The time-dependent variational Monte Carlo (t-VMC) method is a
quantum Monte Carlo
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 ...
approach to study the dynamics of closed, non-relativistic
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 in the context 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 ...
. It is an extension of the
variational Monte Carlo 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.
The basic building block is a generic wave function , \Psi(a) \rangle ...
method, in which a time-dependent
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 ...
is encoded by some variational
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 ...
, generally parametrized as
:
where the complex-valued
are time-dependent variational parameters,
denotes a many-body configuration and
are time-independent operators that define the specific
ansatz. The time evolution of the parameters
can be found upon imposing a
variational principle
In science and especially in mathematical studies, a variational principle is one that enables a problem to be solved using calculus of variations, which concerns finding functions that optimize the values of quantities that depend on those funct ...
to 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 ...
. In particular one can show that the optimal parameters for the evolution satisfy at each time the equation of motion
:
where
is the
Hamiltonian of the system,
are connected averages, and the quantum expectation values are taken over the time-dependent variational
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 ...
, i.e.,
.
In analogy with the
Variational Monte Carlo 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.
The basic building block is a generic wave function , \Psi(a) \rangle ...
approach and following 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 ...
for evaluating integrals, we can interpret
as a
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomeno ...
function over the multi-dimensional space spanned by the many-body configurations
. The
Metropolis–Hastings algorithm
In statistics and statistical physics, the Metropolis–Hastings algorithm is a Markov chain Monte Carlo (MCMC) method for obtaining a sequence of random samples from a probability distribution from which direct sampling is difficult. This seq ...
is then used to sample exactly from this probability distribution and, at each time
, the quantities entering the equation of motion are evaluated as statistical averages over the sampled configurations. The trajectories
of the variational parameters are then found upon numerical integration of the associated
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, a ...
.
References
*
*
* {{cite thesis
, author = G. Carleo
, title = Spectral and dynamical properties of strongly correlated systems
, url = http://www.sissa.it/cm/phdsection/past_phd_thesis/2011/Carleo.pdf
, type= PhD Thesis
, pages = 107–128
, year = 2011
Quantum mechanics
Quantum Monte Carlo