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.

Learning the Ground-State Wave Function

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

$${\displaystyle E(W)=\langle \Psi ;W|{\hat {H}}|\Psi ;W\rangle .}$$

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.

$${\displaystyle E(W)\simeq {\frac {1}{M}}\sum _{i}^{M}E_{\mathrm {loc} }(S^{(i)}).}$$

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

$${\displaystyle {\frac {\partial E(W)}{\partial W_{k}}}\simeq {\frac {1}{M}}\sum _{i}^{M}(E_{\mathrm {loc} }(S^{(i)})-E(W))O_{k}^{\star }(S^{(i)}),}$$

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.

Connection with Tensor Networks

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 ${\displaystyle N}$.

See also

• Differentiable programming

References