Non-Abelian Ising Anyons — Fusion & Braiding

σ×σ = 1+ψ · F-matrix · Topological Qubit from 4 Majorana Modes

Visualization

Fusion Rules

1 × 1 = 1
ψ × ψ = 1
1 × ψ = ψ
σ × σ = 1 + ψ
σ × ψ = σ
σ × 1 = σ
Non-Abelian: σ×σ has two fusion outcomes — the fusion space is 2D. Four σ anyons span a qubit (2-fold degenerate ground state).

F-matrix: Basis change between different fusion orderings. For Ising: F^σσσσ_{1,ψ} = 1/√2 [[1,1],[1,-1]] (Hadamard!).

R-matrix: Braiding phase. R^1_σσ=e^{-iπ/8}, R^ψ_σσ=e^{3iπ/8}. Together with F → universal computation (with ancilla).