Natural Qubit Algebra: clarification of the Clifford boundary and new non-embeddability theorem

• URL: http://arxiv.org/abs/2602.21387v1
• Date: Tue, 24 Feb 2026 21:30:19 GMT
• Title: Natural Qubit Algebra: clarification of the Clifford boundary and new non-embeddability theorem
• Authors: Grigory Koroteev,
• Abstract summary: We introduce Natural Qubit Algebra (NQA), a compact real operator calculus for qubit systems.<n>NQA is based on a $2times2$ block alphabet $I,X,Z,WsubsetmathrmMat (2,mathbbR)$ and tensor-word representations.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We introduce Natural Qubit Algebra (NQA), a compact real operator calculus for qubit systems based on a $2\times2$ block alphabet $\{I,X,Z,W\}\subset\mathrm{Mat}(2,\mathbb{R})$ and tensor-word representations. The resulting multiplication law induces a canonical $(\mathbb{Z}_2)^{2m}$-grading with a bicharacter that controls commutation signs, placing the framework naturally within the theory of color-graded and Clifford-type algebras. Within this language, we provide: (i) an explicit real Clifford normal form for two-qubit operators via the identification $\mathrm{Mat}(4,\mathbb{R})\cong\mathrm{Cl}(2,2;\mathbb{R})$; (ii) a purely algebraic reformulation of the Bell--CHSH scenario, where the quantum violation is expressed as a spectral non-embeddability of a noncommutative spinor algebra into any commutative Kolmogorov algebra; and (iii) compact factored representations of the Bernstein--Vazirani and Grover phase oracles, showing that both Clifford and non-Clifford examples can admit similarly structured symbolic descriptions. We clarify that Grover's iterate remains outside the Clifford group due to its continuous spectral rotation, consistent with the Gottesman--Knill theorem, while retaining a compact tensor-block form in NQA. The framework isolates spectral, algebraic, and syntactic aspects of operator structure, providing a graded operator language compatible with standard quantum mechanics.

Related papers

• The State-Operator Clifford Compatibility: A Real Algebraic Framework for Quantum Information [0.0]
  We introduce a real, grade-preserving framework for $N$-qubit quantum computation based on the tensor product structure $Cell_2,0(mathbbR)otimes N$.
  arXiv  Detail & Related papers  (2025-12-05T22:55:31Z)
• Notes on Real Quantum Mechanics in a Kahler Space [0.0]
  The necessity of complex numbers in quantum mechanics has long been debated.<n>This paper develops a real Kahler space formulation of quantum mechanics asserting equivalence to the standard complex Hilbert space framework.
  arXiv  Detail & Related papers  (2025-06-09T10:58:13Z)
• Volichenko-type metasymmetry of braided Majorana qubits [0.0]
  This paper presents different mathematical structures connected with the parastatistics of braided Majorana qubits. Mixed-bracket Heisenberg-Lie algebras are introduced.
  arXiv  Detail & Related papers  (2024-06-02T21:50:29Z)
• Symmetry-restricted quantum circuits are still well-behaved [45.89137831674385]
  We show that quantum circuits restricted by a symmetry inherit the properties of the whole special unitary group $SU(2n)$. It extends prior work on symmetric states to the operators and shows that the operator space follows the same structure as the state space.
  arXiv  Detail & Related papers  (2024-02-26T06:23:39Z)
• Classification of dynamical Lie algebras for translation-invariant 2-local spin systems in one dimension [44.41126861546141]
  We provide a classification of Lie algebras generated by translation-invariant 2-local spin chain Hamiltonians. We consider chains with open and periodic boundary conditions and find 17 unique dynamical Lie algebras. In addition to the closed and open spin chains, we consider systems with a fully connected topology, which may be relevant for quantum machine learning approaches.
  arXiv  Detail & Related papers  (2023-09-11T17:59:41Z)
• Quantum Current and Holographic Categorical Symmetry [62.07387569558919]
  A quantum current is defined as symmetric operators that can transport symmetry charges over an arbitrary long distance. The condition for quantum currents to be superconducting is also specified, which corresponds to condensation of anyons in one higher dimension.
  arXiv  Detail & Related papers  (2023-05-22T11:00:25Z)
• Duality theory for Clifford tensor powers [0.7826806223782052]
  The representation theory of the Clifford group is playing an increasingly prominent role in quantum information theory. In this paper, we provide a unified framework for the duality approach that also covers qubit systems.
  arXiv  Detail & Related papers  (2022-08-02T18:27:17Z)
• Self-adjoint extension schemes and modern applications to quantum Hamiltonians [55.2480439325792]
  monograph contains revised and enlarged materials from previous lecture notes of undergraduate and graduate courses and seminars delivered by both authors over the last years on a subject that is central both in abstract operator theory and in applications to quantum mechanics. A number of models are discussed, which are receiving today new or renewed interest in mathematical physics, in particular from the point of view of realising certain operators of interests self-adjointly.
  arXiv  Detail & Related papers  (2022-01-25T09:45:16Z)
• Integrable spin chains and the Clifford group [0.0]
  We construct new families of spin chain Hamiltonians that are local, integrable and translationally invariant. For spin chains solvable by algebraic Bethe Ansatz, we explain how conjugating by a matrix product operator affects the underlying integrable structure.
  arXiv  Detail & Related papers  (2021-07-05T18:00:04Z)
• Sub-bosonic (deformed) ladder operators [62.997667081978825]
  We present a class of deformed creation and annihilation operators that originates from a rigorous notion of fuzziness. This leads to deformed, sub-bosonic commutation relations inducing a simple algebraic structure with modified eigenenergies and Fock states. In addition, we investigate possible consequences of the introduced formalism in quantum field theories, as for instance, deviations from linearity in the dispersion relation for free quasibosons.
  arXiv  Detail & Related papers  (2020-09-10T20:53:58Z)
• Models of zero-range interaction for the bosonic trimer at unitarity [91.3755431537592]
  We present the construction of quantum Hamiltonians for a three-body system consisting of identical bosons mutually coupled by a two-body interaction of zero range. For a large part of the presentation, infinite scattering length will be considered.
  arXiv  Detail & Related papers  (2020-06-03T17:54:43Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.