  The Fibonacci topological order is the simplest platform for a universal
topological quantum computer, consisting of a single type of non-Abelian anyon,
$\tau$, with fusion rule $\tau\times\tau=1+\tau$. While it has been proposed
that the anyon spectrum of the $\nu=12/5$ fractional quantum Hall state
includes a Fibonacci sector, a dynamical picture of how a pure Fibonacci state
may emerge in a quantum Hall system has been lacking. Here we use recently
proposed non-Abelian dualities to construct a Fibonacci state of bosons at
filling $\nu=2$ starting from a trilayer of integer quantum Hall states. Our
parent theory consists of bosonic "composite vortices" coupled to fluctuating
$U(2)$ gauge fields, which is related to the standard theory of Laughlin
quasiparticles by duality. The Fibonacci state is obtained by clustering the
composite vortices between the layers, along with flux attachment, a procedure
reminiscent of the clustering picture of the Read-Rezayi states. We further use
this framework to motivate a wave function for the Fibonacci fractional quantum
Hall state.
