Wavelet matrix operations and quantum transforms

Abstract

The currently studied version of the quantum wavelet transform implements the Mallat pyramid algorithm, calculating wavelet and scaling coefficients at lower resolutions from higher ones, via quantum computations. However, the pyramid algorithm cannot replace wavelet transform algorithms, which obtain wavelet coefficients directly from signals. The barrier to implementing quantum versions of wavelet transforms has been the fact that the mapping from sampled signals to wavelet coefficients is not canonically represented with matrices. To solve this problem, we introduce new inner products and norms into the sequence space l2(Z), based on wavelet sampling theory. We then show that wavelet transform algorithms using L2(R) inner product operations can be implemented in infinite matrix forms, directly mapping discrete function samples to wavelet coefficients. These infinite matrix operators are then converted into finite forms for computational implementation. Thus, via singular value decompositions of these finite matrices, our work allows implementation of the standard wavelet transform with a quantum circuit. Finally, we validate these wavelet matrix algorithms on MRAs involving spline and Coiflet wavelets, illustrating some of our theorems.