We present the first quantum version of the one-dimensional integer wavelet S-Transform (QIST). This version is based on a new Quantum Block Representation by Basis States (QBRBS) and the design of a quantum rounding operator. The QBRBS format facilitates storing and manipulating pairs of signal elements, decreasing the use of additional qubits and gates. Also, the quantum rounding operator exploits the features of the S-Transform to avoid nonlinearities. Complete quantum circuits for addition, subtraction, and halving operations were developed. The complexity analysis showed a polynomial quantum complexity, O(n), and a time complexity of O(1). The experiments performed on MATLAB and Qiskit verify the theoretical results coinciding with the results of the classical version. Finally, the new quantum transform’s feasibility and accuracy enable new quantum developments in information hiding, compression, and IoT applications.
We present the first quantum version of the one-dimensional integer wavelet S-Transform (QIST). This version is based on a new Quantum Block Representation by Basis States (QBRBS) and the design of a quantum rounding operator. The QBRBS format facilitates storing and manipulating pairs of signal elements, decreasing the use of additional qubits and gates. Also, the quantum rounding operator exploits the features of the S-Transform to avoid nonlinearities. Complete quantum circuits for addition, subtraction, and halving operations were developed. The complexity analysis showed a polynomial quantum complexity, O(n), and a time complexity of O(1). The experiments performed on MATLAB and Qiskit verify the theoretical results coinciding with the results of the classical version. Finally, the new quantum transform’s feasibility and accuracy enable new quantum developments in information hiding, compression, and IoT applications.
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.
A generalization of the new quantum block representation by basis states (QBRBS) is presented and analyzed. This representation facilitates the manipulation of signals by providing a flexible block size to store signal elements. Furthermore, QBRBS uses quantum superposition and entangled qubits to store signal values on the basis states of qubit registers instead of probability amplitude, enabling accurate information retrieval after finite measurements. A detailed analysis exposed a polynomial growth in qubit complexity and decreased quantum complexity on the presented quantum block format. The geometric, statistical, element-value, element-value block, and block position operations are described. Simulation experiments show the feasibility and capabilities of the block representation to achieve data hiding applications. A quantum steganography scheme exploiting the QBRBS features is presented. Finally, the quantum circuit and simulation verification are performed using Qiskit.
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.
There are several possible ways to encode random variables in a quantum state. The basis encoding of bit strings has paramount importance because it allows to load the values of a random variable through the superposition of corresponding basis states, and to then exploit quantum parallelism in processing algorithms. The basis encoding offers a natural way to represent an unsigned integer random variable, and extends to signed integers, as well as to fixed-point and floating-point variables. Each quantum representation of fractional numbers, however, involves a trade-off between accuracy and depth of manipulation circuits. Here, an efficient hybrid quantum-classical representation of quantum floating points is introduced. It combines a quantum register containing the values, with a classical register storing global information about the variable, namely the range and approximation tolerances. The sum and product operations are defined, in such a way as to ensure they are performed without overflow. By taking advantage of the stored classical information, the precision degradation that occurs due to rounding after repeated data manipulations, can be significantly reduced compared to known strategies. Ad hoc examples show up to around 90% reduction in approximation, compared to previous techniques, after repeated additions. The method finds application in many algorithms of practical relevance and constitutes a significant advance in the design of arithmetic circuits with low depth and high accuracy.
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.
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.