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

Ilaria Svampa, Sonia L'Innocente, Stefano Mancini, Andreas Winter

TL;DR

This work elevates $p$-adic quantum mechanics by formulating qubits as 2-dimensional irreducible representations of $SO(3)_p$ and building a full information-processing program around representations, Clebsch–Gordan analysis, entanglement, and quantum gates. It shows that all finite-dimensional reps of $SO(3)_p$ factor through finite quotients $G_{p^k}$, enabling a finite-group approach to qubits and their interactions. The Clebsch–Gordan problem for two $p$-adic qubits yields singlet and doublet (and, for $p=3$, triplet) structures, with entanglement properties tied to the corresponding subrepresentations; the paper provides explicit CG coefficients for small primes and a general framework. Most notably, at $p=3$ the authors construct a universal set of $3$-adically controlled quantum gates from 4D irreps of $G_3$, proven via Lie-algebraic methods and Givens-rotation decompositions, establishing computational universality in this $p$-adic setting and suggesting a path toward adelic formulations that unify real and $p$-adic quantum theories.

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.

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

TL;DR

This work elevates -adic quantum mechanics by formulating qubits as 2-dimensional irreducible representations of and building a full information-processing program around representations, Clebsch–Gordan analysis, entanglement, and quantum gates. It shows that all finite-dimensional reps of factor through finite quotients , enabling a finite-group approach to qubits and their interactions. The Clebsch–Gordan problem for two -adic qubits yields singlet and doublet (and, for , triplet) structures, with entanglement properties tied to the corresponding subrepresentations; the paper provides explicit CG coefficients for small primes and a general framework. Most notably, at the authors construct a universal set of -adically controlled quantum gates from 4D irreps of , proven via Lie-algebraic methods and Givens-rotation decompositions, establishing computational universality in this -adic setting and suggesting a path toward adelic formulations that unify real and -adic quantum theories.

Abstract

In the context of -adic quantum mechanics, we investigate composite systems of -adic qubits and -adically controlled quantum logic gates. We build on the notion of a single -adic qubit as a two-dimensional irreducible representation of the compact -adic special orthogonal group SO(3). We show that the classification of these representations reduces to the finite case, as they all factorise through some finite quotient SO(3) mod . Then, we tackle the problem of -adic qubit composition and entanglement, fundamental for a -adic formulation of quantum information processing. We classify the representations of SO(3) mod , and analyse tensor products of two -adic qubit representations lifted from SO(3) mod . 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 , we construct a set of gates from -dimensional irreducible representations of SO(3) mod that we prove to be universal for quantum computation.
Paper Structure (16 sections, 28 theorems, 102 equations)

This paper contains 16 sections, 28 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Rapells
  3. The profinite group Lg
  4. Representation theory
  5. About representations of Lg
  6. Factorisation of representations modulo Lg
  7. Structure and irreps of Lg
  8. Clebsch-Gordan problem for Lg-adic qubits
  9. Clebsch-Gordan coefficients
  10. Entanglement for two $p$-adic qubits
  11. Lg-Adically controlled quantum logic gates
  12. Universal quantum computation
  13. Four-dimensional irreps of $G_3$
  14. Gates from the representations Lg and Lg
  15. A universal set of $3$-adically controlled quantum gates
  16. ...and 1 more sections

Key Result

Proposition 2.1

For every prime $p$, up to equivalence, there is a unique non-isotropic quadratic form over $\mathbb{Q}_p^3$: where $v\coloneqq $ for a non-square $u\in\mathbb{U}_p$. Then, for every prime $p$, up to isomorphism, there is a unique compact special orthogonal group on $\mathbb{Q}_p^3$: where $A_+\coloneqq $ is the matrix representation of $Q_+$ with respect to the canonical basis.

Theorems & Definitions (61)

  • Proposition 2.1: our1st, Sec. 2
  • Remark 2.2
  • Definition 2.3
  • Example 2.4
  • Definition 2.5
  • Theorem 2.6: profinite
  • Lemma 2.7: profinite
  • Theorem 2.8: profinite
  • Theorem 2.9: Haar2 and our2nd
  • Definition 2.10: Cf. folland2016course
  • ...and 51 more