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.
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.
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.
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.
We present some basic integer arithmetic quantum circuits, such as adders and multiplier-accumulators of various forms, which operate on multilevel qudits. The integers to be processed are represented in an alternative basis after they have been Fourier transformed. Several arithmetic circuits operating on Fourier-transformed integers have appeared in the literature for two-level qubits. Here we extend these techniques to multilevel qudits, as they may offer some advantages relative to qubit implementations. The arithmetic circuits presented here can be used as basic building blocks for higher level algorithms such as quantum phase estimation, quantum simulation, quantum optimization, etc. Detailed decomposition is given down to elementary two-level single- and two-qudit gates as such gates are the most appropriate for physical implementation. A complexity analysis is given after this decomposition step and it is shown that the depth of the circuits is linear in the number of qudits employed and quadratic in the dimension of each qudit while their quantum cost is quadratic in the number of the qudits and quadratic in the dimension.
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.
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
The quantum Fourier transform (QFT) brings efficiency in many respects, especially usage of resource, for most operations on quantum computers. In this study, the existing QFT-based and non-QFT-based quantum arithmetic operations are examined. The capabilities of QFT-based addition and multiplication are improved with some modifications. The proposed operations are compared with the nearest quantum arithmetic operations. Furthermore, novel QFT-based subtraction, division and exponentiation operations are presented. The proposed arithmetic operations can perform nonmodular operations on all signed numbers without any limitation by using less resources. In addition, novel quantum circuits of two’s complement, absolute value and comparison operations are also presented by using the proposed QFT-based addition and subtraction operations.