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    Area of Science:

    • Quantum computing
    • Image processing
    • Wavelet transforms

    Background:

    • Wavelet transforms are crucial in classical image processing.
    • Existing quantum wavelet transforms (QWTs) are limited to one dimension or direct products of 1-D transforms.
    • A two-dimensional (2-D) QWT is essential for advanced image analysis.

    Purpose of the Study:

    • To develop a general theory for multilevel 2-D QWTs.
    • To construct specific implementations of multilevel 2-D Haar QWT and Daubechies D4 QWT.
    • To provide quantum circuits for these transforms and analyze their efficiency.

    Main Methods:

    • Construction of the general theory for multilevel 2-D QWT.
    • Development of explicit multilevel 2-D Haar QWT and Daubechies D4 QWT.
    • Design of complete quantum circuits using noniterative and iterative approaches.
    • Complexity analysis to compare with classical methods.

    Main Results:

    • The proposed multilevel 2-D QWT involves entanglement between components in different dimensions.
    • Complexity analysis shows an exponential speedup compared to classical wavelet transforms.
    • Quantum image compression was successfully realized using the developed 2-D QWTs.

    Conclusions:

    • The developed multilevel 2-D QWTs are significant for quantum image processing.
    • These quantum transforms achieve results comparable to classical methods but with exponential efficiency gains.
    • The research provides a foundation for further advancements in quantum image analysis and compression.