Composing $p$-adic qubits: from representations of SO(3)$_p$ to entanglement and universal quantum logic gates
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
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2026
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| author | Svampa, Ilaria L'Innocente, Sonia Mancini, Stefano Winter, Andreas |
| author_facet | Svampa, Ilaria L'Innocente, Sonia Mancini, Stefano Winter, Andreas |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_13808 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Composing $p$-adic qubits: from representations of SO(3)$_p$ to entanglement and universal quantum logic gates Svampa, Ilaria L'Innocente, Sonia Mancini, Stefano Winter, Andreas Quantum Physics Mathematical Physics Representation Theory 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. |
| title | Composing $p$-adic qubits: from representations of SO(3)$_p$ to entanglement and universal quantum logic gates |
| topic | Quantum Physics Mathematical Physics Representation Theory |
| url | https://arxiv.org/abs/2601.13808 |