Well, the original artificial neuron by Pitts and McCulloch was a propositional logic circuit- but that strain of research, too, has fallen by the wayside in favour of neural nets that can be trained with backpropagation of error.

One can train binary NN via backprop by using qbits. Samples are taken, and backprop can be done as if the result was X% 1 and (1-X)% 0. It can be noisy and slow but it is a way to do it.