Quantum Algorithms Using the Curvelet Transform

TL;DR

This paper introduces a quantum implementation of the curvelet transform and demonstrates its potential for efficient algorithms to locate centers of geometric objects and functions in high-dimensional spaces, suggesting a quantum speed-up.

Contribution

It provides the first efficient quantum algorithm for the curvelet transform and applies it to develop new quantum algorithms for geometric and functional center-finding tasks.

Findings

Quantum curvelet transform can be implemented efficiently.

Algorithms succeed with constant probability independent of dimension.

Supports conjecture of quantum speed-up for high-dimensional problems.

Abstract

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.