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 $(\mathbb{Z}_M)^n$ is supported on an affine grid line $\{\, jΔ+ v^{\ast}(E) + M_2 k \bmod M : j \in \mathbb{Z},\ k \in \mathcal{K} \,\}$, with $Δ= 2D^2 b$, $M = 2M_2 = 2D^2 Q$, and $Q$ odd. The amplitudes include a quadratic Karst-wave chirp $\exp(-2πi j^2 / Q)$ and an unknown run-dependent offset $v^{\ast}(E)$. We show that Chen's Steps 8-9 can be replaced by a single exact post-processing routine: measure the deterministic residue $τ:= X_1 \bmod D^2$, obtain the run-local class $v_{1,Q} := v_1^{\ast}(E) \bmod Q$ as explicit side information in our access model, apply a $v_{1,Q}$-dependent diagonal quadratic phase on $X_1$ to cancel the chirp, and then apply $\mathrm{QFT}_{\mathbb{Z}_M}^{\otimes n}$ to the coordinate registers. The routine never needs the full offset $v^{\ast}(E)$. Under Additional Conditions AC1-AC5 on the front end, a measured Fourier outcome $u \in \mathbb{Z}_M^n$ satisfies the resonance $\langle b, u \rangle \equiv 0 \pmod Q$ with probability $1 - o(1)$. Moreover, conditioned on resonance, the reduced outcome $u \bmod Q$ is exactly uniform on the dual hyperplane $H = \{\, v \in \mathbb{Z}_Q^n : \langle b, v \rangle \equiv 0 \pmod Q \,\}$.

翻译：本文重新审视了Chen的喀斯特波量子格算法（Chen, 2024）在学习带误差（LWE）参数体系下的后处理阶段。在给定迹$E$的条件下，步骤7后$(\mathbb{Z}_M)^n$上的坐标态支撑在一个仿射网格线$\{\, jΔ+ v^{\ast}(E) + M_2 k \bmod M : j \in \mathbb{Z},\ k \in \mathcal{K} \,\}$上，其中$Δ= 2D^2 b$，$M = 2M_2 = 2D^2 Q$，且$Q$为奇数。振幅包含一个二次喀斯特波线性调频项$\exp(-2πi j^2 / Q)$和一个未知的运行相关偏移量$v^{\ast}(E)$。我们证明Chen算法的步骤8-9可被一个单一精确后处理流程替代：测量确定性余数$τ:= X_1 \bmod D^2$，在我们的访问模型中获取作为显式边信息的运行局部类$v_{1,Q} := v_1^{\ast}(E) \bmod Q$，对$X_1$施加一个依赖于$v_{1,Q}$的对角二次相位以消除线性调频项，随后对坐标寄存器应用$\mathrm{QFT}_{\mathbb{Z}_M}^{\otimes n}$。该流程始终不需要完整的偏移量$v^{\ast}(E)$。在前端满足附加条件AC1-AC5的情况下，测得的傅里叶结果$u \in \mathbb{Z}_M^n$以$1 - o(1)$的概率满足共振条件$\langle b, u \rangle \equiv 0 \pmod Q$。此外，在共振条件下，约化结果$u \bmod Q$在对偶超平面$H = \{\, v \in \mathbb{Z}_Q^n : \langle b, v \rangle \equiv 0 \pmod Q \,\}$上严格均匀分布。