Anyon Braiding — Topological Quantum Gates

Braid Generators (4 anyons)

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Topological Qubit State

(empty braid = identity)

|ψ⟩ = 1.00|0⟩

P(1) = 100%P(ψ) = 0%

Crossings applied

Braid group elt.

How it works

Four non-Abelian Ising anyons pinned to a plane trace worldlines upward through time; the vertical block is a (2+1)D spacetime diagram. Exchanging neighbouring anyons i and i+1 is a generator σ_i of the braid group B₄, and σ_iσ_i+1σ_i = σ_i+1σ_iσ_i+1 — the Yang–Baxter relation.

The 4 anyons together host a 2-dimensional space of fusion outcomes, encoding one topologically protected qubit. In the standard alternating fusion-tree basis the elementary Ising-anyon braid matrices are:

B(odd)  = e^(-iπ/8) · diag(1, i)
B(even) = F · B(odd) · F,   F = (1/√2)[[1, 1],[1,-1]]

Each button applies B(σᵢ) (or its inverse, the conjugate transpose) to the qubit's 2-vector and appends the matching over/under crossing to the worldlines. Because these matrices only depend on which strands cross and how many times — not on wobbles, speed, or exact position — the resulting gate is a topological invariant of the braid: two braids that can be smoothly deformed into each other (same braid-group element) always apply the exact same gate. This is the core promise of topological quantum computation: local noise that jiggles anyon paths without changing the crossing pattern cannot corrupt the logical state.

σᵢ / σᵢ⁻¹ — swap the anyons currently in slots i, i+1 (over vs. under crossing) and apply the corresponding unitary.
Braid word — the ordered list of generators applied so far, exactly as it would appear in a braid-group presentation.
Reset braid — clears the worldlines and returns the qubit to |1⟩ (vacuum fusion channel).

Real systems pursuing this: Majorana zero modes at the ends of topological superconducting nanowires (Microsoft's approach) and non-Abelian quasiparticles in the ν = 5/2 fractional quantum Hall state are the leading experimental candidates for Ising anyons.

anyon 1 anyon 2 anyon 3 anyon 4

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