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Can I in $\tilde{O}(r)$ number of gates (single and double qubit) create a uniform superposition of vertices of Johnson Graph $J(n,r)$? I would like to create a state $|\psi\rangle = \frac{1}{\sqrt{n \choose r}} \sum_{|S|=r, S \subset [n]}|S\rangle$ where $S$ denotes an $r$-sized tuple of indices in ascending order of their value.

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One can create this uniform superposition of vertices of Johnson graph $J(n,r)$ for any $r=n^b$ with $0<b<1$ in $\tilde{O}(r)$ gates using the gate construction mentioned in Andras Gilyen's Master's thesis https://web.cs.elte.hu/blobs/diplomamunkak/msc_mat/2014/gilyen_andras_pal.pdf

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