Pattern Formation in Quantum Hierarchical Cellular Neural Networks

TL;DR

This paper introduces a novel class of quantum neural networks based on p-adic Schrödinger equations, linking hierarchical neural networks with quantum models through mathematical and numerical analysis.

Contribution

It develops a new quantum neural network framework derived from p-adic Schrödinger equations and provides discretization methods and numerical simulations.

Findings

Existence of local solutions for p-adic Schrödinger equations.

Discretization of p-adic equations enables construction of QNNs on simple graphs.

Numerical simulations illustrate the functioning of the new QNNs.

Abstract

We present a new class of quantum neural networks (QNNs) whose states are solutions of $p$-adic Schr\"{o}dinger equations with a non-local potential that controls the interaction between the neurons. These equations are obtained as Wick rotations of the state equations of $p$-adic cellular neural networks (CNNs). The CNNs are continuous limits of discrete hierarchical neural networks (NNs). The CNNs are bio-inspired in the Wilson-Cowan model, which describes the macroscopic dynamics of large populations of neurons. We provide a detailed study of the discretization of the new $p$-adic Schr\"{o}dinger equations, which allows the construction of new QNNs on simple graphs. We also conduct detailed numerical simulations, offering a clear insight into the functioning of the new QNNs. At a mathematical level, we show the existence of local solutions for the new $p$ -adic Schr\"{o}dinger equations.