Exact Coset Sampling for Quantum Lattice Algorithms

We revisit the post-processing phase of Chen's Karst-wave quantum lattice algorithm (Chen, 2024) in the Learning with Errors (LWE) parameter regime. Conditioned on a transcript $E$, the post-Step 7 coordinate state on $\left(\mathbb{Z}/M\mathbb{Z}\right)^n$ is supported on the shifted grid line $\left\{\, jΔ+ \mathbf{v}^\ast(E) + M_2 \mathbf{k} \bmod M \;:\; j\in\mathbb{Z},\ \mathbf{k}\in\mathcal{K} \,\right\}$, where $Δ= 2D^2\mathbf{b}$ and $M = 2M_2 = 2D^2 Q$ with $Q$ odd. The amplitudes contain a quadratic chirp $\exp\!\left(-2πi\, j^2/Q\right)$ and an unknown run-dependent offset $\mathbf{v}^\ast(E)$. We show that Chen's Steps~8-9 can be replaced by a single routine, Step $9^\dagger$: measure $τ:= X_1 \bmod D^2$, apply a $τ$-dependent quadratic phase on the first coordinate register to remove the chirp, and then apply $\mathrm{QFT}_{\mathbb{Z}/M\mathbb{Z}}^{\otimes n}$ to the coordinate registers. The chirp removal uses a canonical gauge promise: conditioned on $E$, the offset $v_1^\ast(E) \bmod M_2$ equals the centered lift $\operatorname{ctr}(τ)$. Under explicit front-end conditions (AC1-AC5), a Fourier sample $\mathbf{u}\in\left(\mathbb{Z}/M\mathbb{Z}\right)^n$ satisfies the resonance $\langle \mathbf{b}, \mathbf{u} \rangle \equiv 0 \pmod Q$ with probability $1-\operatorname{negl}(n)$, and conditioned on resonance the reduced sample $\mathbf{u}\bmod Q$ is exactly uniform on the dual hyperplane. This yields exact recovery of $\mathbf{b}\bmod Q$ and hence the embedded LWE secret.