Neural network quantum states

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^[1] to approximate wave functions of many-body quantum systems.

Given a many-body quantum state ${\displaystyle |\Psi \rangle }$ comprising ${\displaystyle N}$ degrees of freedom and a choice of associated quantum numbers ${\displaystyle s_{1}\ldots s_{N}}$, then an NQS parameterizes the wave-function amplitudes

$${\displaystyle \langle s_{1}\ldots s_{N}|\Psi ;W\rangle =F(s_{1}\ldots s_{N};W),}$$

where ${\displaystyle F(s_{1}\ldots s_{N};W)}$ is an artificial neural network of parameters (weights) ${\displaystyle W}$, ${\displaystyle N}$ input variables (${\displaystyle s_{1}\ldots s_{N}}$) and one complex-valued output corresponding to the wave-function amplitude.

This variational form is used in conjunction with specific stochastic learning approaches to approximate quantum states of interest.