Ising Anyon Braiding Generates Only Clifford Gates, Not Universal Quantum Computation — E8 Intelligence Research

Andrew Stewart Caldin

Published Oct 18, 2026• 0 citations

Abstract

FINDING: Braiding of Ising anyons generates only Clifford gates, not universal quantum computation; two-qubit braiding exhausts the full two-qubit Clifford group. | MATH: Ising anyon braid group representation → Clifford group (generated by H, S, CNOT). Key matrices: H = (1/√2)[[1,1],[1,-1]], S = diag(1, i), CNOT. Braid generators σ_i satisfy σ_i² = -1 (fermion parity), σ_i σ_j = σ_j σ_i for |i-j|≥2. The π/8 gate (T = diag(1, e^{iπ/4})) is *not* in the Clifford group — hence not reachable by braiding alone. | CONNECTION: The Clifford group is the normalizer of the Pauli group, isomorphic to the binary symplectic group Sp(2n, F₂) — a root system of type C_n over F₂. The π/8 gate corresponds to a 45° rotation in the Bloch sphere — a geometric angle of π/4, which is *not* a multiple of the golden-ratio-derived angles (π/5, 2π/5) nor of the crystallographic 60° (π/3) or 90° (π/2). The Ising braid matrices live in SU(2) with level-2 Kac-Moody algebra — whose fusion rules (1/2 ⊗ 1/2 = 0 ⊕ 1) Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI: 10.5281/zenodo.23447092

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