One finite abelian hidden-subgroup Fourier sample

Inputs are a hiding instance from Finite abelian hidden-subgroup promise and output encoding, a fresh $b$-qubit group register and supplied exact circuits with declared access/costs, where $b$ is the group-code width in that definition:

• $P_G$ prepares the normalized uniform state $\mathrm{\mid }G⟩=N^{-1\mathrm{/}2}{\sum }_{g\in G}\mathrm{\mid }{\mathrm{code}}_G(g)⟩$ on valid group labels and returns its workspace to the declared initial state.
• $O_F$ is the supplied coherent XOR oracle for the total extension $\tilde{F}$ in Finite abelian hidden-subgroup promise and output encoding, acting on the group register and a fresh $m$-qubit output register initialized to the all-zero word $0^m$.
• $Q_G$ is the supplied exact product QFT, with the fixed negative-character convention $Q_G\mathrm{\mid }{\mathrm{code}}_G(g)⟩=N^{-1\mathrm{/}2}{\sum }_{\mathbf{k}}\overline{{\chi }_{\mathbf{k}}(g)}\mathrm{\mid }{\mathrm{code}}_{\widehat{G}}(\mathbf{k})⟩$ on the valid group subspace. It is supplied as a unitary on the full $b$-qubit register, so any extension from the valid-label subspace is part of the circuit input.

If $N=1$, return the unique trivial character without a query. Otherwise:

1. Prepare $\mathrm{\mid }G⟩$ with $P_G$.
2. Apply $O_F$ once to the group register and fresh output register.
3. Discard the output register by the declared partial-trace operation; do not measure it.
4. Apply $Q_G$ to the group register and measure the full $b$-qubit register in its ordered computational basis; the prescribed QFT action places the state in the valid character-code subspace.
5. Decode the measured valid character code as the tuple $\mathbf{k}$ and return ${\chi }_{\mathbf{k}}$.

This procedure assumes the preparation, oracle, and QFT circuits are supplied exactly. It makes no existence, uniformity, or efficiency claim for those circuits.