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Wavelet matrix operations and quantum transforms

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  • Zhang, Zhiguo
  • Kon, Mark A.

Abstract

The currently studied version of the quantum wavelet transform implements the Mallat pyramid algorithm, calculating wavelet and scaling coefficients at lower resolutions from higher ones, via quantum computations. However, the pyramid algorithm cannot replace wavelet transform algorithms, which obtain wavelet coefficients directly from signals. The barrier to implementing quantum versions of wavelet transforms has been the fact that the mapping from sampled signals to wavelet coefficients is not canonically represented with matrices. To solve this problem, we introduce new inner products and norms into the sequence space l2(Z), based on wavelet sampling theory. We then show that wavelet transform algorithms using L2(R) inner product operations can be implemented in infinite matrix forms, directly mapping discrete function samples to wavelet coefficients. These infinite matrix operators are then converted into finite forms for computational implementation. Thus, via singular value decompositions of these finite matrices, our work allows implementation of the standard wavelet transform with a quantum circuit. Finally, we validate these wavelet matrix algorithms on MRAs involving spline and Coiflet wavelets, illustrating some of our theorems.

Suggested Citation

  • Zhang, Zhiguo & Kon, Mark A., 2022. "Wavelet matrix operations and quantum transforms," Applied Mathematics and Computation, Elsevier, vol. 428(C).
    • Handle: RePEc:eee:apmaco:v:428:y:2022:i:c:s0096300322002533
    DOI: 10.1016/j.amc.2022.127179
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    References listed on IDEAS

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    1. Saray, Behzad Nemati & Lakestani, Mehrdad, 2020. "On the sparse multi-scale solution of the delay differential equations by an efficient algorithm," Applied Mathematics and Computation, Elsevier, vol. 381(C).
    2. Golmankhaneh, Alireza K. & Tunç, Cemil, 2019. "Sumudu transform in fractal calculus," Applied Mathematics and Computation, Elsevier, vol. 350(C), pages 386-401.
    3. Alipour, Sahar & Mirzaee, Farshid, 2020. "An iterative algorithm for solving two dimensional nonlinear stochastic integral equations: A combined successive approximations method with bilinear spline interpolation," Applied Mathematics and Computation, Elsevier, vol. 371(C).
    4. Barajas-García, Carolina & Solorza-Calderón, Selene & Gutiérrez-López, Everardo, 2019. "Scale, translation and rotation invariant Wavelet Local Feature Descriptor," Applied Mathematics and Computation, Elsevier, vol. 363(C), pages 1-1.
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