One Simon Fourier-sampling sample

Input: Integers $n\ge 1$, $m\ge n-1$, and ordinary access through the Standard bit-flip query convention to a total function satisfying Simon hidden-XOR-period promise. Prepare an ordered $n$-qubit address register and $m$-qubit target register, both in their zero computational-basis states.

Instructions:

1. Apply the exact one-qubit Hadamard gate to every address wire, producing $H^{\otimes n}\mathrm{\mid }{0}^n⟩$, and leave the target at $\mathrm{\mid }{0}^m⟩$.
2. Apply the standard bit-flip oracle $U_f\mathrm{\mid }x⟩\mathrm{\mid }z⟩=\mathrm{\mid }x⟩\mathrm{\mid }z\oplus f(x)⟩$ once. Measure the target with the computational-basis projective measurement Finite projective measurements, viewed as the finite POVM Finite POVMs and their outcome law. Use the Lüders branch maps ${\mathcal{I}}_v(X)=P_{v}X{P}_v$, where $P_v=I_{\mathrm{a}\mathrm{d}\mathrm{d}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{s}}\otimes \mathrm{\mid }v⟩⟨v\mathrm{\mid }$, under the instrument and conditional-state interface Finite instruments and conditional states. Theorem The square-root Lüders maps form an instrument supplies a valid instrument; its outcome weights and positive-branch states are covered by Instrument outcome weights normalize and Positive-probability instrument branches are density operators. Discard the reported value and target register. The POVM effects are these same $P_v$.
3. Apply $H^{\otimes n}$ to the address register and measure it in the computational basis. Return the observed $n$-bit string $y$ as one sample row.

Every branch uses exactly one oracle query and $2n$ Hadamard gates. It makes one finite instrument measurement on the target and one terminal projective measurement on the address. The intermediate target outcome is not used by classical postprocessing. There is no failure branch, controlled-oracle call, or inverse-oracle call. This procedure's sample law is proved separately in Simon samples uniformly from s-perp.