We discuss $p$-adic quantum mechanics, where physics takes place in a three-dimensional $p$-adic (rather than Euclidean) space. In particular, we study the $p$-adic rotation group $SO(3)$, 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, where the $p$-adic qubit arises as a two-dimensional representation. We describe the foundations of this program, starting from the main features of $SO(3)$ (in parallel to its real counterpart), such as a $p$-adic analogue of the Cardano (aka nautical) angles decomposition. We show that all finite-dimensional projective unitary representations of $SO(3)$ factorise on some $SO(3)$ modulo pⁿ, through which we find explicit $p$-adic qubit representations for every prime $p$. We further address the Clebsch-Gordan problem and identify entangled states for composite systems of two $p$-adic qubits. In the realm of quantum computing, we finally 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 $SO(3)$. As in the standard approach, our emphasis lies primarily on p-adic quantum logic gates operating on two qubits (from the known four-dimensional unitary representations of $SO(3)$), 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.