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 comprising degrees of freedom and a choice of associated quantum numbers , then an NQS parameterizes the wave-function amplitudes

where is an artificial neural network of parameters (weights) , input variables () 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 FunctionEdit

One common application of NQS is to find an approximate representation of the ground state wave function of a given Hamiltonian  . 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  , stochastic techniques based, for example, on the Monte Carlo method are used to estimate  , analogously to what is done in Variational Monte Carlo, see for example [2] for a review. More specifically, a set of   samples  , with  , is generated such that they are uniformly distributed according to the Born probability density  . Then it can be shown that the sample mean of the so-called "local energy"   is a statistical estimate of the quantum expectation value  , i.e.


Similarly, it can be shown that the gradient of the energy with respect to the network weights   is also approximated by a sample mean


where   and can be efficiently computed, in deep networks through backpropagation.

The stochastic approximation of the gradients is then used to minimize the energy   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   is generated, in an iterative procedure similar to what done in unsupervised learning.

Connection with Tensor NetworksEdit

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  .

See alsoEdit


  1. ^ Carleo, Giuseppe; Troyer, Matthias (2017). "Solving the quantum many-body problem with artificial neural networks". Science. 355 (6325): 602–606. arXiv:1606.02318. Bibcode:2017Sci...355..602C. doi:10.1126/science.aag2302. PMID 28183973. S2CID 206651104.
  2. ^ Becca, Federico; Sorella, Sandro (2017). Quantum Monte Carlo Approaches for Correlated Systems. Cambridge University Press. doi:10.1017/9781316417041. ISBN 9781316417041.
  3. ^ Chen, Jing; Cheng, Song; Xie, Haidong; Wang, Lei; Xiang, Tao (2018). "Equivalence of restricted Boltzmann machines and tensor network states". Phys. Rev. B. 97 (8): 085104. arXiv:1701.04831. Bibcode:2018PhRvB..97h5104C. doi:10.1103/PhysRevB.97.085104. S2CID 73659611.