Abstract This paper presents a quantum morphological Haar wavelet transform (QMHWT) that integrates the morphological minimum operator into the low-frequency subband of the Haar wavelet, unifying nonlinear morphological processing with multi-directional wavelet decomposition in a quantum arithmetic circuit framework. For a 2 × 2 pixel block [ a , b , c , d ] , the low-frequency subband is defined as L = min ( a , b , c , d ) , while the three directional detail subbands adopt the linear combination form of the classical Haar transform with floor division for integer-domain compatibility. We design a four-stage quantum circuit: a cascade of forward compare-swap (FCS) units extracts the four-way minimum, a reverse compare-swap recovery (FCS − 1 ) stage recycles auxiliary qubits, a modular addition/subtraction stage computes three detail subbands through separate arithmetic chains, and an arithmetic shift right stage performs integer division by two. The resulting circuit has width W ( n ) = 7 n + 12 , and the number of auxiliary qubits does not scale with image size. A D -map metric D = L mean − L min is derived from the morphological low-frequency subband, and its non-negativity, zero-value condition, and upper-bound relationship with the classical morphological gradient are established. Experiments on three standard 512 × 512 test images at 6-bit quantization confirm exact agreement between the quantum circuit output and classical reference results for all 3 × 65536 pixel blocks. The D -map achieves consistently high Pearson correlation with the morphological gradient ( r > 0.95 , mean r = 0.973 ), demonstrating the validity of the proposed transform and its potential use in quantum image edge analysis. The current large-image validation employs a hybrid quantum–classical scheme in which the per-block unitary cost is
Abstract This paper presents a quantum morphological Haar wavelet transform (QMHWT) that integrates the morphological minimum operator into the low-frequency subband of the Haar wavelet, unifying nonlinear morphological processing with multi-directional wavelet decomposition in a quantum arithmetic circuit framework. For a 2 × 2 pixel block [ a , b , c , d ] , the low-frequency subband is defined as L = min ( a , b , c , d ) , while the three directional detail subbands adopt the linear combination form of the classical Haar transform with floor division for integer-domain compatibility. We design a four-stage quantum circuit: a cascade of forward compare-swap (FCS) units extracts the four-way minimum, a reverse compare-swap recovery (FCS − 1 ) stage recycles auxiliary qubits, a modular addition/subtraction stage computes three detail subbands through separate arithmetic chains, and an arithmetic shift right stage performs integer division by two. The resulting circuit has width W ( n ) = 7 n + 12 , and the number of auxiliary qubits does not scale with image size. A D -map metric D = L mean − L min is derived from the morphological low-frequency subband, and its non-negativity, zero-value condition, and upper-bound relationship with the classical morphological gradient are established. Experiments on three standard 512 × 512 test images at 6-bit quantization confirm exact agreement between the quantum circuit output and classical reference results for all 3 × 65536 pixel blocks. The D -map achieves consistently high Pearson correlation with the morphological gradient ( r > 0.95 , mean r = 0.973 ), demonstrating the validity of the proposed transform and its potential use in quantum image edge analysis. The current large-image validation employs a hybrid quantum–classical scheme in which the per-block unitary cost is
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.
Abstract A novel framework for quantum integer wavelet transforms over min–max–plus algebra, termed the QMMPLS-Wavelet, is proposed. This approach extends the advantages of integer-based wavelet transforms into the quantum domain by integrating quantum arithmetic operators derived from the Quantum Fourier Transform (QFT). A key contribution is the design of a new quantum operator capable of comparing three integers, which explicitly produces quantum states representing both minimum and maximum values. This operator is further combined with quantum full addition and subtraction to implement the predict and update steps in the lifting scheme. In addition, quantum circuits are developed for the decomposition (analysis) and reconstruction (synthesis) processes of the QMMPLS-Wavelet Type MaxLS. Complexity analysis shows efficiency in terms of the number of quantum gates, delay time, garbage output, and ancilla qubits, where the overall quantum complexity is O ( n 2 ) based on the number of quantum gates employed. The findings confirm the potential of the QMMPLS-Wavelet as an efficient quantum framework for signal processing while opening new avenues for the development of quantum algorithms based on max-plus algebra.
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.
Edge detection is a vital component of industrial image-processing and evaluation workflows, particularly in X-ray computed tomography (XCT), where it underpins segmentation, feature extraction, and defect localization. Quantum computing provides powerful building blocks for image processing by exploiting superposition and entanglement, even on today’s noisy hardware. In this work, we focus on edge detection for industrial XCT data and translate two widely used classical approaches, which are the Haar wavelet filter and the Sobel operator, into the quantum domain, while keeping the design intentionally simple: overlapping 2 × 2 patches with angle-based RY encoding, a compact bidirectional quantum Haar transform, and small spectral phase shifts (φ = π/8) applied to the HL/LH subbands to mimic Sobel-like directional selectivity. Each patch compiles into a shallow four-qubit circuit with a logical depth of 20 or less, rendering the approach feasible for both simulation and actual quantum hardware implementation. We evaluate the method on three 64 × 64 grayscale XCT radiographs, each decomposed into 63 × 63 = 3969 overlapping 2 × 2 patches, corresponding to one quantum circuit per patch. Across ideal and noise-aware simulations, the quantum Haar and Sobel operators closely follow their classical reference maps for all 2x2 patches. We additionally executed the whole pipeline for Test Case 1 on the IBM EU-DE backend (ibm_aachen) using Runtime V2 (SamplerV2). The experiment required 20 min 59 s for the quantum Haar configuration and 21 min 22 s for the quantum Sobel configuration (3969 circuits each), demonstrating that the shallow-circuit design is compatible with current superconducting hardware. Different simulation backends, as well as the real IBM device, execute the same circuit structure without modification, and edge strength is reconstructed using a hardware-robust and straightforward metric, E = 1 − P(|0 . . . 0⟩), where P(|0 . . . 0⟩) denotes the probability of observing the all-zero measurement outcome. The combination of a small Haar circuit and π/8 phase modulation yields a hardware-friendly quantum edge detector that aligns well with classical benchmarks on industrial XCT data. The results indicate that meaningful edge information can be extracted within NISQ constraints, providing a reproducible pathway for deploying quantum image-processing pipelines on present-day devices.
Convolutional neural networks rely on linear filtering operations that can be reformulated efficiently in suitable transform domains. At the same time, advances in quantum computing have shown that certain structured linear transforms can be implemented with shallow quantum circuits, opening the door to hybrid quantum-classical approaches for enhancing deep learning models. In this work, we introduce WTHaar-Net, a convolutional neural network that replaces the Hadamard Transform used in prior hybrid architectures with the Haar Wavelet Transform (HWT). Unlike the Hadamard Transform, the Haar transform provides spatially localized, multi-resolution representations that align more closely with the inductive biases of vision tasks. We show that the HWT admits a quantum realization using structured Hadamard gates, enabling its decomposition into unitary operations suitable for quantum circuits. Experiments on CIFAR-10 and Tiny-ImageNet demonstrate that WTHaar-Net achieves substantial parameter reduction while maintaining competitive accuracy. On Tiny-ImageNet, our approach outperforms both ResNet and Hadamard-based baselines. We validate the quantum implementation on IBM Quantum cloud hardware, demonstrating compatibility with near-term quantum devices.
Quantum information science is an emerging research field devoted to the use of quantum mechanical systems to devise and implement information processing tasks faster than that possible with classical computers. In this study, two quantum image resolution enhancement (QIRE-I and QIRE-II) schemes are proposed based on quantum wavelet transform and quantum interpolation. Using these, the resolutions of low-resolution (LR) images are enhanced by decomposing them into four frequency sub-bands using a single-level one-dimensional (1-D) quantum Haar wavelet transform (QHWT). Subsequently, to preserve the edges and obtain sharper high-resolution (HR) images, quantum interpolation was applied to three of the high-frequency sub-bands. A few simulation-based demonstrations are presented to illustrate the feasibility and effectiveness of the proposed schemes. The visual and quantitative results demonstrate the superiority of the proposed schemes over those that use only quantum interpolation.