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
Ψ
(
X
,
t
)
=
exp
(
∑
k
a
k
(
t
)
O
k
(
X
)
)
where the complex-valued
a
k
(
t
)
are time-dependent variational parameters,
X
denotes a many-body configuration and
O
k
(
X
)
are time-independent operators that define the specific ansatz. The time evolution of the parameters
a
k
(
t
)
can be found upon imposing a variational principle to the wave function. In particular one can show that the optimal parameters for the evolution satisfy at each time the equation of motion
i
∑
k
′
⟨
O
k
O
k
′
⟩
t
c
a
˙
k
′
=
⟨
O
k
H
⟩
t
c
,
where
H
is the Hamiltonian of the system,
⟨
A
B
⟩
t
c
=
⟨
A
B
⟩
t
−
⟨
A
⟩
t
⟨
B
⟩
t
are connected averages, and the quantum expectation values are taken over the time-dependent variational wave function, i.e.,
⟨
⋯
⟩
t
≡
⟨
Ψ
(
t
)
|
⋯
|
Ψ
(
t
)
⟩
.
In analogy with the Variational Monte Carlo approach and following the Monte Carlo method for evaluating integrals, we can interpret
|
Ψ
(
X
,
t
)
|
2
∫
|
Ψ
(
X
,
t
)
|
2
d
X
as a probability distribution function over the multi-dimensional space spanned by the many-body configurations
X
. The Metropolis–Hastings algorithm is then used to sample exactly from this probability distribution and, at each time
t
, the quantities entering the equation of motion are evaluated as statistical averages over the sampled configurations. The trajectories
a
(
t
)
of the variational parameters are then found upon numerical integration of the associated differential equation.
