  In this paper, we will discuss a formal link between neural networks and
quantum computing. For that purpose we will present a simple model for the
description of the neural network by forming sub-graphs of the whole network
with the same or a similar state. We will describe the interaction between
these areas by closed loops, the feedback loops. The change of the graph is
given by the deformations of the loops. This fact can be mathematically
formalized by the fundamental group of the graph. Furthermore the neuron has
two basic states $|0\rangle$ (ground state) and $|1\rangle$ (excited state).
The whole state of an area of neurons is the linear combination of the two
basic state with complex coefficients representing the signals (with 3
Parameters: amplitude, frequency and phase) along the neurons. Then it can be
shown that the set of all signals forms a manifold (character variety) and all
properties of the network must be encoded in this manifold. In the paper, we
will discuss how to interpret learning and intuition in this model. Using the
Morgan-Shalen compactification, the limit for signals with large amplitude can
be analyzed by using quasi-Fuchsian groups as represented by dessins d'enfants
(graphs to analyze Riemannian surfaces). As shown by Planat and collaborators,
these dessins d'enfants are a direct bridge to (topological) quantum computing
with permutation groups. The normalization of the signal reduces to the group
$SU(2)$ and the whole model to a quantum network. Then we have a direct
connection to quantum circuits. This network can be transformed into operations
on tensor networks. Formally we will obtain a link between machine learning and
Quantum computing.
