Fourier transform has been shown to be a powerful tool in many area of science. However, there is another class of unitary transforms, the wavelet transforms, which are as useful as the Fourier transform. Wavelet transforms are used to expose the multi-scale structure of a signal and very useful for image processing and data compressiyon. In this paper, we construct quantum algorithms for Haar wavelet transforms and show its application in analyzing the multi-scale structure of the dynamical system by the Logistic Map (x! x(1 x)). 1
Fourier transform has been shown to be a powerful tool in many area of science. However, there is another class of unitary transforms, the wavelet transforms, which are as useful as the Fourier transform. Wavelet transforms are used to expose the multi-scale structure of a signal and very useful for image processing and data compressiyon. In this paper, we construct quantum algorithms for Haar wavelet transforms and show its application in analyzing the multi-scale structure of the dynamical system by the Logistic Map (x! x(1 x)). 1
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.
Quantum computing has emerged as a revolutionary paradigm in numerous fields owing to its basis in quantum mechanical concepts such as quantum entanglement and parallelism. This has potential in various applications including computer vision, particularly, image processing and compression. Applications that depend on largescale image data for storing, processing, and transmission have to rely on compression methods for real-time processing. While the classical methods based on discrete cosine transformations (DCT used in JPEG) and discrete wavelet transformations (DWT) perform image compression efficiently, they suffer from trade-off between loss-less compression and compression ratios. In the current work, we would like to explore the usefulness of implementing quantum wavelet transforms for image compression for better PSNR values and execution times. We compare the application of quantum Haar wavelet transform (QHWT) with different quantum encoding methods such as NASS and FRQI and find that NASS-encoded images on application of QHWT gives better PSNR values and shorter runtime with visibly closer details to the original images, showcasing potential for efficient and faster image compression methods.
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 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.
The Haar wavelet algorithm based on quantum computing and its application for color image compression are discussed.Firstly,classical Haar wavelet operator is decomposed into direct sums,direct products and dot products of unitary matrices,so that shuffle permutation matrices is found.Secondly,based on the generalized Kronecker product,Harr functions are decomposed into a sequence of W-H and shuffle permutation matrices.Thirdly,the efficient,complete,quantum circuits for representative quantum wavelet transform are given.Finally,algorithm for color image compression by quantum Haar wavelet is presented,and experiment results are given.The experiment results show that the quantum Haar wavelet has very good feasibility in image compression.
The quantum Fourier transform and quantum wavelet transform have been cornerstones of quantum information processing. However, for non-stationary signals and anomaly detection, the Hilbert transform can be a more powerful tool, yet no prior work has provided efficient quantum implementations for the discrete Hilbert transform. This letter presents a novel construction for a quantum Hilbert transform in polylogarithmic size and logarithmic depth for a signal of length N, exponentially fewer operations than classical algorithms for the same mapping. We generalize this algorithm to create any d-dimensional Hilbert transform in depth O(dlogN). Simulations demonstrate effectiveness for tasks such as power systems control and image processing, with exact agreement with classical results.
Abstract Quantum Haar Wavelet Transform (QHWT) methods connect multiresolution analysis with quantum information processing through reversible averaging and differencing operations that can be implemented with comparatively compact circuits. Their localized basis functions, real-valued binary structure, and compatibility with hierarchical state representations make them attractive for resource-constrained quantum processing. Nevertheless, claims of practical superiority are difficult to generalize because published studies use heterogeneous encodings, simulators, hardware platforms, noise models, and evaluation metrics. This systematic review examines scholarly work on Haar-based quantum methods published between 1997 and March 2025. A PRISMA-oriented protocol was used to search major engineering and computing databases, remove duplicates, screen records, assess eligibility, and construct a qualitative evidence corpus. The review includes peer-reviewed journal and conference studies together with a limited number of historically influential technical reports and preprints; therefore, the corpus is described as scholarly literature rather than as a uniformly peer-reviewed dataset. Because the reported experiments are not sufficiently homogeneous for statistical pooling, numerical results are treated as study-specific evidence rather than global averages. The review makes three analytical contributions. First, it introduces a unified conceptual framework that links mathematical foundations, circuit realization, application design, hardware evaluation, and research translation. Second, it develops a taxonomy spanning transform theory, quantum image and signal processing, hardware implementation, and emerging quantum artificial intelligence and security applications. Third, it critically compares QHWT with Quantum Fourier Transform and higher-order wavelet approaches by distinguishing circuit depth, gate complexity, approximation behavior, hardware assumptions, and empirical validation. The synthesis indicates that QHWT is most compelling when localization, reversible multiresolution processing, and shallow circuit structure are more important than smooth-basis approximation. Individual studies report promising compression, feature-extraction, and hardware-efficiency results, but these findings remain conditional on encoding choices and platform constraints. Important limitations include sparse cross-platform benchmarking, limited fault-tolerant analysis, noise sensitivity, uncertain scaling, and optimization difficulties in variational quantum learning, including barren plateaus. The resulting roadmap prioritizes reproducible benchmarks, hardware-aware circuit design, explicit complexity reporting, error-mitigation studies, and validation on larger quantum systems.