NAA—QUANTUM SYSTEMS & SCIENCE, ENGINEERING

Date & Time: Wednesday, September 30th, 2026 | 11:00 - 13:30
Place: tba
Chairs: Wilson Zuniga-Galindo, Mariia Serdiuk

Abstract:
The mini symposium aims at bringing together researchers in non-archimedean analysis and applications, in order to present the state of the art of new developments in non-archimedean analysis and applications in the Sciences, Engineering and Economics. Having initially diffusion processes over the p-adic number field, now non-linear dynamics (e.g. an analogue of Navier-Stokes or the porous medium equation), as well as other p-adic domains like p-adic analytic manifolds are now included. First applied and further developped in mathematical physics, the methods are now part of computer science, biology, earth sciences and other domains. Bringing together researchers in quantum systems and computing aims at fostering new insights and ideas towards a deeper understanding of general hierarchical structures in this domain overlapping with mathematics. 

Confirmed Speakers:

Speaker: Ilaria Svampa: (University of Cologne)
Title: p-Adic qubits and universal sets of p-adically controlled gates
Abstract: We discuss a p-adic formulation of quantum mechanics and quantum information processing, where the three-dimensional configuration space is p-adic (rather than Euclidean). In particular, we study the p-adic rotation group SO(3)p, and we outline a program aimed at classifying its irreducible projective unitary representations, by exploiting its profinite structure and its Haar measure. These representations can be interpreted as a theory of p-adic angular momentum and spin; specifically, the p-adic qubit arises as a two-dimensional representation. We show that all finite dimensional projective unitary representations of SO(3)p factorise on some SO(3)p modulo \(p^n\), through which we find explicit p-adic qubit representations for every prime p. Interestingly, there are several inequivalent p-adic qubits for p > 3. Then, it is natural to compose systems of multiple p-adic qubits, through the tensor product of their representations. We solve the Clebsch-Gordan problem for systems of two p-adic qubits from SO(3)p modulo p, revealing that the coupled bases decompose into singlet and doublet states. We further study entanglement arising from those stable subsystems: every singlet, doublet (and triplet) can be given by maximally entangled Bell states. However, except for the singlets, the projectors onto doublets (and triplets) are separable quantum states. Lastly, we propose a circuit model of quantum computation where logic gates are driven by the actions of SO(3)p. For p = 3, we construct a set of gates from four-dimensional irreducible representations of SO(3)p modulo p, that we prove to be universal for quantum computation.
The talk is based on Reference 1 and Reference 2

Speaker: Wilson Zuniga-Galindo: University of Texas Rio Grande Valley
Title: p-Adic Quantum Cellular Neural Networks
Abstract: We present a new class of quantum neural networks (QNNs) whose states are solutions to p-adic Schrödinger equations with a non-local potential that governs the interaction between neurons. These equations are obtained as Wick rotations of the state equations of p-adic cellular neural networks (CNNs). The p-adic CNNs arise as continuous limits of large, hierarchical discrete 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 global solutions to the new p-adic Schrödinger equations.

The talk is based on the preprint: W. A. Zúñiga-Galindo, B. A. Zambrano-Luna, Chayapuntika Indoung. Pattern Formation in Quantum Hierarchical Cellular Neural Networks.

Speaker: Andreas Winter
Title: p-Adic qubits and universal sets of p-adically controlled gates.
Abstract: TBA

Speaker: Evgeny Zelenov
Title: \(p\)-adic Gaussification of quantum states.
Abstract: Let \(H\) be a complex separable infinite-dimensional Hilbert space. The state of a quantum system is defined by the density operator \(\rho\).

The state of a quantum system can be described in terms of a characteristic function. Let an irreducible representation of canonical commutation relations in the Weyl form be given in the space \(H\). That is,  the map \(W\) from the phase space of the classical system \((V,\Delta)\) (a two-dimensional symplectic space over the field \(\mathbb Q_p\) for the single-mode case) into a set of unitary operators on \(H\) satisfying the commutation relation
\(W(z)W(z') = \exp\left(i\pi\{\Delta(z,z')\}_p\right)W(z+z')\), \,z,\(z'\in V\).

The quantum state of \(\rho\) is uniquely determined by its characteristic function \)(h_\rho(z)=\Tr\left(\rho W(z)\right)\).

We will say that the state \(\rho\boxplus\sigma\) is an independent sum of the states \(\rho\) and \(\sigma\) if the equality

\(h_{\rho\boxplus\sigma}(z) = h_\rho(z)h_\sigma(z)\)  is fulfilled.
 
 The main result is that (under certain constraints on  \(\rho\)) the sequence of states \(\rho^{\boxplus n}\) weakly converges to the \(p\)-adic Gaussian state \(\rho_0\), that is
\(\lim_{n\to\infty}Tr(\rho^{\boxplus n}B)=Tr(\rho_0B)\)

for any bounded operator \(B\).

Speaker: Brian Zambrano-Luna: (U Alberta)
Title: p-Adic Neural Models for Images and Hierarchical Data.
Abstract: Non-Archimedean metrics offer a natural mathematical language for hierarchical relations, encoding tree-like structure directly in the p-adic distance rather than imposing it externally. In a series of works, we have shown that this structure can be exploited computationally: p-adic cellular neural networks for image processing, p-adic statistical field theory linked to convolutional deep Boltzmann machines, and hierarchical Wilson–Cowan models built on p-adic PDEs, applied to real relational matrices such as the cat cortical connectivity matrix. This talk presents an overview exploring how p-adic analysis can be used to formulate neural models, process images, and describe complex systems organized through hierarchical interaction structures. These results illustrate how suitable hierarchical representations can connect real-valued observations and relational data with p-adic mathematical models. We will conclude by outlining a possible future research direction in which p-adic partial differential and neural models could be used to represent the spatial, temporal, and multiscale structure of satellite imagery for environmental monitoring.

Speaker: Angel Alfredo Moran Ledezma: (KIT)
Title: Spectral Geometry and Heat Kernels on Phylogenetic Trees
Abstract: The shape of a phylogenetic tree has been analyzed through the spectrum of a matrix derived from its geometry. In this talk, we introduce a different operator, the ultrametric Laplacian, built directly from the ultrametric structure of the tree, which generates a stochastic process on its leaves. This construction yields closed-form formulas for its eigenvalues and eigenvectors, in contrast to existing approaches that require numerically diagonalizing an arbitrary matrix, an operation that scales cubically with the number of taxa. Using this explicit spectral structure, we show how to analyze the geometry of a phylogenetic tree through spectral and heat kernel techniques, leading to several applications: an exact reconstruction of the tree from its spectrum, a geometric interpretation of spectral gaps which we use to separate regions of the tree associated with different rates of diversification, a decomposition of trait variance across the phylogeny, and a closed-form centrality index for evolutionary distinctiveness. We close by outlining how this spectral framework opens the door to new approaches for comparing the shape of phylogenetic trees, a direction we are currently exploring.

Speaker: Angel Alfredo Moran Ledezma: (KIT)
Title: A \(p\)-adic reaction-diffusion model of branching coral growth and calcification dynamics
Abstract: Branching corals display self-similar, hierarchical architectures shaped by internal biochemical processes. We present a reaction-diffusion model formulated over the \(p \)-adic integers Zp, whose ultrametric, tree like structure offers a natural mathematical substrate for representing such ramified biological systems. The model couples reactions between calcium and bicarbonate ions, driving the precipitation of calcium carbonate, with nonlocal diffusion governed by the Vladimirov operator, a \(p \)-adic analog of the fractional Laplacian. Discretizing the space into \(p\)-adic balls yields a system of coupled ordinary differential equations, which we solve numerically to explore how environmental and kinetic parameters influence morphogenetic outcomes. We introduce a branching rule based on local calcium carbonate accumulation and a halting condition tied to saturation thresholds, together reproducing coral-like branching patterns. This approach bridges non-Archimedean analysis with morphogenesis modeling, offering a new mathematical perspective on hierarchical structure formation in developmental biology, with possible extensions to other ramified biological systems.