Quantum Error Correction as Parallel Transport on the Grassmannian

Ian Richard C Buckley

Published Jul 9, 2026• 0 citations

Abstract

Shows that stabiliser QEC has a natural geometric home on the Grassmannian Gr(2^k,2^n). Code subspace = point, distance = geodesic, syndrome = boundary map, logical gates = loops, threshold = beta* snap. Eastin-Knill theorem derived as pi_2(Gr)=Z obstruction requiring BIND.

DOI: 10.5281/zenodo.21284199

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Publisher

Abstract

DOI: 10.5281/zenodo.21284199

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\mathcal{P}_n

via generators

S_i

with

S_i^2 = I

[S_i,S_j]=0

. Logical qubit encoded in

2^k

-dimensional code space defined by

S_i|\psi\rangle = |\psi\rangle

. Surface code: distance-

d

lattice, threshold ~1% per gate, logical error rate

\epsilon_L \propto (\epsilon/\epsilon_{th})^{(d+1)/2}

. Riverlane's 10,000× reduction implies

\epsilon_L

scaling from

10^{-3}

10^{-7}

— a factor of

10^4 = (10^2)^2

, consistent with doubling code distance

d \to 2d

in the sub-threshold regime. | CONNECTION: Surface codes are **planar graphs on square lattices Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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

G_2

Topologies

\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})

d

yields logical error rate

P_L \propto (p/p_{th})^{d/2}

where

p

is physical error rate,

p_{th} \approx 0.57\%

for standard depolarizing noise; threshold condition

p < p_{th}

. Stabilizer formalism uses Pauli group

\mathcal{P}_n

and syndrome extraction via

n

ancilla qubits. | CONNECTION: Surface code lattice is a 2D square grid (crystallographic symmetry

p4m

wallpaper group). Error correction cycles correspond to toric code on a planar boundary — topological order with anyonic excitations. No direct golden ratio or base-60 link. | DEPTH: 8 — milestone demonstrates fault-tolerant quantum memory below threshold, a necessary condition for scalable quantum computing. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com