Exact Coset Sampling for Quantum Lattice Algorithms
Yifan Zhang
TL;DR
The paper presents Step $9^ obreakdash{\dagger}$, a corrected procedure that completes Chen's windowed-QFT lattice algorithm by resolving affine-offset-induced conflicts in Steps 8–9. By leveraging a center-referenced phase correction on the first coordinate and a retuned Karst-wave front end with width $\sigma_J\asymp Q\log n$, it enables exact Fourier sampling over $\mathbb{Z}_M^n$ that concentrates on samples with $\langle \mathbf{b},\bm u\rangle \equiv 0 \pmod Q$ and near-uniform distribution on the dual fibre. Under explicit front-end assumptions A1–A5, the step yields samples from which the LWE secret can be recovered via a CRT-based modular reconstruction in polynomial time, effectively extending Chen’s results to standard LWE. The work also clarifies the role of offset coherence and shows how a single-shot sampling strategy can be combined with multi-run access to achieve robust dual-sampling outcomes. Overall, it provides a concrete path to translating structured quantum lattice sampling into practical LWE solving within a rigorous parameter regime.
Abstract
In this work, we give a new completion of Chen's windowed-QFT lattice algorithm~\citep{chen2024quantum}. This extra step, called Step~$9^\dagger$, replaces the domain extension stage in Steps~8--9. The published Step~9 calls an amplitude periodicity lemma, yet its hypotheses break in the presence of affine offsets $\boldsymbol{v}^*$. Our analysis finds a basic conflict between two design constraints. The lattice problem asks for high spectral resolution, so the method prefers wide time windows. The quadratic phase error of the state prefers narrow time windows. Assumption~A5 packages the spectral concentration and near-uniformity properties that we require from the front end. Under~A5, a direct $\mathbb{Z}_M^n$ Fourier transform of the chirp-corrected coordinate state produces samples $\boldsymbol{u}$ that satisfy $\langle \boldsymbol{b}, \boldsymbol{u} \rangle \equiv 0 \pmod{Q}$ with probability $1-\mathrm{negl}(n)$ and are nearly uniform on the dual hyperplane $\{\boldsymbol{u} : \langle \boldsymbol{b}, \boldsymbol{u} \rangle \equiv 0 \pmod{Q}\}$. The new procedure does not require internal access to control wires. It uses the normalization $b_1=-1$ to apply a center-referenced phase correction directly on the first coordinate register. The scaling parameter $D$ ensures that this physical operation can be implemented by arithmetic on $X_1$ alone and does not read the hidden loop index. For Chen's complex-Gaussian Karst-wave window, we isolate a parameter regime, formalized in Assumption~A5, in which a polynomial retuning of the parameters gives a one-dimensional envelope for the loop index with width $σ_J \asymp Q\log n$.