Topological Quantum Codes

Anyons, braiding, and topologically protected logical operators

ANYON WORLD-LINES (click to add, drag to braid)

ANYON TYPES (Z₂ Toric Code)

The toric code supports 4 anyon types:

1 — vacuum (no anyon)
e — electric charge (end of X string)
m — magnetic flux (end of Z string)
ε — fermion (e × m, composite)

e × e = 1 m × m = 1 e × m = ε ε × ε = 1 Braiding e around m → phase: e^{iπ} = −1 (mutual statistics)

● e-anyon
bosonic self

● m-anyon
bosonic self

● ε = e×m
fermionic

○ vacuum
trivial

Place anyons and braid them to see the phase.

TORIC CODE — qubits on edges of torus

TORIC CODE STRUCTURE

The toric code is defined on a torus (periodic boundary). Qubits live on edges of a square lattice.

Vertex operators Aᵥ: XXXX on 4 edges meeting at vertex v

Plaquette operators Bₚ: ZZZZ on 4 edges of face p

Hamiltonian: H = −Σ Aᵥ − Σ Bₚ

Ground state: simultaneous +1 eigenstate of all A and B.

Logical operators: non-contractible loops around the torus. Two independent loops → encode 2 logical qubits!

Topological gap: Error must span full system to flip logical qubit. No local perturbation can do damage. Code distance = L (system size)

FIBONACCI ANYONS — UNIVERSAL QUANTUM COMPUTING

Fibonacci anyons have just 2 types: 1 and τ.

Fusion rules:

τ × τ = 1 + τ 1 × τ = τ 1 × 1 = 1 F-matrix (associativity): φ = (1+√5)/2 (golden ratio) [F^τττ_τ] = [φ⁻¹ φ⁻½ ] [φ⁻½ -φ⁻¹] R-matrices (braiding): R¹_ττ = e^{-i4π/5} R^τ_ττ = e^{i3π/5}

3 τ-anyons can store 1 qubit in fusion channel. Braiding implements dense unitary gates — universal quantum computation!

Key property: local decoherence cannot distinguish fusion channels → topological protection.