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，可用于估计欧氏空间中波传播的位置和方向。我们针对n维径向波函数证明了一个不确定性原理，该原理为这些估计的方差设定了下界。作为该不确定性原理的应用，我们展示了一种构造量子算法求解格问题（如近似最短向量问题和有界距离解码）的方法的不可行性。这揭示了近似最短向量问题计算难解性的内在机理——该问题在整数规划算法和后量子密码系统中均具有重要地位。在该格问题求解方法中，需要制备以格点为中心的类高斯波函数的量子叠加态，其关键步骤是通过量子曲线变换定位每个类高斯波函数的中心。我们证明：无论选择何种形式的类高斯波函数，该步骤的误差都将超过求解有界距离解码和近似最短向量问题所需的阈值。