Paper: 257 Portfolio C — Quantum Hardware

DOI: 10.5281/zenodo.20088536

Abstract

Standard quantum error correction is formulated over matrix algebras, where the Knill-Laflamme (KL) conditions reduce to orthogonality of error syndromes. We lift the code space from stabiliser codes over $\mathbb{F}_2$ to the exceptional Jordan algebra $J_3(\mathbb{O})$ — the unique 27-dimensional formally real Jordan algebra that is not a matrix algebra — and ask whether the KL conditions can be satisfied by algebraic structure alone, independent of code distance.

The central result is Theorem 3.2 (Exceptional Jordan–KL Condition): for the Furey projector $P = \mathrm{diag}(1,0,0) \in J_3(\mathbb{O})$ and any error operators ${E_a}$ supported on the Peirce-1$(P)$ subspace, the Jordan triple product satisfies ${P, E_a^\dagger \circ E_b, P} = c_{ab} \cdot P$, where $c_{ab} \in \mathbb{R}$ is a scalar. This collapses the KL conditions to a scalar constraint, giving exact error correction by algebraic structure. We also propose Conjecture 4.1 (Eastin-Knill Evasion): because $J_3(\mathbb{O})$ is not a matrix algebra, the Eastin-Knill no-go theorem may not apply, potentially permitting transversal universal gates.

Key Results

  • Theorem 3.2 (Exceptional Jordan–KL Condition): ${P, E_a^\dagger \circ E_b, P} = c_{ab} \cdot P$ for errors supported on Peirce-1$(P)$. Proved via: (1) Hermitian reality of the Jordan triple product; (2) U-operator $U_P(X) = 2P \circ (P \circ X) - P^2 \circ X$ acts as a Peirce-1 projector; (3) $U_P$ has 1-dimensional range on Peirce-1; (4) scalar value $c_{ab} = \langle P, E_a^\dagger \circ E_b \rangle$.
  • Conjecture 4.1 (Eastin-Knill Evasion): Transversal universal gates may exist in $J_3(\mathbb{O})$ because the non-matrix-algebra structure invalidates the standard proof.

Relation to Portfolio

  Stabiliser QEC FTC (Paper 206) NA-QEC (this paper)
Code space $\mathbb{F}_2^n$ $G_2$ fibre bundle $J_3(\mathbb{O})$
KL mechanism Syndrome orthogonality Holonomy distance Jordan triple product
Eastin-Knill Blocked Partially evaded Conjectured evaded

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