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 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.
Quantum communication has achieved significant performance gains compared to classical systems but remains sensitive to channel noise and decoherence. These limitations become especially critical in quantum image transmission, where high-dimensional visual data must be preserved with both structural fidelity and robustness. In this context, transform-based quantum encoding methods have emerged as promising approaches, yet their relative performance under noisy conditions has not been fully explored. This paper presents a comparative study of two such methods, the quantum Fourier transform (QFT) and the quantum Haar wavelet transform (QHWT), within an image transmission framework. The process begins with source coding (JPEG/HEIF), followed by channel coding to enhance error resilience. The bitstreams are then mapped into quantum states using variable qubit encoding and transformed using either QFT or QHWT prior to transmission over noisy quantum channels. At the receiver, the corresponding decoding operations are applied to reconstruct the images. Simulation results demonstrate that the QFT achieves superior performance under noisy conditions, consistently delivering higher Peak Signal-to-Noise Ratio (PSNR), Structural Similarity Index Measure (SSIM), and Universal Quality Index (UQI) values across different qubit sizes and image formats compared to the QHWT. This advantage arises because QFT uniformly spreads information across all basis states, making it more resilient to noise. By contrast, QHWT generates localized coefficients that capture structural details effectively but become highly vulnerable when dominant coefficients are corrupted. Consequently, while QHWT emphasizes structural fidelity, QFT provides superior robustness, underscoring a fundamental trade-off in quantum image communication.
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.
Quantum Image Representation (QIR) has become a key component of Quantum Image Processing (QIP), enabling the encoding and manipulation of visual data within quantum computing environments.Its integration with Quantum Artificial Intelligence (QAI) and Quantum Machine Learning (QML) has attracted significant research interest due to the potential advantages of quantum parallelism and compact data representation.This survey reviews major QIR techniques, analyzes their encoding strategies and computational characteristics, and examines their opportunities, challenges, and future relevance in quantum AI and machine learning applications.A systematic review of 70 peer-reviewed journal and conference publications published between 1992 and 2025 was conducted.The selected studies were evaluated based on encoding mechanisms, qubit requirements, circuit complexity, retrieval capability, scalability, and suitability for quantum image processing and learning tasks.The review covers prominent QIR models, including FRQI, NEQR, MCQI, QPIE, GQIR, QRMW, and tensor-network-based representations.Comparative analysis reveals important trade-offs between qubit efficiency, encoding complexity, retrieval accuracy, and scalability.Amplitude-based approaches provide compact image encoding but suffer from probabilistic retrieval, whereas basis-state representations offer deterministic recovery at the expense of increased circuit complexity.QIR demonstrates significant potential in image classification, feature extraction, image compression, and hybrid quantum-classical learning frameworks.QIR provides a promising foundation for next-generation quantum-enhanced visual intelligence systems.However, practical implementation remains constrained by state preparation overhead, noise, decoherence, limited qubit availability, and the lack of standardized evaluation frameworks.Future research should focus on scalable encoding methods, fault-tolerant quantum architectures, benchmarking standards, and deeper integration with advanced QML models to accelerate the development of practical quantum image analytics.
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
Modern agriculture generates large, heterogeneous and high-dimensional data streams whose volume and combinatorial complexity increasingly strain classical computation. Quantum and quantum-inspired computing have been proposed as a means of representing and searching such spaces more efficiently. This review synthesises the emergence of quantum-inspired artificial intelligence and computing in agriculture, maps where these methods have been applied, appraises the certainty of the reported evidence, and evaluates the conditions under which the methods are likely to add value over classical approaches. Following the PRISMA 2020 guideline, four bibliographic databases (Scopus, Web of Science, IEEE Xplore and PubMed) were searched for peer-reviewed studies published between 2017 and 2026, supplemented by citation chaining and a constrained grey-literature scan from which preprints reporting a fully specified method and result were also eligible. After removal of 149 duplicates, 478 records were screened on title and abstract, 82 were assessed in full text, and 17 studies met the eligibility criteria and were carried into a thematic, task-oriented synthesis; no quantitative meta-analysis was performed because reported outcomes are not commensurable. The 17 included studies concentrate in five domains including the quantum-inspired optimisation (2 studies), quantum machine learning (5), plant disease detection (3), genomics and bioinformatics (6), and precision-agriculture sensing (1). By task, the corpus clusters on classification (7 studies) and sequence assembly/alignment (6), with optimisation (2), clustering (1) and sensing (1) thinly represented, and prediction/regression and dynamic, time-resolved phenotyping absent (0 studies each). Reported gains are frequently large in headline terms. For example, quantum image-descriptor disease classifiers reporting accuracies near 99% on curated leaf datasets but rest predominantly on simulators or noisy intermediate-scale quantum (NISQ) hardware, curated benchmarks and weak classical baselines, so the certainty of any quantum advantage is low. Quantum-inspired methods are a credible near-term toolset that can run on classical hardware while quantum processors mature, but current evidence is simulation-heavy, weakly baselined and difficult to reproduce. The clearest research gap is dynamic disease and phenotype-progression modelling. The paper outline a roadmap toward Agriculture 5.0, quantum-enabled breeding and quantum phenomics that prioritises temporal, uncertainty-aware modelling supported by stronger baselines and field validation.
The nondecimated or translation-invariant wavelet transform (NDWT) is a central tool in classical multiscale signal analysis, valued for its stability, redundancy, and shift invariance. This paper develops two complementary quantum formulations of the NDWT that embed these classical properties coherently into quantum computation. The first formulation is based on the epsilon-decimated interpretation of the NDWT and realizes all circularly shifted wavelet transforms simultaneously by promoting the shift index to a quantum register and applying controlled circular shifts followed by a wavelet analysis unitary. The resulting construction yields an explicit, fully unitary quantum representation of redundant wavelet coefficients and supports coherent postprocessing, including quantum shrinkage via ancilla-driven completely positive trace preserving maps. The second formulation is based on the Hadamard test and uses diagonal phase operators to probe scale-shift wavelet structure through interference, providing direct access to shift-invariant energy scalograms and multiscale spectra without explicit coefficient reconstruction. Together, these two approaches demonstrate that redundancy and translation invariance can be exploited rather than avoided in the quantum setting. Applications to denoising, feature extraction, and spectral scaling illustrate how quantum NDWTs provide a flexible and physically meaningful foundation for multiscale quantum signal processing.