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It was recently proposed to leverage the representational power of artificialneural networks, in particular Restricted Boltzmann Machines, in order to modelcomplex quantum states of many-body systems [Science, 355(6325), 2017]. Statesrepresented in this way, called Neural Network States (NNSs), were shown todisplay interesting properties like the ability to efficiently capturelong-range quantum correlations. However, identifying an optimal neural networkrepresentation of a given state might be challenging, and so far this problemhas been addressed with stochastic optimization techniques. In this work weexplore a different direction. We study how the action of elementary quantumoperations modifies NNSs. We parametrize a family of many body quantumoperations that can be directly applied to states represented by UnrestrictedBoltzmann Machines, by just adding hidden nodes and updating the networkparameters. We show that this parametrization contains a set of universalquantum gates, from which it follows that the state prepared by any quantumcircuit can be expressed as a Neural Network State with a number of hiddennodes that grows linearly with the number of elementary operations in thecircuit. This is a powerful representation theorem (which was recently obtainedwith different methods) but that is not directly useful, since there is nogeneral and efficient way to extract information from this unrestricteddescription of quantum states. To circumvent this problem, we propose astep-wise procedure based on the projection of Unrestricted quantum states toRestricted quantum states. In turn, two approximate methods to perform thisprojection are discussed. In this way, we show that it is in principle possibleto approximately optimize or evolve Neural Network States without relying onstochastic methods such as Variational Monte Carlo, which are computationallyexpensive.

Neural Network Operations and Susuki-Trotter evolution of Neural Network States