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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.

Use the available unitaries to preparein the first register andin the answer qubit. The quantum phase kickback identity givesChoose a unitary $F$ with $F|0⟩=|s⟩$, apply $F^{-1}$ to the first register, and measure it. Its amplitude at $|0⟩$ isThis equals $+1$ or $-1$ for a constant string and zero for a balanced string. Thus outcome $0$ means constant, while every other outcome means balanced, with certainty after one oracle query. This is the Deutsch-Jozsa test with an arbitrary uniform-state unitary.