Generation of Novel Wavelet Transformations towards applications in Tensor Network Algorithms
Date
2024
Authors
Dayton, Aaron
Journal Title
Journal ISSN
Volume Title
Publisher
University of Victoria
Abstract
Wavelets can be formulated in a pyramidal gate structure of unitary rotation matrices which satisfy a set of vanishing moment equations. The vanishing moment equations can be satisfied by passing them into a cost function and minimizing with the Nelder-Mead algorithm. Barren plateau-like features exist in the solution-space of the vanishing moment equations which make it difficult to solve for greater circuit depths. The basins of these barren plateaus can be widened by introducing a parameter, β, in the exponent of each vanishing moment equation for a given circuit.
Wavelets up to depth 3, β=1, are generated to high precision and shown to match the known Daubechies wavelets. Solutions to the altered vanishing moment equations are shown to exist on a continuum for a domain of β-values for the depth 2 circuit.
An improved algorithm for solving for wavelet circuits of greater depths is proposed. These wavelets will be used in the multi-scale entanglement renormalization ansatz (MERA) tensor network algorithm to solve for the ground state energy of gapless systems.
Description
Keywords
Wavelet, Tensor-Network, quantum, algorithm, MERA, Gapless-System