One common application of NQS is to find an approximate representation of the ground state wave function of a given Hamiltonian ${\displaystyle {\hat {H}}}$ . The learning procedure in this case consists in finding the best neural-network weights that minimize the variational energy

Since, for a general artificial neural network, computing the expectation value is an exponentially costly operation in ${\displaystyle N}$ , stochastic techniques based, for example, on the Monte Carlo method are used to estimate ${\displaystyle E(W)}$ , analogously to what is done in Variational Monte Carlo, see for example ^[2] for a review. More specifically, a set of ${\displaystyle M}$  samples ${\displaystyle S^{(1)},S^{(2)}\ldots S^{(M)}}$ , with ${\displaystyle S^{(i)}=s_{1}^{(i)}\ldots s_{N}^{(i)}}$ , is generated such that they are uniformly distributed according to the Born probability density ${\displaystyle P(S)\propto |F(s_{1}\ldots s_{N};W)|^{2}}$ . Then it can be shown that the sample mean of the so-called "local energy" ${\displaystyle E_{\mathrm {loc} }(S)=\langle S|{\hat {H}}|\Psi \rangle /\langle S|\Psi \rangle }$  is a statistical estimate of the quantum expectation value ${\displaystyle E(W)}$ , i.e.

Similarly, it can be shown that the gradient of the energy with respect to the network weights ${\displaystyle W}$  is also approximated by a sample mean

where ${\displaystyle O(S^{(i)})={\frac {\partial \log F(S^{(i)};W)}{\partial W_{k}}}}$  and can be efficiently computed, in deep networks through backpropagation.

The stochastic approximation of the gradients is then used to minimize the energy ${\displaystyle E(W)}$  typically using a stochastic gradient descent approach. When the neural-network parameters are updated at each step of the learning procedure, a new set of samples ${\displaystyle S^{(i)}}$  is generated, in an iterative procedure similar to what done in unsupervised learning.

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.^[3] 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 ${\displaystyle N}$ .

• Differentiable programming