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Consider the uniform superposition of all length-$n$ bit-strings of Hammming weight $w$, $$ |\phi_w\rangle =\sum_{x\in \{0,1\}^n,|x|=w} |x\rangle$$

What is known or conjectured about the stabilizer rank of such states? Clearly it is at most $\binom{n}{w}$, because the decomposition above is, in particular, a decomposition into $\binom{n}{w}$ stabilizer states.

If I understand correctly, there are currently no techniques for proving exponential lower bounds on the stabilizer rank of specific states, so I would also be interested references which merely speculate on the stabilizer rank of this (or related) states.

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