Quaternionic and Octonionic Frameworks for Quantum Computation: Mathematical Structures, Models, and Fundamental Limitations
Abstract
We develop detailed quaternionic and octonionic frameworks for quantum computation grounded on normed division algebras. Our central result is to prove the polynomial computational equivalence of quaternionic and complex quantum models: Computation over ℍ is polynomially equivalent to the standard complex quantum circuit model and hence captures the same complexity class BQP up to polynomial reductions. Over ℍ, we construct a complete model---quaternionic qubits on right ℍ-modules with quaternion-valued inner products, unitary dynamics, associative tensor products, and universal gate sets---and establish polynomial equivalence with the standard complex model; routes for implementation at fidelities exceeding 99% via pulse-level synthesis on current hardware are discussed. Over 𝕆, non-associativity yields path-dependent evolution, ambiguous adjoints/inner products, non-associative tensor products, and possible failure of energy conservation outside associative sectors. We formalize these obstructions and systematize four mitigation strategies: Confinement to associative subalgebras, G2-invariant codes, dynamical decoupling of associator terms, and a seven-factor algebraic decomposition for gate synthesis. The results delineate the feasible quaternionic regime from the constrained octonionic landscape and point to applications in symmetry-protected architectures, algebra-aware simulation, and hypercomplex learning.
Versions
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BibTeX
@misc{munoz2026quaternionic,
title = {{Quaternionic and Octonionic Frameworks for Quantum Computation: Mathematical Structures, Models, and Fundamental Limitations}},
author = {Johan Heriberto Rúa Muñoz and Jorge Eduardo Mahecha Gómez and Santiago Pineda Montoya},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03348},
eprint = {2611.03348},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03348},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Quaternionic and Octonionic Frameworks for Quantum Computation: Mathematical Structures, Models, and Fundamental Limitations
AU - Johan Heriberto Rúa Muñoz
AU - Jorge Eduardo Mahecha Gómez
AU - Santiago Pineda Montoya
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03348
UR - https://arxiv.org/abs/2611.03348
AB - We develop detailed quaternionic and octonionic frameworks for quantum computation grounded on normed division algebras. Our central result is to prove the polynomial computational equivalence of quaternionic and complex quantum models: Computation over ℍ is polynomially equivalent to the standard complex quantum circuit model and hence captures the same complexity class BQP up to polynomial reductions. Over ℍ, we construct a complete model---quaternionic qubits on right ℍ-modules with quaternion-valued inner products, unitary dynamics, associative tensor products, and universal gate sets---and establish polynomial equivalence with the standard complex model; routes for implementation at fidelities exceeding 99% via pulse-level synthesis on current hardware are discussed. Over 𝕆, non-associativity yields path-dependent evolution, ambiguous adjoints/inner products, non-associative tensor products, and possible failure of energy conservation outside associative sectors. We formalize these obstructions and systematize four mitigation strategies: Confinement to associative subalgebras, G_2-invariant codes, dynamical decoupling of associator terms, and a seven-factor algebraic decomposition for gate synthesis. The results delineate the feasible quaternionic regime from the constrained octonionic landscape and point to applications in symmetry-protected architectures, algebra-aware simulation, and hypercomplex learning.
ER -