Composing $p$-adic qubits: from representations of SO(3)$_p$ to entanglement and universal quantum logic gates

TL;DR

This paper explores the structure and entanglement of $p$-adic qubits modeled as representations of SO(3)$_p$, demonstrating their classification, tensor product decomposition, and constructing universal quantum gates for $p=3$.

Contribution

It introduces a classification of $p$-adic qubits via finite quotient representations of SO(3)$_p$, analyzes their entanglement, and constructs universal quantum gates in the $p$-adic setting.

Findings

Classified $p$-adic qubits as representations of SO(3)$_p$ mod $p$.

Solved the Clebsch-Gordan problem for $p$-adic qubits.

Constructed universal quantum gates for $p=3$.

Abstract

In the context of $p$-adic quantum mechanics, we investigate composite systems of $p$-adic qubits and $p$-adically controlled quantum logic gates. We build on the notion of a single $p$-adic qubit as a two-dimensional irreducible representation of the compact $p$-adic special orthogonal group SO(3)$_p$. We show that the classification of these representations reduces to the finite case, as they all factorise through some finite quotient SO(3)$_p$ mod $p^k$. Then, we tackle the problem of $p$-adic qubit composition and entanglement, fundamental for a $p$-adic formulation of quantum information processing. We classify the representations of SO(3)$_p$ mod $p$, and analyse tensor products of two $p$-adic qubit representations lifted from SO(3)$_p$ mod $p$. We solve the Clebsch-Gordan problem for such systems, revealing that the coupled bases decompose into singlet and doublet states. We further study entanglement arising from those stable subsystems. 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.