Braiding and Computation

If the degenerate fusion space of non-abelian anyons is the memory, braids are the program. This lesson makes the computation-by-braiding picture concrete: how worldlines form braid-group elements, how those elements become unitary gates, and why the resulting gates are intrinsically protected.

The braid group as a circuit

Track $n$ anyons through time and their worldlines trace out strands in $2+1$ dimensions. The set of distinct ways to braid them is the braid group $B_n$ , generated by the elementary swaps $\sigma_i$ (exchange neighbors $i$ and $i+1$ counterclockwise) subject to

\sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1}, \qquad \sigma_i \sigma_j = \sigma_j \sigma_i \;\;(|i-j| \ge 2).

The first relation is the Yang–Baxter relation — the algebraic statement that the order of "third-Reidemeister" slides of strands does not matter. A computation is a word in the $\sigma_i$ ; reading the braid bottom-to-top is running the circuit.

From braids to unitaries

A topological model supplies a representation $\rho: B_n \to U(\mathcal{V}_n)$ sending each generator $\sigma_i$ to a unitary $\rho(\sigma_i)$ acting on the fusion space. These unitaries are built from two pieces of local data:

the $R$ -matrix, $R^{ab}_c$ , the phase/operator for exchanging neighbors $a,b$ that fuse to $c$ ;
the $F$ -matrix, $F^{abc}_d$ , the change of basis between the two ways of fusing three anyons ( $((ab)c)$ versus $(a(bc))$ ).

Braiding non-adjacent fusion channels requires first changing basis with $F$ , applying $R$ , then changing back. The product of generator unitaries for a chosen braid word is the gate the braid implements.

A worked phase: Ising $\sigma$ self-exchange

For Ising anyons the exchange of two $\sigma$ 's in a definite fusion channel is, up to an overall phase,

R_{\sigma\sigma} \;=\; e^{-i\pi/8} \begin{pmatrix} 1 & 0 \\ 0 & i \end{pmatrix} \quad\text{in the } \{1,\psi\}\ \text{fusion basis,}

so braiding two $\sigma$ 's applies a relative phase of $i = e^{i\pi/2}$ between the vacuum and fermion fusion channels. Composing such braids on four $\sigma$ 's (one qubit) generates the single-qubit Clifford rotations; an entangling two-qubit Clifford comes from braiding $\sigma$ 's belonging to different qubits. These braids alone yield the Clifford group — powerful, but not universal.

Why braided gates are protected

The unitary $\rho(\text{braid})$ depends only on the topological class of the braid — its pattern of over/under crossings — not on the speed, shape, or small wiggles of the worldlines. Two physical processes that are continuous deformations of each other implement exactly the same gate. There is no continuous "how far did I rotate" parameter to mis-set: the gate is digital and exact, set by integer crossing data. This is the topological origin of fault tolerance at the gate level, complementary to the error protection of the degenerate memory.

What to take away

A braiding computation is a word in the braid group $B_n$ , mapped by a representation $\rho$ — built from the $R$ - and $F$ -matrices — to a unitary on the fusion space. Because $\rho$ depends only on the topological class of the braid, the gates are exact and protected. Ising braids realize the Clifford group; achieving universality requires either a richer anyon (Fibonacci) or supplementing braids, as the next lessons discuss.