 Solution

This is the Deutsch-Jozsa test with an arbitrary uniform-state unitary, and does not require $N$ to be a power of two. Choose a known unitary operator $F$ such that $F|0⟩=|s⟩=N^{-1/2}{\sum }_i|i⟩$. Prepare the target qubit in $|-⟩=(|0⟩-|1⟩)/\sqrt{2}$. One Boolean-oracle call gives quantum phase kickback:Apply $F^{†}$ to the index register. The amplitude on $|0⟩$ isIt equals $+1$ or $-1$ for the two constant strings, and equals zero for a balanced string. Consequently a single query decides the promised problem exactly: measure the index register and report constant for outcome zero, balanced for any other outcome. This is the Deutsch-Jozsa algorithm with the available exact state-preparation operation.