Generation of Novel Wavelet Transformations towards applications in Tensor Network Algorithms




Dayton, Aaron

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University of Victoria


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.



Wavelet, Tensor-Network, quantum, algorithm, MERA, Gapless-System