The curvelet transform is a special type of wavelet transform, which is useful for estimating the locations and orientations of waves propagating in Euclidean space. We prove an uncertainty principle that lower-bounds the variance of these estimates, for radial wave functions in n dimensions. As an application of this uncertainty principle, we show the infeasibility of one approach to constructing quantum algorithms for solving lattice problems, such as the approximate shortest vector problem (approximate-SVP), and bounded distance decoding (BDD). This gives insight into the computational intractability of approximate-SVP, which plays an important role in algorithms for integer programming, and in post-quantum cryptosystems. In this approach to solving lattice problems, one prepares quantum superpositions of Gaussian-like wave functions centered at lattice points. A key step in this procedure requires finding the center of each Gaussian-like wave function, using the quantum curvelet transform. We show that, for any choice of the Gaussian-like wave function, the error in this step will be above the threshold required to solve BDD and approximate-SVP.

翻译：曲线变换是一种特殊类型的小波变换，可用于估计欧几里得空间中传播波的定位与方向。我们证明了该估计方差下界的不确定性原理，适用于n维径向波函数。基于该不确定性原理，我们展示了构建求解格问题（如近似最短向量问题（approximate-SVP）和有界距离解码（BDD））的量子算法的一种方法不可行性。这揭示了近似-SVP的计算难解性——该问题在整数规划算法和后量子密码系统中具有重要作用。在该求解格问题的方法中，需制备以格点为中心的高斯型波函数的量子叠加态。该程序的关键步骤需要利用量子曲线变换找到每个高斯型波函数的中心。我们证明：对任何高斯型波函数的选取，该步骤的误差将超过求解BDD和approximate-SVP所需的阈值。