Classical collision query bound for Simon's problem ($\Theta (2^{n/2})$)

For an unknown nonzero period, a collision $f(x)=f(y)$ reveals the period $x\oplus y$. Birthday problem sampling finds one using $O(2^{n/2})$ queries with bounded error. A matching lower bound follows by choosing the period and output labels randomly: without a collision, $q$ queries rule out at most $\left(\genfrac{}{}{0ex}{}{q}{2}\right)$ candidate periods. For $q=o(2^{n/2})$, almost all periods remain consistent with the transcript. The exponential separation is in query complexity, compared with Simon's algorithm's linear quantum query count.