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Multimodal Image Exploitation and Learning

New method of computing the quantum cosine transforms

Artyom M. Grigoryan, Alexis A. Gomez, Sos С. Agaian

Published May 28, 2025• 0 citations
Publisher

Abstract

The Quantum Signal Induced Heap Transform (QsiHT) is an emerging transform in the field of quantum computing, designed to enhance efficiency in signal processing tasks. This transform offers a novel approach for handling large-scale data structures in quantum algorithms, particularly in applications related to quantum image processing and signal decomposition. In this paper, we explore the use of QsiHT in generating various types of the quantum discrete cosine transforms (DCT), including the II-DCT and III-DCT, commonly used in image compression and signal processing. In QsiHT, different paths can be used in the computation, and in this work, we analyze a specific path to make the transformation computation most efficient. The proposed quantum circuits and algorithms for the QsiHT with 3 qubits are described in detail.

DOI
10.1117/12.3054033

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The Quantum Signal Induced Heap Transform (QsiHT) is an emerging transform in the field of quantum computing, designed to enhance efficiency in signal processing tasks. This transform offers a novel approach for handling large-scale data structures in quantum algorithms, particularly in applications related to quantum image processing and signal decomposition. In this paper, we explore the use of QsiHT in generating various types of the quantum discrete cosine transforms (DCT), including the II-DCT and III-DCT, commonly used in image compression and signal processing. In QsiHT, different paths can be used in the computation, and in this work, we analyze a specific path to make the transformation computation most efficient. The proposed quantum circuits and algorithms for the QsiHT with 3 qubits are described in detail.
The multi-qubit transformation is the important step in quantum computation in many applications. In recent works, this problem is addressed by solving by using the QR decomposition with elementary rotations with one- and two-qubit CNOT gates. Reducing the number of these gates is critical since operations on three or more qubits are difficult to implement on the current quantum computers. The quantum signal induced heap transform (QsiHT) provides an efficient method for state preparation in the realm of quantum image representation by amplitudes. This paper presents a detailed analysis of how the QsiHT, particularly through the strong path, is utilized to construct quantum states to represent images. The algorithm behind this method and the realization of the corresponding quantum circuits is outlined. Practical examples of the QsiHT on 2 and 3-qubit systems are given using the Qiskit Framework for quantum circuit simulation.
This article presents the general method of QR decomposition of r-qubit operations, r≥ 3, by means of quantum signal-induced heap transformations (QsiHT). These are quantum analogues of discrete signal-induced heap transformations, which are generated by given signals and paths, or orders, of processing the data. The case of the 5-qubit operations is described in detail, and a recurrent form of calculation of all 5-qubit QsiHTs from the 4-qubit QsiHTs is given. For that, roadmaps and quantum circuits are presented for all 31 5-point QsiHTs that are used in the QR decomposition. New roadmaps, namely the schemes with paths for performing basic operations on qubits, and corresponding quantum circuits, are also described for the 4- and 3-qubit transformations. All QsiHTs use fast paths, which allow us to calculate the QR decomposition only on disjoint bit planes. As a result, we build the quantum circuits for 5- and more-qubit operations without permutations, only elementary rotations. Unitary operations with real numbers are considered. In the general case, the method of compositing all the roadmaps and quantum schemes for the calculation of any r-qubit operation by only (2r−1)2r−1 elementary rotations is described.
This article presents a novel approach to the decomposition of unitary operations for 3-qubit systems by 28 controlled rotations and no permutations. The QR decomposition is described, which is based on the concept of the discrete signal-induced heap transform (DsiHT) and its quantum analogue. This transform is generated by a given signal and may use different paths, or orders, of processing the data, and, among them, one can find paths that allow one to construct efficient quantum circuits for implementing multi-qubit unitary gates. The case of real unitary matrices is considered. The proposed approach is described in detail, and quantum circuits are presented for computing 3-qubit operations. This approach allowed us to write simple Qiskit codes to implement the decomposition of 3-qubit operations. Examples with quantum circuits for the quantum 3-qubit quantum cosine and Hartley transforms are described.
The article presents the quantum signal-induced heap transform (QsiHT) method of the QR-decomposition of multi-qubit operations. This transform can be generated by a given signal, by using different paths, or orders, of processing the data. We propose using the concept of the fast path of calculation of the QsiHT and applying such transforms on each stage of the matrix decomposition. This allows us to build quantum circuits for multi-qubit unitary operation without permutations. Unitary operations with real and complex matrices are considered. The cases of 3- and 4-qubit operations are described in detail with quantum circuits. These circuits use a maximum of 28 and 120 Givens rotation gates for 3- and 4-qubit real operations, respectively. All rotations are performing only on adjacent bit planes. For complex unitary operation, each of the Givens gates is used in pairs with two Z-rotation gates. These two types of rotations and the global phase gate are the universal gate set for multi-qubit operations. The presented approach can be used for implementing quantum circuits for n-qubits when n≥2, with a maximum of (4n/2−2n−1) Givens rotations and no permutations.
The article presents a new approach to the decomposition of unitary operations by controlled rotations.A new method of quantum signal-induced heap transform based QR decomposition is described.This transform is an analogue of the discrete transform which is generated by a given signal and uses different paths of processing the data.It is shown that among such paths we can find paths that allow constructing efficient quantum circuits for implementing multiqubit unitary gates.The case of real unitary matrices is considered.The proposed approach is described in detail in examples and quantum circuits are presented for 2-qubit operations.The general case of 𝑟-qubit unitary operations, when 𝑟 ≥ 2, is considered similarly.
The slant transformation extends the Fourier transformation by incorporating a linear phase modulation that introduces directional sensitivity and localized frequency response. This makes the slant transformation a valuable tool for image processing, pattern recognition, and signal encoding. In this work, we present a new implementation for the slant transformation using the quantum signal-induced heap transformation (QsiHT) and its corresponding QsiHT-based QR decomposition to realize the quantum slant transformation (QST). The QsiHT-QR approach decomposes the slant transformation into a sequence of unitary heap-based rotational transformations that encode the required linear phase modulation. This decomposition minimizes the use of multi-controlled rotations and eliminates redundant CNOT gates, which significantly reduces the circuit depth. The quantum algorithms for the 2- and 3-qubit cases are described in detail, providing the necessary foundations for generalizing the implementation to higher-order qubit systems. Using IBM’s Qiskit framework for quantum simulation, the proposed quantum algorithms are validated through a 4-qubit quantum neural network (QNN) architecture.