From the Verlinde Formula to Quantum Gates

Braiding Fibonacci anyons, derived from the modular data of SU(2)₃ and checked numerically at every step.

This repository follows a single chain of ideas from conformal field theory to quantum computation:

S-matrix of SU(2)_k  →  fusion rules (Verlinde)  →  F and R symbols (pentagon, hexagon)
                     →  braid representation on 3 anyons  →  gate approximation by braid search

The Verlinde formula turns the S-matrix of SU(2)_k into fusion rules; at level k = 3 the integer-spin sector is the Fibonacci category, with the single fusion rule τ ⊗ τ = 1 ⊕ τ. Its associativity and braiding data reduce to a 2×2 matrix F and two phases R₁, R_τ, fixed by the pentagon and hexagon equations. Three τ anyons with total charge τ span a qubit on which the braid group B₃ acts, densely in PU(2). A brute-force search over braid words then approximates the Hadamard, NOT and T gates.

Contents

File	What it is
fibonacci_anyons.py	The whole computation in one file: Verlinde formula, F/R symbols, braid representation, gate search
fibonacci-anyons.tex / .pdf	Expository note (15 pages) with full derivations and proofs
hadamard_error.png	Approximation error vs braid length, produced by the script

Running the code

Requires Python 3 and NumPy; Matplotlib only for the plot.

python fibonacci_anyons.py               # all checks + braid search to length 13 (seconds)
python fibonacci_anyons.py -L 16 --plot  # deeper search, saves hadamard_error.png
python fibonacci_anyons.py -L 20         # a few minutes, ~2.6 million distinct gates

The script runs four parts and asserts every identity it relies on:

1. Verlinde formula. Builds the S-matrix of SU(2)_k for k = 1…6, computes fusion multiplicities N_ab^c = Σ_x S_axS_bxS̄_cx/S_0x, and checks they are non-negative integers matching the truncated Clebsch–Gordan rule. At k = 3 it extracts τ ⊗ τ = 1 ⊕ τ and d_τ = φ.
2. F and R symbols. Verifies the pentagon equation over all 2⁹ label assignments and both hexagon equations over all 2⁶, to ~10⁻¹⁶. Also shows that perturbing φ or R_τ breaks them: the golden ratio is forced, not chosen.
3. Braid representation. Constructs σ₁ = diag(R₁, R_τ) and σ₂ = F σ₁ F, checks unitarity, the braid relation, σ₁σ₂σ₁ = R₁F, the full twist (σ₁σ₂)³ = e^2πi/5I, and that fusion-space dimensions are Fibonacci numbers.
4. Gate search. Breadth-first search over braid words, deduplicated by gate (up to global phase), reporting the best approximation to H, X and T at each length bound.

Results

Best error d(U, V) = √(1 − |tr(U^†V)|/2) among braids of length ≤ L:

L	distinct gates	H	X	T
3	44	0.084	0.275	0.056
9	2,516	0.084	0.080	0.053
13	31,561	0.021	0.080	0.053
14	59,367	0.021	0.048	0.023
20	2,645,329	0.012	—	—

Two of the short optima have closed forms: the length-3 approximation of H is the half twist σ₁σ₂σ₁ = R₁F, and the length-3 approximation of T is σ₁³, which acts as diag(1, e^iπ/5) up to phase. The number of distinct gates grows like ~1.9^L, so the error decays roughly like |gates|^−1/3, the ε-net scaling for the 3-dimensional group SO(3).

Conventions

• Labels a = 0, …, k for SU(2)_k, with a = 2 × spin; in the Fibonacci category, 0 = 1 and 1 = τ.
• F- and R-symbols follow Bonderson's thesis: |(ab)_e c; d⟩ = Σ_f [F^abc_d]_ef |a (bc)_f; d⟩. Symbols with a forbidden channel are 0 and symbols with a vacuum label are 1, so the pentagon and hexagon can be checked by looping over all labels.
• F = [[φ⁻¹, φ^−1/2], [φ^−1/2, −φ⁻¹]], R₁ = e^−4πi/5, R_τ = e^3πi/5 (the choice matching the conformal weight h = 2/5 of the spin-1 primary of ŝu(2)₃).
• Braid words are read left to right and mapped to matrix products in the same order.

Possible extensions

• General level k via quantum 6j-symbols; universality holds for k = 3 and k ≥ 5.
• Two-qubit gates with six anyons and leakage measurement (Bonesteel–Hormozi–Zikos–Simon).
• Meet-in-the-middle search to reach braid lengths of 30 or more.

References

• J. Preskill, Lecture Notes for Physics 219, Chapter 9 (topological quantum computation).
• P. Bonderson, Non-Abelian Anyons and Interferometry, PhD thesis, Caltech, 2007.
• E. Verlinde, "Fusion rules and modular transformations in 2D conformal field theory," Nucl. Phys. B 300 (1988).
• M. Freedman, M. Larsen, Z. Wang, "A modular functor which is universal for quantum computation," Comm. Math. Phys. 227 (2002).
• N. Bonesteel, L. Hormozi, G. Zikos, S. Simon, "Braid topologies for quantum computation," Phys. Rev. Lett. 95 (2005).
• E. Rowell, Z. Wang, "Mathematics of topological quantum computing," Bull. AMS 55 (2018).

License

MIT