  Classical artificial neural networks, built from perceptrons as their
elementary units, possess enormous expressive power. Here we investigate a
quantum neural network architecture, which follows a similar paradigm. It is
structurally equivalent to so-called (1+1)D quantum cellular automata, which
are two-dimensional quantum lattice systems on which dynamics takes place in
discrete time. Information transfer between consecutive time slices -- or
adjacent network layers -- is governed by local quantum gates, which can be
regarded as the quantum counterpart of the classical perceptrons. Along the
time-direction an effective dissipative evolution emerges on the level of the
reduced state, and the nature of this dynamics is dictated by the structure of
the elementary gates. We show how to construct the local unitary gates to yield
a desired many-body dynamics, which in certain parameter regimes is governed by
a Lindblad master equation. We study this for small system sizes through
numerical simulations and demonstrate how collective effects within the quantum
cellular automaton can be controlled parametrically. Our study constitutes a
step towards the utilisation of large-scale emergent phenomena in large quantum
neural networks for machine learning purposes.
