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

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 $\mathfrak{J}_3(\mathbb{O})$.