# How many single and double qubit gates are required to create a uniform superposition of vertices of a Johnson Graph J(n,r)?

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.

One can create this uniform superposition of vertices of Johnson graph $$J(n,r)$$ for any $$r=n^b$$ with $$0 in $$\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