%0 Journal Article
%J Proc. ACM Symposium on Theory of Computing (STOC)
%D 2009
%T Quantum Algorithms Using the Curvelet Transform
%A Yi-Kai Liu
%X The curvelet transform is a directional wavelet transform over R^n, which is used to analyze functions that have singularities along smooth surfaces (Candes and Donoho, 2002). I demonstrate how this can lead to new quantum algorithms. I give an efficient implementation of a quantum curvelet transform, together with two applications: a single-shot measurement procedure for approximately finding the center of a ball in R^n, given a quantum-sample over the ball; and, a quantum algorithm for finding the center of a radial function over R^n, given oracle access to the function. I conjecture that these algorithms succeed with constant probability, using one quantum-sample and O(1) oracle queries, respectively, independent of the dimension n -- this can be interpreted as a quantum speed-up. To support this conjecture, I prove rigorous bounds on the distribution of probability mass for the continuous curvelet transform. This shows that the above algorithms work in an idealized "continuous" model.
%B Proc. ACM Symposium on Theory of Computing (STOC)
%P 391-400
%8 2009/10/28
%G eng
%U http://arxiv.org/abs/0810.4968v2
%! Proc. ACM Symposium on Theory of Computing (STOC)