Quantum Wavelet Transforms Generated by the Product of the Sine Polynomial and the Gaussian Envelope on the Tetrahedral Graph

摘要

In this paper we present a novel technique that permits to extract the essential on information embedded in the product of sine polynomial and Gaussian envelope by simply knowing the vertices of the tetrahedral graph. The study proves that the matrix of vertices of the tetrahedral graph and its variants are the building block of both Haar wavelets, Hadamard-Walsh transform, wavelets sets and tight frames. We also prove that the Berkeley B Gate is a function of the degree matrix and the adjacency matrix of the tetrahedral graph. The latter is the Hermitian part of the unitary polar decomposition in terms of elementary gates for quantum computation [68] which reveals interesting properties of the tetrahedral graph in both quantum group, Lie group and Pauli group for wavelets sets, quantum image processing and quantum data compression. We explore the connection existing among graphs theory, wavelets, tight frames and quantum logic gates.