Exact constant-depth adaptive protocols prepare any Dicke-(n,k) state with O(n^2 + k^2 log^2 n + k n log n log log n) ancillas, and any n-qubit symmetric state with O(n^3 sqrt(log n)) ancillas.
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Constant-depth adaptive preparation of Dicke and symmetric states
Exact constant-depth adaptive protocols prepare any Dicke-(n,k) state with O(n^2 + k^2 log^2 n + k n log n log log n) ancillas, and any n-qubit symmetric state with O(n^3 sqrt(log n)) ancillas.