Efficient Quantum Algorithm for All Quantum Wavelet Transforms

• URL: http://arxiv.org/abs/2309.09350v2
• Date: Mon, 22 Apr 2024 15:48:58 GMT
• Title: Efficient Quantum Algorithm for All Quantum Wavelet Transforms
• Authors: Mohsen Bagherimehrab, Alan Aspuru-Guzik,
• Abstract summary: We develop a simple yet efficient quantum algorithm for executing any wavelet transform on a quantum computer. Our proposed quantum wavelet transforms could be used in quantum computing algorithms in a similar manner to their well-established counterpart, the quantum Fourier transform.
• Score: 0.08968838300743379
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Wavelet transforms are widely used in various fields of science and engineering as a mathematical tool with features that reveal information ignored by the Fourier transform. Unlike the Fourier transform, which is unique, a wavelet transform is specified by a sequence of numbers associated with the type of wavelet used and an order parameter specifying the length of the sequence. While the quantum Fourier transform, a quantum analog of the classical Fourier transform, has been pivotal in quantum computing, prior works on quantum wavelet transforms~(QWTs) were limited to the second and fourth order of a particular wavelet, the Daubechies wavelet. Here we develop a simple yet efficient quantum algorithm for executing any wavelet transform on a quantum computer. Our approach is to decompose the kernel matrix of a wavelet transform as a linear combination of unitaries (LCU) that are compilable by easy-to-implement modular quantum arithmetic operations and use the LCU technique to construct a probabilistic procedure to implement a QWT with a \textit{known} success probability. We then use properties of wavelets to make this approach deterministic by a few executions of the amplitude amplification strategy. We extend our approach to a multilevel wavelet transform and a generalized version, the packet wavelet transform, establishing computational complexities in terms of three parameters: the wavelet order $M$, the dimension $N$ of the transformation matrix, and the transformation level $d$. We show the cost is logarithmic in $N$, linear in $d$ and superlinear in $M$. Moreover, we show the cost is independent of $M$ for practical applications. Our proposed quantum wavelet transforms could be used in quantum computing algorithms in a similar manner to their well-established counterpart, the quantum Fourier transform.

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