Neural Network Quantum States (NQS or NNQS) is a general class of variational quantum states parameterized in terms of an artificial neural network. It was first introduced in 2017 by the physicists Giuseppe Carleo and Matthias Troyer to approximate wave functions of many-body quantum systems.
|\Psi\rangle
N
s1\ldotssN
where
F(s1\ldotssN;W)
W
N
s1\ldotssN
This variational form is used in conjunction with specific stochastic learning approaches to approximate quantum states of interest.
\hat{H}
Since, for a general artificial neural network, computing the expectation value is an exponentially costly operation in
N
E(W)
M
S(1),S(2)\ldotsS(M)
S(i)
(i) | |
=s | |
1\ldots |
(i) | |
s | |
N |
P(S)\propto|F(s1\ldotssN;W)|2
Eloc(S)=\langleS|\hat{H}|\Psi\rangle/\langleS|\Psi\rangle
E(W)
Similarly, it can be shown that the gradient of the energy with respect to the network weights
W
where
O(S(i))=
\partiallogF(S(i);W) | |
\partialWk |
The stochastic approximation of the gradients is then used to minimize the energy
E(W)
S(i)
Neural-Network representations of quantum wave functions share some similarities with variational quantum states based on tensor networks. For example, connections with matrix product states have been established. These studies have shown that NQS support volume law scaling for the entropy of entanglement. In general, given a NQS with fully-connected weights, it corresponds, in the worse case, to a matrix product state of exponentially large bond dimension in
N