  It was recently proposed to leverage the representational power of artificial
neural networks, in particular Restricted Boltzmann Machines, in order to model
complex quantum states of many-body systems [Science, 355(6325), 2017]. States
represented in this way, called Neural Network States (NNSs), were shown to
display interesting properties like the ability to efficiently capture
long-range quantum correlations. However, identifying an optimal neural network
representation of a given state might be challenging, and so far this problem
has been addressed with stochastic optimization techniques. In this work we
explore a different direction. We study how the action of elementary quantum
operations modifies NNSs. We parametrize a family of many body quantum
operations that can be directly applied to states represented by Unrestricted
Boltzmann Machines, by just adding hidden nodes and updating the network
parameters. We show that this parametrization contains a set of universal
quantum gates, from which it follows that the state prepared by any quantum
circuit can be expressed as a Neural Network State with a number of hidden
nodes that grows linearly with the number of elementary operations in the
circuit. This is a powerful representation theorem (which was recently obtained
with different methods) but that is not directly useful, since there is no
general and efficient way to extract information from this unrestricted
description of quantum states. To circumvent this problem, we propose a
step-wise procedure based on the projection of Unrestricted quantum states to
Restricted quantum states. In turn, two approximate methods to perform this
projection are discussed. In this way, we show that it is in principle possible
to approximately optimize or evolve Neural Network States without relying on
stochastic methods such as Variational Monte Carlo, which are computationally
expensive.
