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The multi-level and multi-dimensional quantum wavelet packet transforms.

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This study introduces efficient quantum wavelet packet transforms (QWPTs) for quantum information processing. The novel Haar QWPT (HQWPT) and D4 QWPT (DQWPT) demonstrate high efficiency and correctness through simulations.

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Area of Science:

  • Quantum Information Science
  • Quantum Computing
  • Signal Processing

Background:

  • Classical wavelet packet transform is vital in information processing.
  • Quantum wavelet packet transform (QWPT) shows potential for quantum information processing.

Purpose of the Study:

  • To design and propose novel multi-level and multi-dimensional QWPTs.
  • To introduce Haar QWPT (HQWPT) and D4 QWPT (DQWPT) for the first time.
  • To analyze quantum costs and time complexities of proposed QWPTs.

Main Methods:

  • Design of quantum circuits for generalized tensor product (GTP) and perfect shuffle permutation (PSP).
  • Development of 1D, 2D, and 3D QWPTs, including inverse transforms.
  • Proof of correctness based on GTP and PSP.

Main Results:

  • Precise analysis of quantum costs and time complexities.
  • Demonstration of high-efficiency for HQWPTs with time complexity at most 6 on 2n elements.
  • Validation of QWPT correctness and effectiveness through simulations.

Conclusions:

  • The proposed QWPTs are correct and effective for quantum information processing.
  • The developed QWPTs offer high efficiency, particularly the HQWPT.
  • This work lays a foundation for advanced quantum signal processing applications.