Wavelet matrix operations and quantum transforms

Highlights：

• Such infinite matrix operations can be stably transformed into implementable finite ones, suitable for practical applications.

• This constructed finite matrix operation can greatly improve accuracy of wavelet coefficients from low density samples, and can yield a huge computational savings as compared with classic inner products estimated via quadrature.

• This finite matrix operation also allows for a quantization of the wavelet transform (quantum wavelet transform) in its true sense (as compared with current version of quantum wavelet transforms implementing the Mallat pyramid algorithm).

• The proposed quantum wavelet transform is the first re-formulation of wavelet transforms into quantum algorithms, formulated in parallel with the standard quantum Fourier transform.

摘要

•Using constructed new inner products on l2(Z), we re-formulate the wavelet transform in terms of operations with infinite matrices. This allows for direct mapping of discrete function samples into wavelet coefficients.•Such infinite matrix operations can be stably transformed into implementable finite ones, suitable for practical applications.•This constructed finite matrix operation can greatly improve accuracy of wavelet coefficients from low density samples, and can yield a huge computational savings as compared with classic inner products estimated via quadrature.•This finite matrix operation also allows for a quantization of the wavelet transform (quantum wavelet transform) in its true sense (as compared with current version of quantum wavelet transforms implementing the Mallat pyramid algorithm).•The proposed quantum wavelet transform is the first re-formulation of wavelet transforms into quantum algorithms, formulated in parallel with the standard quantum Fourier transform.