...

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

systems, while providing an approximate description of interacting

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 quantum mechanics, a density matrix (or density operator) is a matrix that describes the quantum state of a physical system. It allows for the calculation of the probabilities of the outcomes of any measurement performed upon this system, usin ...

. 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

, 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.

Quantum Monte Carlo methods

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

Zero-temperature (only ground state)

* Variational Monte Carlo: A good place to start; it is commonly used in many sorts of quantum problems. **

Diffusion Monte Carlo 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 ene ...

: 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 Gaussian 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 m ...

Path integral ground state A path is a route for physical travel – see Trail. Path or PATH may also refer to: Physical paths of different types * Bicycle path * Bridle path, used by people on horseback * Course (navigation), the intended path of a vehicle * Desire ...

: Mainly used for boson systems; for those it allows calculation of physical observables exactly, i.e. with arbitrary accuracy

Finite-temperature (thermodynamic)

* 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 In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...

or Hirsch–Fye quantum Monte Carlo *

Hybrid quantum Monte Carlo Hybrid may refer to: Science * Hybrid (biology), an offspring resulting from cross-breeding ** Hybrid grape, grape varieties produced by cross-breeding two ''Vitis'' species ** Hybridity, the property of a hybrid plant which is a union of two dif ...

* Path integral Monte Carlo: Finite-temperature technique mostly applied to bosons where temperature is very important, especially superfluid helium. * Stochastic Green function algorithm: An algorithm designed for bosons that can simulate any complicated lattice Hamiltonian that does not have a sign problem. *

World-line quantum Monte Carlo The world line (or worldline) of an object is the path that an object traces in 4-dimensional spacetime. It is an important concept in modern physics, and particularly theoretical physics. The concept of a "world line" is distinguished from con ...

Real-time dynamics (closed quantum systems)

* Time-dependent variational Monte Carlo: An extension of the variational Monte Carlo to study the dynamics of

pure quantum state In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in t ...

Notes

References

* * * * * *

External links

QMC in Cambridge and around the world
Large amount of general information about QMC with links.
Quantum Monte Carlo simulator (Qwalk)
{{Authority control Quantum chemistry Electronic structure methods