  We demonstrate that it is possible to implement a quantum perceptron with a
sigmoid activation function as an efficient, reversible many-body unitary
operation. When inserted in a neural network, the perceptron's response is
parameterized by the potential exerted by other neurons. We prove that such a
quantum neural network is a universal approximator of continuous functions,
with at least the same power as classical neural networks. While engineering
general perceptrons is a challenging control problem --also defined in this
work--, the ubiquitous sigmoid-response neuron can be implemented as a
quasi-adiabatic passage with an Ising model. In this construct, the scaling of
resources is favorable with respect to the total network size and is dominated
by the number of layers. We expect that our sigmoid perceptron will have
applications also in quantum sensing or variational estimation of many-body
Hamiltonians.
