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

Efficient State Preparation for Metrology and Quantum Error Correction with Global Control