Exact Coset Sampling for Quantum Lattice Algorithms

We give a simple and provably correct replacement for the contested ``domain-extension'' in Step 9 of a recent windowed-QFT lattice algorithm with complex-Gaussian windows~\citep{chen2024quantum}. As acknowledged by the author, the reported issue is due to a periodicity/support mismatch when applying domain extension to only the first coordinate in the presence of offsets. Our drop-in subroutine replaces domain extension by a pair-shift difference that cancels all unknown offsets exactly and synthesizes a uniform cyclic subgroup (a zero-offset coset) of order $P$ inside $(\mathbb{Z}_{M_2})^n$. A subsequent QFT enforces the intended modular linear relation by plain character orthogonality. The sole structural assumption is a residue-accessibility condition enabling coherent auxiliary cleanup; no amplitude periodicity is used. The unitary is reversible, uses $\mathrm{poly}(\log M_2)$ gates, and preserves upstream asymptotics.