We give a general proof for the existence and realizability of Clifford gates in the Ising topological quantum computer. We show that all quantum gates that can be implemented by braiding of Ising anyons are Clifford gates. We find that the braiding gates for two qubits exhaust the entire two-qubit Clifford group. Analyzing the structure of the Clifford group for $n\ensuremath{\ge}3$ qubits we prove that the image of the braid group is a nontrivial subgroup of the Clifford group so that not all Clifford gates could be implemented by braiding in the Ising topological quantum computation scheme. We also point out which Clifford gates cannot in general be realized by braiding.
We demonstrate that the two inequivalent spinor representations of the braid group , describing the exchanges of 2 n +2 non-Abelian Ising anyons in the Pfaffian topological quantum computer, are equivalent from the computational point of view, i.e., the sets of topologically protected quantum gates that could be implemented in both cases by braiding exactly coincide. We give the explicit matrices generating almost all braidings in the spinor representations of the 2 n +2 Ising anyons, as well as important recurrence relations. Our detailed analysis allows us to understand better the physical difference between the two inequivalent representations and to propose a process that could determine the type of representation for any concrete physical realization of the Pfaffian quantum computer.
FINDING: Fibonacci anyons — whose braiding statistics are governed by the golden ratio — have been experimentally realized (Quantinuum) and theoretically compiled into universal single-qubit gates, confirming the golden ratio as a physical constant of topological quantum computation. | MATH: Fibonacci anyon quantum dimension d=ϕ=21+5≈1.618; fusion rule τ×τ=1+τ (non-Abelian); braid matrices satisfy σi2=1 but Yang-Baxter σiσi+1σi=σi+1σiσi+1; single-qubit gate compilation uses F-matrix with entries involving ϕ−1=ϕ−1≈0.618 and ϕ−1/2. The Ising anyon has d=2, but Fibonacci is universal for topological QC while Ising is not (without magic state distillation). | CONNECTION: Direct geometric harmony — the golden ratio ϕ appears as the quantum dimension, and its inverse ϕ−1=0.618 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com