South African - Japanese Discrete Homotopy Meeting 2025 (Cape Town, South Africa, 10-11 September 2025)

11th of September 2025  - 09.50/10.30  

Venue: MAM111 - Department of Mathematics and Applied Mathematics - University of Cape Town

Speaker:   Ilaria Svampa (Universität zu Köln, Köln, Germany)

Title:          Representations of the p-adic rotation group, towards p-adic quantum computing

Abstract:  We investigate the structure of the p-adic rotation group SO(3)ₚ, and initiate a program aimed at classifying its finite-dimensional irreducible projective unitary representations. Our approach relies on the profinite nature of SO(3)ₚ, together with its Haar measure. These representations can be interpreted as a theory of p-adic angular momentum and spin, where the p-adic qubit arises as a two-dimensional representation. Indeed, p-adic numbers find fruitful applications in p-adic formulations of quantum mechanics, and in dynamical systems, where their ultrametric topology naturally models hierarchical and fractal-like structures. We describe the foundations of our program, starting from the main features of SO(3)ₚ (in parallel to its real counterpart), such as a p-adic analogue of the Cardano (aka nautical) angles decomposition. We characterise the profinite group SO(3)ₚ as an inverse limit of the inverse family of groups SO(3)ₚ modulo pⁿ, and we exploit the inverse-limit machinery to express the Haar measure on SO(3)ₚ and to induce its representations. In fact, as a key result, we show that all finite-dimensional projective unitary representations of SO(3)ₚ factorise on some SO(3)ₚ modulo pⁿ, and we find explicit p-adic qubit representations for every prime p. As an application, in the realm of quantum computing, we outline how these representations can be used to define algebraic operations on qubits. In particular, we propose to construct p-adically controlled quantum logic gates on the single-, two- and n-qubit levels, using elements from the same 2ⁿ-dimensional unitary representations of SO(3)ₚ. The main focus is placed on the four-dimensional representations, with the ultimate aim to provide a universal set of gates. (Based on arXiv:2104.06228, arXiv:2306.07110, arXiv:2401.14298, and arXiv:2112.03362.)