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Black Holes as Brains: Neural Networks with Area Law Entropy

Dvali, G.

Published Jan 1, 2018• 27 citations
OpenAlex

Abstract

Motivated by the potential similarities between the underlying mechanisms of the enhanced memory storage capacity in black holes and in brain networks, we construct an artificial quantum neural network based on gravity-like synaptic connections and a symmetry structure that allows to describe the network in terms of geometry of a d-dimensional space. We show that the network possesses a critical state in which the gapless neurons emerge that appear to inhabit a (d-1)-dimensional surface, with their number given by the surface area. In the excitations of these neurons, the network can store and retrieve an exponentially large number of patterns within an arbitrarily narrow energy gap. The corresponding micro-state entropy of the brain network exhibits an area law. The neural network can be described in terms of a quantum field, via identifying the different neurons with the different momentum modes of the field, while identifying the synaptic connections among the neurons with the interactions among the corresponding momentum modes. Such a mapping allows to attribute a well-defined sense of geometry to an intrinsically non-local system, such as the neural network, and vice versa, it allows to represent the quantum field model as a neural network.

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arXiv:1812.08338•EPJ Web of Conferences•Jan 1, 2019•3 citations
Motivated by the potential similarities between the underlying mechanisms of the enhanced memory storage capacity in black holes and in brain networks, we construct an artificial quantum neural network based on gravity-like synaptic connections and a symmetry structure that allows to describe the network in terms of geometry of a d-dimensional space. We show that the network possesses a critical state in which the gapless neurons emerge that appear to inhabit a (d-1)-dimensional surface, with their number given by the surface area. In the excitations of these neurons, the network can store and retrieve an exponentially large number of patterns within an arbitrarily narrow energy gap. The corresponding micro-state entropy of the brain network exhibits an area law. The neural network can be described in terms of a quantum field, via identifying the different neurons with the different momentum modes of the field, while identifying the synaptic connections among the neurons with the interactions among the corresponding momentum modes. Such a mapping allows to attribute a well-defined sense of geometry to an intrinsically non-local system, such as the neural network, and vice versa, it allows to represent the quantum field model as a neural network.
Abstract Motivated by the potential similarities between the underlying mechanisms of the enhanced memory storage capacity in black holes and in brain networks, we construct an artificial quantum neural network based on gravity‐like synaptic connections and a symmetry structure that allows to describe the network in terms of geometry of a d‐dimensional space. We show that the network possesses a critical state in which the gapless neurons emerge that appear to inhabit a ‐dimensional surface, with their number given by the surface area. In the excitations of these neurons, the network can store and retrieve an exponentially large number of patterns within an arbitrarily narrow energy gap. The corresponding micro‐state entropy of the brain network exhibits an area law. The neural network can be described in terms of a quantum field, via identifying the different neurons with the different momentum modes of the field, while identifying the synaptic connections among the neurons with the interactions among the corresponding momentum modes. Such a mapping allows to attribute a well‐defined sense of geometry to an intrinsically non‐local system, such as the neural network, and vice versa, it allows to represent the quantum field model as a neural network.
We implement a mechanism - originally proposed as a model for the large memory storage capacity of black holes - in quantum neural networks and show that an exponentially increased capacity of pattern storage and recognition is achieved in certain critically excited states, without involvement of synaptic plasticity. We consider a simple network of N interconnected quantum neurons with weak excitatory synaptic connections. We show that for frozen synaptic weights there exist the critical states of enhanced memory storage capacity. These states are achieved thanks to the high excitation levels of some of the neurons, which - despite of feeble synaptic connections - dramatically lower the response threshold of the remaining weaker-excited neurons. As a results, the latter neurons acquire a capacity to store an exponentially large number of patterns within a narrow energy gap. The stored patterns can be recognized and retrieved with perfect response under the influence of arbitrarily soft input stimuli. In sharp contrast, under the same stimuli the recall is absent in the ground-state of the system. The lesson is that the state with the highest micro-state entropy and memory storage capacity is not necessarily a local minimum of energy, but rather an excited critical state. The considered phenomenon has a smooth classical limit and can serve for achieving an enhanced memory storage capacity in classical brain networks.
It is often assumed that the area law of microstate entropy and the holography are intrinsic properties exclusively of the gravitational systems, such as black holes. We construct a nongravitational model that exhibits an entropy that scales as area of a sphere of one dimension less. It is represented by a nonrelativistic bosonic field living on a ddd-dimensional sphere of radius RRR and experiencing an angular-momentum-dependent attractive interaction. We show that the system possesses a quantum critical point with the emergent gapless modes. Their number is equal to the area of a $d\ensuremath{-}1$-dimensional sphere of the same radius RRR. These gapless modes create an exponentially large number of degenerate microstates with the corresponding microstate entropy given by the area of the same $d\ensuremath{-}1$-dimensional sphere. Thanks to a double-scaling limit, the counting of the entropy and of the number of the gapless modes is made exact. The phenomenon takes place for arbitrary number of dimensions and can be viewed as a version of holography.
An explicit microscopic realization of the phenomenon of holography is provided by a class of simple quantum theories of a bosonic field inhabiting a d-dimensional space and experiencing a momentum dependent attractive interaction. An exact mode counting reveals a family of holographic states. In each a set of gapless modes emerges with their number equal to the area of a (d-1)-dimensional sphere. These modes store an exponentially large number of patterns within a microscopic energy gap. The resulting micro-state entropy obeys the area-law reminiscent of a black hole entropy. We study the time-evolution of the stored patterns and observe the following phenomenon: Among the degenerate micro-states the ones with heavier loaded memories survive longer than those that store emptier patterns. Thus, a state gets stabilized by the burden of its own memory. From time to time the information pattern gets off-loaded from one holographic state into another but cannot escape the system. During this process the pattern becomes highly entangled and scrambled. We suggest that this phenomenon is universal in systems with enhanced memory storage capacity, such as black holes or critical neural networks. This universality sheds an interesting light on the puzzle of why, despite the evaporation, is a black hole forced to maintain information internally for a very long time.
To address Quantum Artificial Neural Networks as quantum dynamical computing systems, a formalization of quantum artificial neural networks as dynamical systems is developed, expanding the concept of unitary map to the neural computation setting and introducing a quantum computing field theory on the network. The formalism is illustrated in a simulation of a quantum recurrent neural network and the resulting field dynamics is researched upon, showing emergent neural waves with excitation and relaxation cycles at the level of the quantum neural activity field, as well as edge of chaos signatures, with the local neurons operating as far-from-equilibrium open quantum systems, exhibiting entropy fluctuations with complex dynamics including complex quasiperiodic patterns and power law signatures. The implications for quantum computer science, quantum complexity research, quantum technologies and neuroscience are also addressed.
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 groupS 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 groupS 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.