Representations of the -adic rotation group, towards -adic quantum computing.

We investigate the structure of the p-adic rotation group , and initiate a program aimed at classifying its finite-dimensional irreducible projective unitary representations. Our approach relies on the profinite nature of , together with its Haar measure. These representations can be interpreted as a theory of -adic angular momentum and spin, where the p-adic qubit arises as a two-dimensional representation. Indeed, -adic numbers find fruitful applications in -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 (in parallel to its real counterpart), such as a -adic analogue of the Cardano (aka nautical) angles decomposition. We characterise the profinite group as an inverse limit of the inverse family of groups modulo , and we exploit the inverse-limit machinery to express the Haar measure on and to induce its representations. In fact, as a key result, we show that all finite-dimensional projective unitary representations of factorise on some modulo , and we find explicit -adic qubit representations for every prime .

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 -adically controlled quantum logic gates on the single-, two- and -qubit levels, using elements from the same -dimensional unitary representations of . 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)