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This paper introduces a novel class of quantum neural networks grounded in $p$-adic Schrödinger equations derived from hierarchical cellular models. By establishing the existence of local solutions and providing numerical simulations, the work advances theoretical frameworks for bio-inspired QNN architectures on simple graphs. The research represents significant progress at the intersection of mathematical physics and emerging quantum machine learning paradigms.
Pattern Formation in Quantum Hierarchical Cellular Neural Networks
arXiv:2603.27063v3 Announce Type: replace 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 by 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.
Abstract:We present a new class of quantum neural networks (QNNs) whose states are solutions of $p$-adic Schrödinger 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 by 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 Schrödinger 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 Schrödinger equations.
The paper was fully revised. Some new sections were added
arXiv:2603.27063 [quant-ph]
(or arXiv:2603.27063v3 [quant-ph] for this version)
From: W. A. Zuniga-Galindo [view email] [v1] Sat, 28 Mar 2026 00:39:53 UTC (566 KB) [v2] Thu, 9 Jul 2026 00:46:55 UTC (529 KB) [v3] Mon, 17 Aug 2026 14:37:26 UTC (545 KB)
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