  Even the most sophisticated artificial neural networks are built by
aggregating substantially identical units called neurons. A neuron receives
multiple signals, internally combines them, and applies a non-linear function
to the resulting weighted sum. Several attempts to generalize neurons to the
quantum regime have been proposed, but all proposals collided with the
difficulty of implementing non-linear activation functions, which is essential
for classical neurons, due to the linear nature of quantum mechanics. Here we
propose a solution to this roadblock in the form of a small quantum circuit
that naturally simulates neurons with threshold activation. Our quantum circuit
defines a building block, the "quantum neuron", that can reproduce a variety of
classical neural network constructions while maintaining the ability to process
superpositions of inputs and preserve quantum coherence and entanglement. In
the construction of feedforward networks of quantum neurons, we provide
numerical evidence that the network not only can learn a function when trained
with superposition of inputs and the corresponding output, but that this
training suffices to learn the function on all individual inputs separately.
When arranged to mimic Hopfield networks, quantum neural networks exhibit
properties of associative memory. Patterns are encoded using the simple Hebbian
rule for the weights and we demonstrate attractor dynamics from corrupted
inputs. Finally, the fact that our quantum model closely captures (traditional)
neural network dynamics implies that the vast body of literature and results on
neural networks becomes directly relevant in the context of quantum machine
learning.
