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

Abstract The brain can be seen as a dynamical graph with electrical signals having amplitude, frequency and phase. Because of the complexity of the brain, it is hopeless to include the whole graph. Instead we form areas of... [ view full abstract ]

Abstract The brain can be seen as a dynamical graph with electrical signals having amplitude, frequency and phase. Because of the complexity of the brain, it is hopeless to include the whole graph. Instead we form areas of neurons having nearly the same state (ground state or excited state). Inspired by the Ising model, we describe the interaction between these areas by closed loops, the feedback loops. The change of the graph is given by deformations of the loops. At first view, the neuron area interaction as represented by loops cannot be neglected. It is equivalent to say that the loops cannot be contracted by deformations. 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 SL(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 talk, we will discuss how to interpret learning and intuition in this model. Using the Morgan-Shalen compactification, the limit for large graphs can be analyzed by using quasi-Fuchsian groups as represented by dessins d'enfants (graphs to analyze Riemannian surfaces). As shown by M. Planat and collaborators, these dessins d'enfants are a direct bridge to (topological) Quantum computing with permutation groups. The normalization of the signal reduces the group to 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 products of states. By using the work “quantum circuit by one step method and neural network” by Germano and Nagata, one can directly show the relation to quantum computing. Formally, we will obtain a link between machine learning and Quantum computing.