Representations of the p-adic three-dimensional rotation group: towards p-adic quantum computing

This dissertation investigates geometry, structure and representation theory of the compact p-adic special orthogonal groups, with particular attention to that of degree three, SO(3)p. In addition to its mathematical significance, SO(3)p and its unitary representations are predicted to play a central role in the development of angular momentum and spin in p-adic quantum mechanics. In particular, those representations of dimension two provide a suitable model of p-adic qubit, at the foundations of a burgeoning p-adic theory of quantum information and computation. Thus, we propose to build quantum information processing using elements from the same representations of SO(3)p as quantum logic gates. The study begins with the classification of p-adic quadratic forms, according to which, compact p-adic special orthogonal groups exist only of degree two, three and four. This yields a unique group SO(3)p of rotations on Q3 p , a unique group of degree four, but several incarnations of the group of rotations on the p-adic plane. SO(3)p shows similarities with its real counterpart, while also revealing differences due to the number-theoretic properties of Qp, depending on the prime p. The even prime p = 2 exhibits some peculiarities, therefore it occasionally necessitates a separate and cautious treatment. The entire group SO(3)p admits a representation in terms of Cardano (aka nautical) “angles”, however, this works only for certain orderings of the product of rotations around the reference axes, depending on the prime; furthermore, there is no general Euler decomposition. For p = 2, no Euler or Cardano decomposition exists. We express the Haar measure on SO(3)p, as well as on the other compact p-adic special orthogonal groups, employing two approaches: (1) an inverse-limit machinery of counting measures, since these groups are profinite, and (2) a gen- eral integral formula for the Haar measure on p-adic Lie groups, to be exploited together with the quaternion realisations of p-adic rotations. This paves the way for harmonic analysis on these groups, and specifically for their representations by invoking the Peter-Weyl theorem. Since all the finite-dimensional projective unitary representations of SO(3)p factorise on some quotient modulo p k , k ∈ N, we embark on the path of studying the representations of SO(3)p starting from those induced by SO(3)p mod p. In particular, we explicitly find p-adic qubits for every prime p. We further address the Clebsch-Gordan problem and identify entangled states for composite systems of two p-adic qubits. We finally begin to work on logic gates operating on two qubits, from the known four-dimensional unitary representations of SO(3)p, with the ultimate aim to provide a universal set of gates.