Improved bounds on stabilizer extent and Clifford rank

We prove that every pure state of stabilizer rank at most $k$ has stabilizer extent at most $2^{O(\sqrt{k\log(k+1)})}$, and establish the analogous bound for the squared Clifford coefficient norm of Clifford-rank-$k$ operators. This implies stabilizer fidelity at least $2^{-O(\sqrt{k\log(k+1)})}$, r