Abstract:

An important computational tool in quantum information is the quantum Fourier transform, with its broad set of applications in this area. We discuss extending the scope of quantum transforms, and consider in particular the wavelet transform. Currently studied implementations of quantum wavelet transforms encode only the Mallat pyramid algorithm, calculating wavelet and scaling coefficients at lower resolutions from higher ones. However, such methods cannot replace the functionality of direct wavelet transform methods, which obtain coefficients directly from signals. We introduce new inner products and norms on sequences motivated by wavelet sampling theory, and implement transforms using inner product operations in infinite matrix forms, which directly map discrete function samples to wavelet coefficients. Via singular value decompositions we implement this signalbased wavelet transform in terms of unitary matrix operations, as a step towards their direct computation using quantum algorithms. We validate such quantum wavelet algorithms on spline and Coiflet MRAs.
