Shows that the Fano plane encodes a fault-tolerant logical qubit at room temperature via molecular orbit structure. The seven Fano lines correspond to seven stabiliser generators; the G2 automorphism group provides transversal logical gates. Identifies candidate molecular platforms for physical realisation and connects to the 731-ISA BIND opcode as the error-correction primitive.
Shows that the Fano plane encodes a fault-tolerant logical qubit at room temperature via molecular orbit structure. The seven Fano lines correspond to seven stabiliser generators; the G2 automorphism group provides transversal logical gates. Identifies candidate molecular platforms for physical realisation and connects to the 731-ISA BIND opcode as the error-correction primitive.
PENDING QUALITY AUDIT (2026-08). The file is restricted while this record is reviewed as part of a systematic audit of the author's corpus. It has not yet been assessed. Metadata and DOI remain public, and access can be requested. The Fano plane PG(2,2) — seven points, seven lines, three points per line — appears three times in quantum error correction: as the stabiliser geometry of the 3-qubit GHZ state, as the parity-check matrix of the 7-qubit Steane ⟦7,1,3⟧ code, and as the constraint graph of a device-independent self-testing protocol. We prove that these three appearances are facets of a single geometric object. The central result is a three-way identification: the non-trivial elements of the GHZ stabiliser group (Z₂)³\{0} are canonically bijective with the 7 points of PG(2,2) = GF(2)³\{0}, which are the columns of the [7,4,3] Hamming parity-check matrix. The 3-qubit GHZ stabiliser group generates the 7-qubit Steane code's parity-check matrix. We prove an orbit decomposition of 3-qubit phase space: PG(5,2)\PG(2,2) = ⊔_{L=0}^{6} O_L, seven orbits of eight phase-space points (one per Fano line), giving a new Clifford-invariant magic label {p_L} that refines Wigner negativity into seven independent channels. The Fano game — measuring the seven GHZ stabilisers — achieves quantum maximum w=1, classical bound w≤5/7, and self-testing robustness C=7/8. The value 7/8 = |PG(2,2)|/2³ is a Fano-geometric invariant. Three practical tools follow: a device-independent GHZ certification protocol, the orbit-label magic invariant for magic state distillation, and a hardware coupling specification for a Fano-structured quantum processor (6 of 7 Fano lines at full coupling, one at r≈0.18, matching the FMO efficiency η≈0.1825). Keywords: quantum error correction, Fano plane, Steane code, GHZ state, magic states, self-testing, Wigner function, orbit decomposition, stabiliser codes, device-independent, PSL(2,7), quantum computing
Quantum error correction (QEC) is cohomological encoding made explicit. We show that the three central codes in fault-tolerant quantum computation correspond precisely to three levels of the H^k tower of the Origami instruction-set architecture (ISA): the Steane [[7,1,3]] code is the Fano plane PG(2,2) at the H^1 tier; the surface code encodes logical qubits as generators of H^1(surface;Z_2); and the 15-to-1 magic state distillation factory is an H^1 Clifford circuit that bridges H^1 and~H^2 by purifying the non-Clifford TWIST\ resource. A fourth result is new: dark-magic errors are syndrome-invisible. A unitary with TWIST>0 that commutes with all stabiliser generators corrupts the logical orbit label while leaving every syndrome bit at +1. Standard randomised benchmarking is insensitive to orbit-label decay; we propose orbit-sensitive randomised benchmarking (ORBIT-RB) as the missing diagnostic. The unified picture is captured in a single sentence: a QEC code of depth k specifies which H^k classes are logical and which are trivial; syndrome measurement computes the coboundary ; error correction finds the minimum-weight representative of the observed H^k class.
PENDING QUALITY AUDIT (2026-08). The file is restricted while this record is reviewed as part of a systematic audit of the author's corpus. It has not yet been assessed. Metadata and DOI remain public, and access can be requested. A practitioner's primer for quantum computing researchers. No knowledge of octonions, sheaf theory, or category theory is assumed. Knowledge of stabiliser states, the Clifford group, and T-gates is assumed. The central message: the structure of the 3-qubit Pauli group is secretly governed by the Fano plane PG(2,2), and the Fano plane is the geometry of the 731 instruction set architecture (731 ISA) — a unified framework for quantum computation that organises every opcode as a Pachner move on a simplicial complex. The framework reveals three levels of quantum computational resource:• Level 0 (Clifford): stabiliser states; Gottesman-Knill simulable; the Fano lines are the free sector.• Level 1 (Standard magic): T-gates; Wigner function negativity; non-Clifford but accessible to the Origami ISA (regime 2).• Level 2 (Associamancy): the SPIN opcode; the Schur boundary; genuinely complex irreps of PSL(2,7); not accessible to any associative hardware. The Origami ISA (five opcodes: SPLIT, SPLAT, TWIST, FLIP, FLOP) subsumes Bell sampling, qudit stabiliser learning, and the abelian hidden subgroup problem as special cases. The 731 ISA adds SPIN and BIND, unlocking Level 2. This primer is a companion to "The Topology of Risk" (doi:10.5281/zenodo.20642983), which covers the same H⁰/H¹/H² framework applied to financial risk. Keywords: quantum computing, Fano plane, Origami ISA, 731 ISA, stabiliser states, magic states, associamancy, Schur boundary, primer, tutorial, hidden subgroup problem, MBQC, photonics
Building fault-tolerant quantum computers is a current priority for the quantum computing industry and academic counterparts. We present a quantum error correction code encoding a single logical qubit into n=L2 physical qubits with distance d=L as a subsystem code, and n−O(n) logical qubits with distance d=3 as a standard stabiliser code. The logical single-qubit gates in the subsystem code are transversal. Under independent X,Z noise we find a threshold of 1.4%±0.1% for odd distance and 1.8%±0.2% for even distance for the subsystem variant. The code detects every 1- and 2-qubit error, and additionally every error of weight 3 and higher (constrained by the distance) that is not a product of the 3-qubit check operations. These operations act on the gauge qubits leaving the subsystem code space invariant. The code can be implemented by qubits in a triangular configuration with the plaquettes as 3-qubit check operations.
CORRECTION NOTICE (2026-08). The central mechanism does not work. The quantum dimension sin(pπβ)/sin(πβ) vanishes at β = 1/p for every integer p, not only p = 3, 5, 7, so it cannot select those three codes. The conventional explanations (quantum Hamming bound, Fano-plane structure) are not superseded. A revised version is in preparation. Three quantum codes are foundational in fault-tolerant architecture: triangular colour codes, the [[5,1,3]] perfect code, and the Steane [[7,1,3]] code. Their canonical status is usually explained by counting arguments (quantum Hamming bound) or algebraic coincidence (Fano plane geometry). We give a unified structural explanation: all three sit at snap events of the beta-deformed G2 spider, where quantum dimensions vanish at roots of unity beta=1/p for p=3,5,7. At each snap, a continuous G2 symmetry breaks and a discrete symmetry crystallises — spontaneous symmetry making (SSM). The snap at beta=1/5 forces BIND self-annihilation, the algebraic condition for saturation of the quantum Hamming bound. The snap at beta=1/7 crystallises PSL(2,7) as the transversal T-gate group of the Steane code.
The 7 nodes of the Fano plane — the combinatorial geometry underlying octo- nion multiplication — map exactly onto 7 orbital quantum dot (QD) modes in the Transdimensional Anomalous Hall Effect (TDAHE) ground state of rhombohedral graphene. The adjacency graph of the Fano plane is the complete graph K7: every QD couples to all 6 others, with the coupling sign and mediation structure deter- mined by the unique Fano line through each pair. The ternary gate F({i, j}) = |k⟩ (output is the unique third node on the Fano line through i and j) gives a complete 7-gate primitive set closed under G2. We prove that the Fano-Toffoli associator FT(i, j, k) = [(eiej)ek] −[ei(ejek)] is zero if and only if (i, j, k) lies on a Fano line — precision 1.000, recall 1.000, confirmed exhaustively over all 210 ordered triples. This makes FT a perfect hardware error detector: TDAHE ground states are Fano- line correlated (FT = 0 always); any perturbation exits the Fano line (FT ̸= 0 immediately). The chiral Bell measurement uses ABC/CBA-stacked interferometer pairs; fidelity per 3-step teleportation protocol is (1 −fbulk)3 = 89.3% from pion kinematics alone, versus 82±1% for Laneve et al. (2025) using engineered dissimilar quantum dots. Contents Part of the One-Octonion Brane-Bulk Framework series. Anchor DOI: 10.5281/zenodo.19120873. Community: one-octonion-brane-bulk. Author: Bharathi Dasan Jagadeesan, M.D., University of Minnesota. ORCID: 0000-0002-1143-941X.