Quantum computing and the brain: quantum nets, dessins d’enfants and neural networks

Abstract

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〉 (ground state) and |1〉 (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. If something changed in this area, we need a transformation which will preserve this general form of a state (mathematically, this transformation must be an element of the group S L(2; C)). The same argumentation must be true for the feedback loops, i.e. a general transformation of states along the feedback loops is an assignment of this loop to an element of the transformation group. 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 S U(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.