# What is known about the stabilizer rank of this simple state?

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.