50 research outputs found
Quantum computing and the brain: quantum nets, dessins d'enfants and neural networks
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 (ground state) and (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
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.Comment: 17 pages, 3 Figures, accepted for the proceedings of the QTech 2018
conference (September 2018, Paris
Differential Structures - the Geometrization of Quantum Mechanics
The usual quantization of a classical space-time field does not touch the
non-geometrical character of quantum mechanics. We believe that the deep
problems of unification of general relativity and quantum mechanics are rooted
in this poor understanding of the geometrical character of quantum mechanics.
In Einstein's theory gravitation is expressed by geometry of space-time, and
the solutions of the field equation are invariant w.r.t. a certain equivalence
class of reference frames. This class can be characterized by the differential
structure of space-time. We will show that matter is the transition between
reference frames that belong to different differential structures, that the set
of transitions of the differential structure is given by a Temperley-Lieb
algebra which is extensible to a -algebra comprising the field operator
algebra of quantum mechanics and that the state space of quantum mechanics is
the linear space of the differential structures. Furthermore we are able to
explain the appearance of the complex numbers in quantum theory. The strong
relation to Loop Quantum Gravity is discussed in conclusion.Comment: ReVTeX4, 13 pages, 2 figures,major corrections in the definition of
the singular form, the trace and the complex structur
Dark energy and 3-manifold topology
We show that the differential-geometric description of matter by differential
structures of spacetime leads to a unifying model of the three types of energy
in the cosmos: matter, dark matter and dark energy. Using this model we are
able to calculate the ratio of dark energy to the total energy of the cosmos.Comment: 7 pages, no figure