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Non-Associative Quantum Error Correction (NA-QEC): The Exceptional Jordan-KL Condition and Rank-1 Idempotents in $G_2$ Topologies

Ian Richard C Buckley

Published May 8, 2026• 0 citations
Publisher

Abstract

CORRECTION NOTICE (2026-08). No error-correction simulation was performed for this work; the Knill–Laflamme conditions are asserted rather than verified. Treat the construction as a proposal. A revised version is in preparation. Standard quantum error correction (QEC) is founded on the Knill-Laflamme (KL) conditions, which implicitly assume that physical error operators compose associatively. Over the non-associative Octonions $\mathbb{O}$ and the exceptional Lie group $G_2$, this assumption fails: the elementary identity $(PE_a^\dagger)(E_b P) = P(E_a^\dagger E_b)P$ no longer holds, generating a chiral shear that breaks the Hermitian reality of observable eigenvalues. We resolve this obstruction by replacing ordinary matrix multiplication with the Jordan product $A\circ B = \frac{1}{2}(AB+BA)$, which restores Hermiticity by symmetrisation while retaining non-associativity. Working over the 27-dimensional Exceptional Jordan Algebra $\mathfrak{J}_3(\mathbb{O})$ (the Albert algebra), we prove the Exceptional Jordan-KL Condition: for the Furey projector $P = \mathrm{diag}(1,0,0) \in \mathfrak{J}_3(\mathbb{O})$, the Jordan U-operator satisfies $$\{P, E_a^\dagger \circ E_b, P\} = c_{ab} P$$ for every symmetrised error element $E_a^\dagger \circ E_b \in \mathfrak{J}_3(\mathbb{O})$, where $c_{ab} \in \mathbb{R}$ is a scalar determined by the normalised trace. The proof rests entirely on the standard Peirce decomposition of $\mathfrak{J}_3(\mathbb{O})$ and McCrimmon's classification of primitive idempotents; no novel algebraic machinery is required. We apply this result to the 731-QPU: the Steane $[[7,1,3]]$ code operates as the hardware-native QEC layer, and the Fano-plane incidence structure enforces distance $d=3$ geometrically via the Associator Penalty. NA-QEC monitors Jordan-algebraic coherence continuously rather than measuring discrete Pauli parities, trading measurement-induced decoherence for a geometric error-detection channel. We carefully delineate what is proved from what is conjectured: the claim that NA-QEC evades the Eastin-Knill theorem is stated as an open conjecture, motivated by the observation that the Eastin-Knill proof requires the logical subspace to carry a continuous Lie-group representation, a condition that is absent when the logical space is the 1-dimensional Peirce-1 subspace of $\mathfrak{J}_3(\mathbb{O})$.

DOI
10.5281/zenodo.20088536

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Non-Associative Quantum Error Correction (NA-QEC): The Exceptional Jordan-KL Condition and Rank-1 Idempotents in G2G_2G2​ Topologies
CORRECTION NOTICE (2026-08). No error-correction simulation was performed for this work; the Knill–Laflamme conditions are asserted rather than verified. Treat the construction as a proposal. A revised version is in preparation. Standard quantum error correction (QEC) is founded on the Knill-Laflamme (KL) conditions, which implicitly assume that physical error operators compose associatively. Over the non-associative Octonions O\mathbb{O}O and the exceptional Lie group G2G_2G2​, this assumption fails: the elementary identity (PEa†)(EbP)=P(Ea†Eb)P(PE_a^\dagger)(E_b P) = P(E_a^\dagger E_b)P(PEa†​)(Eb​P)=P(Ea†​Eb​)P no longer holds, generating a chiral shear that breaks the Hermitian reality of observable eigenvalues. We resolve this obstruction by replacing ordinary matrix multiplication with the Jordan product A∘B=12(AB+BA)A\circ B = \frac{1}{2}(AB+BA)A∘B=21​(AB+BA), which restores Hermiticity by symmetrisation while retaining non-associativity. Working over the 27-dimensional Exceptional Jordan Algebra J3(O)\mathfrak{J}_3(\mathbb{O})J3​(O) (the Albert algebra), we prove the Exceptional Jordan-KL Condition: for the Furey projector P=diag(1,0,0)∈J3(O)P = \mathrm{diag}(1,0,0) \in \mathfrak{J}_3(\mathbb{O})P=diag(1,0,0)∈J3​(O), the Jordan U-operator satisfies {P,Ea†∘Eb,P}=cabP\{P, E_a^\dagger \circ E_b, P\} = c_{ab} P{P,Ea†​∘Eb​,P}=cab​P for every symmetrised error element Ea†∘Eb∈J3(O)E_a^\dagger \circ E_b \in \mathfrak{J}_3(\mathbb{O})Ea†​∘Eb​∈J3​(O), where cab∈Rc_{ab} \in \mathbb{R}cab​∈R is a scalar determined by the normalised trace. The proof rests entirely on the standard Peirce decomposition of J3(O)\mathfrak{J}_3(\mathbb{O})J3​(O) and McCrimmon's classification of primitive idempotents; no novel algebraic machinery is required. We apply this result to the 731-QPU: the Steane [[7,1,3]][[7,1,3]][[7,1,3]] code operates as the hardware-native QEC layer, and the Fano-plane incidence structure enforces distance d=3d=3d=3 geometrically via the Associator Penalty. NA-QEC monitors Jordan-algebraic coherence continuously rather than measuring discrete Pauli parities, trading measurement-induced decoherence for a geometric error-detection channel. We carefully delineate what is proved from what is conjectured: the claim that NA-QEC evades the Eastin-Knill theorem is stated as an open conjecture, motivated by the observation that the Eastin-Knill proof requires the logical subspace to carry a continuous Lie-group representation, a condition that is absent when the logical space is the 1-dimensional Peirce-1 subspace of J3(O)\mathfrak{J}_3(\mathbb{O})J3​(O).
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FINDING: Quantum error correction (QEC) is the mathematical scaffold for fault-tolerant quantum computation, with the stabilizer formalism as its core algebraic structure. | MATH: Stabilizer codes are defined by an abelian subgroup S \mathcal{S} S of the Pauli group Pn \mathcal{P}_n Pn​; codewords are simultaneous +1+1+1 eigenstates of all S∈S S \in \mathcal{S} S∈S. The code space dimension is 2n−k 2^{n-k} 2n−k where k=n−rank(S) k = n - \text{rank}(\mathcal{S}) k=n−rank(S). Distance d d d corrects t=⌊(d−1)/2⌋ t = \lfloor (d-1)/2 \rfloor t=⌊(d−1)/2⌋ errors. Key bound: k≤n−log2∣S∣ k \le n - \text{log}_2 |\mathcal{S}| k≤n−log2​∣S∣. The Knill–Laflamme condition: ⟨ψi∣Ea†Eb∣ψj⟩=δijCab \langle \psi_i | E_a^\dagger E_b | \psi_j \rangle = \delta_{ij} C_{ab} ⟨ψi​∣Ea†​Eb​∣ψj​⟩=δij​Cab​, where Ea E_a Ea​ are error operators. | CONNECTION: The stabilizer group S \mathcal{S} S is a **binary linear code** over F22n \mathbb{F}_2^{2n} F22n​ — a **lattice structure** in symplectic space. The symplectic inner product ⟨(a∣b),(a′∣b′)⟩=a⋅b′+a′⋅b(mod2) \langle (a|b), (a'|b') \rangle = a \cdot b' + a' \cdot b \pmod{2} ⟨(a∣b),(a′∣b′)⟩=a⋅b′+a′⋅b(mod2) defines a **self-d Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
We introduce an intrinsic formulation of quantum error correction based on representation theory, in which error-protection structure is encoded directly in a unitary group representation, rather than being tied to a particular embedding into a larger Hilbert space. In this framework, error models are classified according to the isotypic decomposition of the conjugation action on the operator algebra. Our main result, the \emph{Schur bootstrap}, shows that if an intrinsic code satisfies the Knill--Laflamme conditions on a given symmetry sector, then the same error-protection relations hold for every extrinsic realization obtained from a group-equivariant isometric embedding into a larger Hilbert space. Thus a single intrinsic verification certifies the corresponding symmetry-resolved error-correction conditions across an entire family of physical realizations. We further introduce an intrinsic notion of distance, called depth, defined via adjoint order. For standard multi-qudit systems this coincides with conventional code distance, while for more general representations it refines the usual weight-based notion. We also prove an intrinsic Eastin--Knill theorem: any intrinsic code of depth at least two has a discrete logical symmetry group, with the obstruction to continuous covariant gates arising from the representation-theoretic structure of the adjoint action. We illustrate the framework with several examples, including a minimal SU(2)\mathrm{SU}(2)SU(2) construction that unifies permutation-invariant qubit codes and bosonic codes, and higher-dimensional constructions exhibiting transversal Clifford symmetries and realizations beyond qubit systems.
The Non-Associative No-Cloning Theorem and the Fano-Token: Target Obstruction, Map Collapse, and Active Forgery Detection via the Exceptional Jordan Algebra J3(O)\mathfrak{J}_{3}(\mathbb{O})J3​(O)
Wiesner's quantum money protocol was conceived around 1970, over a decade before Wootters and Zurek established the No-Cloning Theorem. Its security was argued operationally; No-Cloning later supplied the proof-theoretic foundation. This paper asks whether the Exceptional Jordan Algebra J3(O)\mathfrak{J}_3(\mathbb{O})J3​(O) (the Albert algebra) admits a non-associative No-Cloning theorem that is strictly stronger than the standard one, and what it implies for quantum money. The standard No-Cloning proof uses only linearity and the bilinear tensor structure; it does not use associativity, so a direct translation to J3(O)\mathfrak{J}_3(\mathbb{O})J3​(O) recovers only the standard result in new notation. The genuinely non-associative obstruction is of a different character: it concerns not the impossibility of a cloning map but the impossibility of a cloning target. We prove two theorems. Theorem 3.1 (NA No-Cloning --- Target Obstruction): the two-copy system of J3(O)\mathfrak{J}_3(\mathbb{O})J3​(O) does not exist as a Jordan algebra. Any Jordan algebra containing two commuting copies of J3(O)\mathfrak{J}_3(\mathbb{O})J3​(O) would be special (embeddable in an associative algebra), contradicting the exceptionality of J3(O)\mathfrak{J}_3(\mathbb{O})J3​(O) (Albert). There is no valid target into which an octonionic state can be cloned while remaining in the same algebraic category. Theorem 3.3 (NA No-Cloning --- Map Collapse): any Jordan algebra homomorphism Φ ⁣:J3(O)→Asa\Phi\colon \mathfrak{J}_3(\mathbb{O}) \to A_{sa}Φ:J3​(O)→Asa​ into the self-adjoint part of an associative algebra AAA satisfies Φ([x,y,z])=0\Phi([x, y, z]) = 0Φ([x,y,z])=0 for all x,y,z∈J3(O)x, y, z \in \mathfrak{J}_3(\mathbb{O})x,y,z∈J3​(O). Since non-Fano associators span the 16-dimensional Peirce-12\tfrac{1}{2}21​ subspace of J3(O)\mathfrak{J}_3(\mathbb{O})J3​(O), Φ\PhiΦ collapses 16 of the 27 dimensions of J3(O)\mathfrak{J}_3(\mathbb{O})J3​(O). A dimension-collapsing map cannot clone. Together these theorems say: for octonionic quantum states, No-Cloning holds for a reason the standard theorem does not reach. The obstruction is structural, not just linear-algebraic. The Fano-Token protocol encodes a quantum banknote as a rank-1 projector in J3(O)\mathfrak{J}_3(\mathbb{O})J3​(O), the Furey projector PℓP_\ellPℓ​ stabilised by the G2G_2G2​ automorphism group. As a Corollary 4.1 of Theorem 3.3 , any associative measurement operator MMM applied to PℓP_\ellPℓ​ produces a defect equal to the associator [Pℓ,M,Pℓ][P_\ell, M, P_\ell][Pℓ​,M,Pℓ​], with norm 222 for non-Fano-compatible MMM. This converts Wiesner's passive defense into an active alarm: unauthorized measurement is detectable instantaneously, without token retrieval. The physical claim that this defect manifests as a measurable energy transition in the 731-QPU hardware is stated as Conjecture and kept separate from the algebraic results.
Fibrational Tensor Codes (FTCs): Holonomic Quantum Computation, G2G_2G2​ Fiber Bundles, and the Cayley-Dickson Hardware Hierarchy
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