  We introduce a simple, experimentally realizable protocol that can prepare
any specific superposition of permutationally invariant qubit states, also
known as Dicke states. The protocol is comprised entirely of global rotations
and globally applied non-linear phase gates -- it does not require local
addressability or ancilla qubits -- and hence can be readily implemented in a
variety of experimental platforms, including trapped-ion quantum simulators and
cavity QED systems. We demonstrate the utility of our protocol by numerically
preparing several states with theoretical infidelities $1-\mathcal{F}<10^{-4}$:
(i) metrologically useful $N$-qubit Dicke states in $\mathcal{O}(1)$ gate
steps, (ii) the $N = 9$ qubit codewords of the Ruskai code with $P = 4$ gate
steps, and (iii) the $N = 13$ qubit Gross codewords in $P = 7$ gate steps.
Focusing on trapped-ion platforms, we estimate that the protocol achieves
fidelities $\gtrsim 95\%$ in the presence of typical experimental noise levels,
thus providing a pathway to the preparation of a variety of useful
highly-entangled quantum states.
