Non-Abelian Anyons

Non-abelian anyons are the prize of the field. Unlike abelian anyons, which respond to braiding with a mere phase, a fixed collection of non-abelian anyons spans a degenerate Hilbert space, and braiding implements noncommuting unitary matrices on that space. This degeneracy is the protected memory, and the braids are the protected gates, of a topological quantum computer.

A topological degeneracy with no local witness

Place $n$ identical non-abelian anyons at well-separated, fixed positions. Their joint state is not unique: it lives in a space $\mathcal{V}_n$ whose dimension grows with $n$ . Crucially, no local measurement near any single anyon can reveal which state of $\mathcal{V}_n$ the system occupies — the information is stored nonlocally, in the collective fusion data of the whole set. That is precisely what makes it robust: local noise cannot read or corrupt it.

Quantum dimension

The growth rate of $\dim \mathcal{V}_n$ defines the quantum dimension $d_a$ of an anyon type $a$ :

\dim \mathcal{V}_n \;\sim\; d_a^{\,n} \quad (n \to \infty).

Abelian anyons have $d_a = 1$ (no degeneracy growth). A genuinely non-abelian anyon has $d_a > 1$ , and in general $d_a$ need not be an integer. The total quantum dimension is $\mathcal{D} = \sqrt{\sum_a d_a^2}$ , the quantity that sets the topological entanglement entropy $\gamma = \ln\mathcal{D}$ from the first lesson.

Two canonical models

Ising anyons. Three sectors $\{1, \sigma, \psi\}$ with $\sigma$ the non-abelian anyon. The fusion rules include $\sigma \times \sigma = 1 + \psi$ — two $\sigma$ 's can fuse to either vacuum or a fermion — which is the hallmark of a multiple fusion channel. The quantum dimension is

d_\sigma = \sqrt{2}, \qquad d_\psi = 1, \qquad \mathcal{D} = \sqrt{1^2 + (\sqrt2)^2 + 1^2} = 2.

Ising anyons arise for Majorana zero modes and the Moore–Read $\nu=5/2$ quantum Hall state. Their braids generate the Clifford gates but are not universal by braiding alone.

Fibonacci anyons. Two sectors $\{1, \tau\}$ with $\tau \times \tau = 1 + \tau$ . The dimension obeys $d_\tau^2 = 1 + d_\tau$ , giving the golden ratio

d_\tau = \varphi = \frac{1+\sqrt5}{2} \approx 1.618 .

Fibonacci anyons are remarkable because braiding them is computationally universal — any quantum gate can be approximated to arbitrary accuracy by an appropriate braid.

Why braiding is matrix-valued

With several fusion channels available, the multi-anyon state is labeled by the intermediate fusion outcomes (a fusion tree). Braiding two anyons rotates the system within this degenerate basis. Two such operations applied in different orders generically give different results,

R_{12}\,R_{23} \neq R_{23}\,R_{12},

because they act as distinct noncommuting unitaries — the source of "non-abelian." The matrices must be consistent: they obey the Yang–Baxter / braid relations and are fixed by the underlying fusion data ( $F$ - and $R$ -matrices), the subject of the fusion-rules lesson.

What to take away

Non-abelian anyons carry a nonlocal, locally unmeasurable degeneracy quantified by a quantum dimension $d_a>1$ (e.g. $\sqrt2$ for Ising $\sigma$ , the golden ratio for Fibonacci $\tau$ ). Braiding acts as noncommuting unitaries on that space, providing protected memory and gates. Ising braiding gives the Clifford group; Fibonacci braiding is universal.