Ilaria Svampa - Universität zu Köln

p-Adic quantum computing and universal sets of p-adically controlled gates

We develop a model of p-adic quantum information processing, where the p-adic qubit emerges from a two-dimensional representation of the p-adic rotation group SO(3)ₚ.

All finite-dimensional projective unitary representations of SO(3)ₚ factorise on some SO(3)ₚ mod pⁿ, through which we find explicit p-adic qubit representations for every prime p. Interestingly, there are several p-adic qubits for every prime 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 mod 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 4-dimensional irreducible representations of SO(3)ₚ mod p, that we prove to be universal for quantum computation.

(Based on arXiv:2601.13808, arXiv:2112.03362)