Quantum Wave Atom Transforms

TL;DR

This paper introduces the first quantum algorithm for wavelet packet transforms with a parabolic scaling tree, enabling faster computations for differential operators and PDEs compared to classical methods.

Contribution

It develops a quantum algorithm for wave atom transforms with a flexible tree structure, significantly improving efficiency over classical implementations.

Findings

Quantum algorithm has O(poly(n)) gate complexity for 2^n-dimensional transforms.

Enables faster quantum solutions for hyperbolic PDEs like wave equations.

Supports a broader class of wavelet and wave atom transforms.

Abstract

This paper constructs the first quantum algorithm for wavelet packet transforms with a "parabolic scaling" tree structure, sometimes called wave atom transforms. Classically, wave atoms are used to construct sparse representations of differential operators, which enable fast numerical algorithms for partial differential equations. Compared to previous work, our quantum algorithm can implement a larger class of wavelet and wave atom transforms, by using an efficient representation for a larger class of possible tree structures. Our quantum implementation has O(poly(n)) gate complexity for applying a transform of dimension 2^n, while classical implementations use O(n*2^n) floating point operations. The result can be used to improve existing quantum algorithms for solving hyperbolic partial differential equations, such as wave equations.