Exact Coset Sampling for Quantum Lattice Algorithms

We give a simple replacement for the contested "domain-extension" in Step 9 of a recent windowed-QFT lattice algorithm with complex-Gaussian windows (Chen, 2024). As acknowledged by the author, the reported issue is due to a periodicity/support mismatch when extending only the first coordinate in the presence of offsets, which breaks the intended $\mathbb{Z}_P$-fiber. Our new subroutine replaces domain extension by a pair-shift difference that cancels unknown offsets exactly and synthesizes a uniform cyclic subgroup (a zero-offset coset) of order $P$ inside $(\mathbb{Z}_{M_2})^n$. We adopt a gate-level access model and run a short prepass that measures the designated outcome registers (Chen's Steps 1, 3, and 5), fixing $E=(y',z',h^{\ast})$. We then identify a concrete program point $t^{\star}$ at which an index wire $J \in \mathbb{Z}_P$ is preserved and the coordinate block equals $\mathbf{X}(j)\equiv 2D^2 j\,\mathbf{b}^{\ast}+\mathbf{v}^{\ast}\ (\bmod M_2)$. A compute-copy-uncompute sandwich on the prefix up to $t^{\star}$ yields a reversible evaluator that we call only on basis inputs $j=0,1$ to harvest $V=\mathbf{X}(0)$ and $\Delta=\mathbf{X}(1)-\mathbf{X}(0)\equiv 2D^2\mathbf{b}^{\ast}$ within the same run. We never invert a measurement, and we do not claim the circuit suffix after $t^{\star}$. The default Step $9^{\dagger}$ uses only $\Delta$ (no foreknowledge of $\mathbf{b}^\ast$): set $\mathbf{Z}\leftarrow -\,T\cdot \Delta\ (\bmod M_2)$ for uniform $T\in\mathbb{Z}_P$ and erase $T$ coherently primewise by modular inversion and CRT.