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

Grigory Koroteev

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.

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

Abstract

We introduce Natural Qubit Algebra (NQA), a compact real operator calculus for qubit systems based on a block alphabet and tensor-word representations. The resulting multiplication law induces a canonical -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 ; (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.
Paper Structure (56 sections, 12 theorems, 113 equations, 1 algorithm)

This paper contains 56 sections, 12 theorems, 113 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. NQA block basis and the $(I,X,Z,W)$ calculus
  3. Single--qubit blocks
  4. Subalgebras generated by $X$ and $Z$
  5. $m$--qubit NQA basis (tensor blocks)
  6. Normalized Frobenius product (orthonormality of the block basis)
  7. Phase--lane realification (embedding complex gates into real matrices)
  8. Controlled-Z (CZ).
  9. CNOT (control = qubit 1, target = qubit 2).
  10. CNOT (control = qubit 2, target = qubit 1).
  11. SWAP.
  12. $\mathbf{Z}\!\otimes\!\mathbf{Z}$-parity projectors.
  13. Computational-basis projectors.
  14. General form (real matrices, phase lane).
  15. C.1 Single--qubit phase gates.
  16. ...and 41 more sections

Key Result

Lemma 2.1

The multiplicative set generated by $\{X,Z\}$ closes on and is stable under conjugation. In particular, $XZX=-Z$ and $ZXZ=-X$.

Theorems & Definitions (25)

  • Lemma 2.1: Finite closure of the multiplicative $X/Z$ generators DehaeneDeMoor2003Clifford
  • Proposition 2.2: $\Phi$ is a real $*$--embedding
  • Proposition 3.1: Asymptotic equivalence under structured oracle access
  • Theorem 3.2: Asymptotic equivalence in real NQA vs. quantum computation
  • proof
  • Definition 3.3: Gate set
  • Lemma 3.4: Elementary gates in block form
  • proof
  • Lemma 3.5: Classical black-box vs. structured NQA access
  • Remark 4.1: Relation to the Gottesman--Knill theorem
  • ...and 15 more