Fusion Tree
Gates
Fibonacci anyons:
Two anyon types: {1, τ}. Fusion: τ⊗τ = 1⊕τ.
Golden ratio: φ=(1+√5)/2=1.618...
F-matrix:
[F^τ_ττ]^τ_τ = 1/φ, [F^τ_ττ]^1_1 = -1/φ
R-matrix:
R^1_ττ = e^{-4πi/5}, R^τ_ττ = e^{3πi/5}
Pentagon equation:
Consistency condition F·R·F = R·F·R
Unique solution → golden ratio!
Universal:
Any unitary can be approximated by Fibonacci braids (Solovay-Kitaev).
Two anyon types: {1, τ}. Fusion: τ⊗τ = 1⊕τ.
Golden ratio: φ=(1+√5)/2=1.618...
F-matrix:
[F^τ_ττ]^τ_τ = 1/φ, [F^τ_ττ]^1_1 = -1/φ
R-matrix:
R^1_ττ = e^{-4πi/5}, R^τ_ττ = e^{3πi/5}
Pentagon equation:
Consistency condition F·R·F = R·F·R
Unique solution → golden ratio!
Universal:
Any unitary can be approximated by Fibonacci braids (Solovay-Kitaev).