Super-Constant Weight Dicke States in Constant Depth Without Fanout

An $n$-qubit Dicke state of weight $k$, is the uniform superposition over all $n$-bit strings of Hamming weight $k$. Dicke states are an entanglement resource with important practical applications in the NISQ era and, for instance, play a central role in Decoded Quantum Interferometry (DQI). Furthermore, any symmetric state can be expressed as a superposition of Dicke states. First, we give explicit constant-depth circuits that prepare $n$-qubit Dicke states for all $k \leq \text{polylog}(n)$, u