It was recently proposed to leverage the representational power of arti¯cial neural networks, in particular Restricted Boltzmann Machines, in order to model complex quantum states of many-body systems [G. Carleo and M. Troyer, Science 355(6325) (2017) 602.]. States represented in this way, called Neural Network States (NNSs), were shown to display interesting properties like the ability to e±ciently capture long-range quantum correlations. However, identifying an optimal neural network

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... ation of a given state might be challenging, and so far this problem has been addressed with st€ ochastic optimization techniques. In this work, we explore a di®erent direction. We study how the action of elementary quantum operations modi¯es 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 di®erent methods) but that is not directly useful, since there is no general and e±cient way to extract information from this unrestricted description of quantum states. To circumvent this problem, we propose a stepwise 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.