Exact Coset Sampling for Quantum Lattice Algorithms

We give a simple, fully correct, and assumption-light replacement for the contested "domain-extension" in Step 9 of a recent windowed-QFT lattice algorithm with complex-Gaussian windows~\citep{chen2024quantum}. The published Step~9 suffers from a periodicity/support mismatch. We present a pair-shift difference construction that coherently cancels all unknown offsets, produces an exact uniform CRT-coset state over $\mathbb{Z}_{P}$, and then uses the QFT to enforce the intended modular linear relation. The unitary is reversible, uses $\mathrm{poly}(\log M_2)$ gates, and preserves the algorithm's asymptotics. Project Page: https://github.com/yifanzhang-pro/quantum-lattice.