Braiding: Fibonacci and Ising

Fibonacci anyons: one fusion rule

The Fibonacci (“golden”) theory has two particle types, 1 and τ, and one nontrivial rule: τ × τ = 1 + τ. Fusing n τ's, the number of fusion paths ending in total charge 1 is Fib(n−1), and in τ is Fib(n); so the Hilbert space of n anyons with trivial total charge has dimension Fib(n−1), growing like φⁿ, and the quantum dimension of τ is the golden ratio φ = (1 + √5)/2, which satisfies φ² = φ + 1.

Changing the order of fusion is the basis change F = [[1/φ, 1/√φ], [1/√φ, −1/φ]], unitary with F² = I (its rows are unit vectors because 1/φ² + 1/φ = 1). Exchanging two anyons is R = diag(e^−4πi/5, e^3πi/5) in the basis where the pair fuses to 1 or τ. Then σ₁ = R and σ₂ = FRF are the braid generators on the two-dimensional space of four anyons, and the consistency condition that makes braiding well defined, the hexagon equation, appears here as the braid relation σ₁σ₂σ₁ = σ₂σ₁σ₂.

The distinction the register makes, checked

The corpus's claims register says braiding is universal for topological quantum computation only model by model: Fibonacci yes (stated in Nayak et al.; the density theorem behind it is Freedman–Larsen–Wang’s, as explained below), Ising no — Clifford gates only, needing a supplementary non-topological gate. This chapter checks the part of that which a script can reach. A set of gates is universal only if the group it generates is dense in SU(2) (mod phase); a finite group is the opposite extreme.

Ising. For four σ anyons with total charge 1, σ₁ = diag(e^−iπ/8, e^3iπ/8) and σ₂ = Hσ₁H (Hadamard-conjugated). They satisfy the braid relation, and the group they generate, modulo overall phase, closes at exactly 24 elements: the one-qubit Clifford group. A finite group is not dense, so braiding alone cannot be universal.

Fibonacci. The group generated by σ₁, σ₂ above does not close: breadth-first search layers hold 5, 17, 42, 90, 184, 360, 690, 1314, 2488, 4698, 8862, 16704, 31468 … distinct elements (mod phase), and the search passes 20,000 without stopping. That is evidence for density and no more; the theorem in the Freedman–Larsen–Wang paper is about a different object from the one tested here. It concerns the Jones representation of the six-strand braid group at q = e^2πi/5, and its Theorem 4.1 says that the closure of the image in U(5)×U(8) contains SU(5)×SU(8). From that they build the simulation of any BQP computation. The paper never uses the words Fibonacci or Ising. Reading its SU(2) Chern–Simons theory at the fifth root of unity as the Fibonacci model is the standard identification, but this page has not checked it in a source, and the script does not reproduce the density theorem.

What passed, and a typo in the source

paths of n tau's: to charge 1 is Fib(n-1), to tau is Fib(n), n = 1..24;   Fib(n+1)/Fib(n) -> 1.618033988750
F unitary, F^2 = I, rows unit; R unitary
sigma_1 sigma_2 sigma_1 = sigma_2 sigma_1 sigma_2          gap 1.6e-16 (Fibonacci);  Ising: same relation
Ising closure mod phase: 24 elements, closes.    Fibonacci closure mod phase: > 20,000, does not close.
controls: sigma_2 = R F R breaks the braid relation; a wrong R phase breaks it;
          with e^{-i pi/7} in place of e^{-i pi/8} the Ising-style closure is NOT finite

A sign typo in the review, found by the braid relation. In Nayak et al., equation (133) and the three-anyon matrix give the τ-channel braiding phase as −e^−2πi/5 (= e^3πi/5, the value used here), but the sentence right after (133) reads R^τ_ττ = −e^+2πi/5; the PDF's own text layer shows the missing minus in that sentence. The braid relation settles it: with the review's F and R¹_ττ = e^−4πi/5, the value −e^−2πi/5 satisfies σ₁σ₂σ₁ = σ₂σ₁σ₂ to 10⁻¹⁶, and −e^+2πi/5 misses by 0.326.

Run it: python3 quantum-maths/qm5-verify.py (standard library only; the sourcing block says SKIP, never PASS, if the source PDF is not in ~/Downloads).