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<p>We present the construction of a quantum integer wavelet transform for one-dimensional signals utilizing max-plus algebraic operations, which we term the QMP-Wavelet Type IVa. This approach leverages the inherent ’max’ and ’+’ operations of max-plus algebra to enable efficient, direct processing of quantum superposition signals. This facilitates seamless integration into quantum computing workflows. The MP-Wavelet Type IVa serves as the foundation for signal decomposition and reconstruction, which enables the development of the QMP-Wavelet Type IVa. First, we introduce a quantum integer comparison operation based on the quantum Fourier transform (QFT) as the foundation for our quantum wavelet transform. We develop a quantum circuit to compare two integers, generating quantum state outputs that represent their maximum and minimum values. This approach facilitates the integration of quantum integer comparison with other quantum circuits, including those for quantum arithmetic operations. Furthermore, we develop a quantum circuits for the analysis and synthesis operators of the QMP-Wavelet Type IVa by leveraging our quantum integer comparison operation, QFT-based quantum arithmetic, and quantum full subtraction. To validate our theoretical results, we implemented and tested the QMP-Wavelet Type IVa using IBM Qiskit. Additionally, we demonstrate that our proposed quantum integer comparison and QMP-Wavelet Type IVa exhibit a complexity of <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40819_2025_1951_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">\(O(n^2)\)</EquationSource> </InlineEquation> and a time complexity of <i>O</i>(<i>n</i>), both calculated based on the number of quantum gates.</p>

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Quantum Integer Wavelet Transform over Max-Plus Algebra

  • 期刊论文

  • 发表日期：

    2025年7月16日

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  • Mohamad Ilham Dwi Firmansyah,
  • Mahmud Yunus,
  • Subiono

摘要

We present the construction of a quantum integer wavelet transform for one-dimensional signals utilizing max-plus algebraic operations, which we term the QMP-Wavelet Type IVa. This approach leverages the inherent ’max’ and ’+’ operations of max-plus algebra to enable efficient, direct processing of quantum superposition signals. This facilitates seamless integration into quantum computing workflows. The MP-Wavelet Type IVa serves as the foundation for signal decomposition and reconstruction, which enables the development of the QMP-Wavelet Type IVa. First, we introduce a quantum integer comparison operation based on the quantum Fourier transform (QFT) as the foundation for our quantum wavelet transform. We develop a quantum circuit to compare two integers, generating quantum state outputs that represent their maximum and minimum values. This approach facilitates the integration of quantum integer comparison with other quantum circuits, including those for quantum arithmetic operations. Furthermore, we develop a quantum circuits for the analysis and synthesis operators of the QMP-Wavelet Type IVa by leveraging our quantum integer comparison operation, QFT-based quantum arithmetic, and quantum full subtraction. To validate our theoretical results, we implemented and tested the QMP-Wavelet Type IVa using IBM Qiskit. Additionally, we demonstrate that our proposed quantum integer comparison and QMP-Wavelet Type IVa exhibit a complexity of \(O(n^2)\) and a time complexity of O(n), both calculated based on the number of quantum gates.