量子小波变换：快速算法和完整电路
Quantum Wavelet Transforms: Fast Algorithms and Complete Circuits
摘要：
量子傅里叶变换（QFT）是经典傅里叶变换的一种量子模拟，已被证明是发展量子算法的有力工具。然而，在经典计算中，还有另一类酉变换，小波变换，它和傅立叶变换一样有用。小波变换用于揭示信号的多尺度结构，并且可能用于量子图像处理和量子数据压缩。在本文中，我们推导出有效的，完整的，量子线路的两个代表性的量子小波变换，量子Haar和量子Daubechies $D^{(4)}$变换。我们的方法是将这些变换的算子分解为酉矩阵的直和、直积和点积。在此过程中，我们发现置换矩阵，一类特殊的酉矩阵，起着关键的作用。令人惊讶的是，我们发现，经典实现起来容易且便宜的操作并不总是量子力学实现起来容易且便宜，反之亦然。特别是，执行某些置换矩阵的计算成本通常被忽略，因为它们可以显式地避免。然而，在量子力学中，这些置换操作必须显式执行，因此它们的成本进入量子变换的完整复杂性度量。我们考虑在量子小波变换中产生的特定的一组置换矩阵，并开发实现它们的有效的量子线路。这使我们能够设计出有效的、完整的量子线路。
英文摘要：
The quantum Fourier transform (QFT), a quantum analog of the classical Fourier transform, has been shown to be a powerful tool in developing quantum algorithms. However, in classical computing there is another class of unitary transforms, the wavelet transforms, which are every bit as useful as the Fourier transform. Wavelet transforms are used to expose the multi-scale structure of a signal and are likely to be useful for quantum image processing and quantum data compression. In this paper, we derive efficient, complete, quantum circuits for two representative quantum wavelet transforms, the quantum Haar and quantum Daubechies $D^{(4)}$ transforms. Our approach is to factor the operators for these transforms into direct sums, direct products and dot products of unitary matrices. In so doing, we find that permutation matrices, a particular class of unitary matrices, play a pivotal role. Surprisingly, we find that operations that are easy and inexpensive to implement classically are not always easy and inexpensive to implement quantum mechanically, and vice versa. In particular, the computational cost of performing certain permutation matrices is ignored classically because they can be avoided explicitly. However, quantum mechanically, these permutation operations must be performed explicitly and hence their cost enters into the full complexity measure of the quantum transform. We consider the particular set of permutation matrices arising in quantum wavelet transforms and develop efficient quantum circuits that implement them. This allows us to design efficient, complete quantum circuits for the quantum wavelet transform.
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