Approximate quantum counting ($k=|\{x:f(x)=1\}|$)

Approximate quantum counting estimates the number $k$ of satisfying assignments of a Boolean function, using a quantum oracle and quantum measurement. Controlled repetitions of a Grover search algorithm iteration let quantum interference reveal the count. For one input bit, a Hadamard test can give Pauli Z gate expectation value $1-k$: with the complemented phase oracle $U_f=\mathrm{diag}(2f(0)-1,2f(1)-1)$ and $G=-X{U}_f$, use $⟨+|X=⟨+|$ to obtain $⟨+|G|+⟩=-[(2f(0)-1)+(2f(1)-1)]/2=1-k$. Repeated preparations estimate this expectation value; one binary measurement result does not distinguish all three counts $0,1,2$.