Uniform coprime state for a semiprime ($|{\xi }_N⟩=\phi (N{)}^{-1/2}{\sum }_{1\le k<N,\gcd (k,N)=1}|k⟩$)

For a semiprime $N=pq$ with distinct known primes, this quantum state is uniform over the reduced residue classes. With $n$ binary digits in $N$, its fraction among all $2^n$ computational labels is $a=(p-1)(q-1)/2^n\ge 1/4$. An ancilla qubit whose one probability is $1/(4a)$ dilutes the joint good probability to exactly $1/4$. A reversible range-and-greatest common divisor predicate marks the good labels with ancilla qubit one, and a single exact amplitude amplification iteration produces $|{\xi }_N⟩|1⟩$. Known prime factors provide the cardinality; the membership test itself needs only $N$. Polynomial gate complexity presumes ideal one-qubit rotations at the specified amplitudes.