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 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 QWTs could be used in quantum computing algorithms in a similar manner to their well-established counterpart, the quantum Fourier transform.

中文翻译：

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适用于所有量子小波变换的高效量子算法

小波变换作为一种数学工具被广泛应用于科学和工程的各个领域，其特点是揭示傅里叶变换忽略的信息。与独特的傅立叶变换不同，小波变换由与所使用的小波类型相关联的数字序列和指定序列长度的阶参数来指定。虽然量子傅里叶变换（经典傅里叶变换的量子模拟）在量子计算中一直至关重要，但之前有关量子小波变换 (QWT) 的工作仅限于特定小波（Daubechies 小波）的二阶和四阶。在这里，我们开发了一种简单而有效的量子算法，用于在量子计算机上执行任何小波变换。我们的方法是将小波变换的核矩阵分解为可通过易于实现的模块化量子算术运算进行编译的酉线性组合 (LCU)，并使用 LCU 技术构建概率过程来实现 QWT已知成功概率。然后，我们利用小波的属性，通过执行几次幅度放大策略来使该方法具有确定性。我们将我们的方法扩展到多级小波变换和广义版本，即分组小波变换，根据三个参数确定计算复杂性：小波阶数 M、变换矩阵的维数 N 和变换级别 d。我们表明成本在 N 中是对数的，在 d 中是线性的，在 M 中是超线性的。此外，我们表明在实际应用中成本与 M 无关。我们提出的 QWT 可以以与其成熟的对应物（量子傅里叶变换）类似的方式用于量子计算算法。