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arXiv:2603.27063

Pattern Formation in Quantum Hierarchical Cellular Neural Networks

W. A. Zúñiga-Galindo, B. A. Zambrano-Luna, Chayapuntika Indoung

quant-phSubmitted Mar 28, 2026• Announced Mar 31, 2026
v1 · Mar 28, 2026v2 · Jul 9, 2026v3 · Aug 17, 2026
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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 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 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.

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We present a new class of quantum neural networks (QNNs) whose states are solutions of ppp-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 ppp-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 ppp-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 ppp -adic Schrödinger equations.
Abstract Classical artificial neural networks, built from elementary units, possess enormous expressive power. Here we investigate a quantum neural network (QNN) architecture, which follows a similar paradigm. It is structurally equivalent to so-called (1+1)D quantum cellular automata, which are two-dimensional quantum lattice systems on which dynamics takes place in discrete time. Information transfer between consecutive time slices—or adjacent network layers—is governed by local quantum gates, which can be regarded as the quantum counterpart of the classical elementary units. Along the time-direction an effective dissipative evolution emerges on the level of the reduced state, and the nature of this dynamics is dictated by the structure of the elementary gates. We show how to construct the local unitary gates to yield a desired many-body dynamics, which in certain parameter regimes is governed by a Lindblad master equation. We study this for small system sizes through numerical simulations and demonstrate how collective effects within the quantum cellular automaton can be controlled parametrically. Our study constitutes a step towards the utilization of large-scale emergent phenomena in large QNNs for machine learning purposes.
Central to the field of quantum machine learning is the design of quantum perceptrons and neural network architectures. A key question in this regard is the impact of quantum effects on the way such models process information. Here, we establish a connection between (1+1)D quantum cellular automata, which implement a discrete nonequilibrium quantum many-body dynamics through successive applications of local quantum gates, and quantum neural networks (QNNs), which process information by feeding it through perceptrons interconnecting adjacent layers. Exploiting this link, we construct a class of QNNs that are highly structured-aiding both interpretability and helping to avoid trainability issues in machine learning tasks-yet can be connected rigorously to continuous-time Lindblad dynamics. We further analyze the universal properties of an example case, identifying a change of critical behavior when quantum effects are varied, showing their potential impact on the collective dynamical behavior underlying information processing in large-scale QNNs.
This paper introduces the Quantum Mechanics Simulation of Neurons Model (QMNN), a novel approach to simulating neuronal behavior based on fundamental principles of quantum mechanics. Traditional artificial neural networks (ANNs) often struggle to accurately replicate the complex dynamics observed in biological neurons due to their reliance on classical computational methods. QMNN leverages concepts such as superposition and entanglement, key features of quantum mechanics, to represent neuronal states and interactions. The model employs a discrete quantum system, where neuronal activity is represented by quantum states, allowing for inherent parallelism and potentially capturing phenomena currently inaccessible to classical neural simulations. This work details the core concepts of QMNN, outlining its mathematical formulation, key components, and potential advantages over conventional ANN approaches. We present a simplified model of a single neuron utilizing a quantum bit (qubit) to represent its state. The model's equations reflect the Schrödinger equation and the evolution of quantum states under influence of external stimuli. The simulation explores scenarios demonstrating the potential for increased computational efficiency and the ability to model phenomena like quantum coherence within neuronal networks. Further research will focus on scaling the model to multi-neuron networks and investigating the model's applicability to various neurological conditions.
We introduce a new class of p-adic Dirac equations, formulated in the standard axiomatic framework of quantum mechanics, in which the ordinary momentum operators are replaced by bounded non-local operators built from arbitrary integrable radial kernels. We diagonalize the free Dirac Hamiltonian in momentum space, construct its plane-wave solutions, and determine its spectrum, which is pure point, with eigenvalues of infinite multiplicity; in the massless case we describe the zero-energy subspace explicitly. We identify the charge-conjugation symmetry of the model: it is given by an antiunitary operator whose square is minus the identity, it exchanges the positive- and negative-energy subspaces, and in an external electromagnetic field it reverses the sign of both the charge and the vector potential. We then discretize the free equation in two ways, both giving genuine continuous-time quantum walks rather than discrete-time, coined walks: one on a countable graph and one on a finite, tree-like graph. In both cases the discretization is exact: the walk is the restriction of the p-adic Dirac evolution to an invariant subspace, and its Hamiltonian is a Dirac-type combination of weighted graph Laplacians with ultrametric weights. We prove that the Born-rule transition probabilities, summed over the internal degrees of freedom and averaged over the initial internal state, form a doubly stochastic matrix at every instant, and that, as for any non-trivial continuous-time quantum walk, this family of matrices is not a Markov semigroup. We discuss how the construction can serve as the foundation of a quantum network with four-component spinor degrees of freedom, complementing earlier, non-relativistic p-adic quantum neural networks, and prove global well-posedness of the associated nonlinear network equation. We close with a discussion of open problems.
Biological neuronal networks represent a fundamental component of complex systems, including the brain. Simulating these networks accurately and efficiently is a significant challenge, largely due to the intricate, non-linear interactions between neurons. Traditional simulation methods, often relying on numerical integration techniques like the Hodgkin-Huxley model, are computationally intensive, particularly when scaling to large networks. This paper proposes a novel approach to neuronal network simulation inspired by quantum mechanics, specifically leveraging concepts of superposition and entanglement. We introduce a framework that utilizes quantum-inspired algorithms to represent and evolve neuronal states, aiming to dramatically reduce computational overhead. The core idea involves mapping neuronal dynamics onto a quantum system, where neurons are represented by qubits, and synaptic connections are modeled as quantum gates. The simulation process then utilizes quantum algorithms to evolve the system, effectively capturing the parallel and interconnected nature of neuronal networks. We present a mathematical formulation of the simulation process, focusing on the evolution equation for neuronal states. We explore key concepts like wave function collapse and measurement in the context of neuronal network simulation, offering a potential pathway for more realistic and efficient modeling of biological neural systems. The ultimate goal is to provide a computationally tractable method for simulating large-scale neuronal networks, leading to improved understanding of brain function and potentially facilitating the development of advanced neurotechnologies. The presented framework offers a distinct alternative to conventional methods, promising significant advantages in terms of speed and scalability.