The graph Fourier transform (GFT) is a vital tool for processing irregular data collected across diverse structured systems, including sensor, social, transportation, and brain networks. Given its wide utility in graph signal applications like compression, prediction, classification, denoising, detection, and estimation, developing efficient methods for computing the GFT over graphs and networks is a crucial task. This paper presents a quantum computation method for calculating the GFT of a graph signal, executed through five main steps: (1) The graph signal is first normalized to a unit norm during preprocessing. (2) A quantum state preparation circuit loads this unit-norm data into a quantum register, generating a superposition state. (3) The quantum circuit representing the unitary GFT matrix is obtained using the IBM Qiskit transpiler, which is then applied to transform the vertex-domain data into the spectral domain. (4) Repeated quantum measurements are performed to generate a histogram that encodes the spectral-domain data. (5) Finally, the histogram's probability data is scaled by the original input signal's norm to yield the complete GFT result. The effectiveness of this proposed quantum method is demonstrated through the GFT computation of temperature data from a sensor network.
The graph Fourier transform (GFT) is a vital tool for processing irregular data collected across diverse structured systems, including sensor, social, transportation, and brain networks. Given its wide utility in graph signal applications like compression, prediction, classification, denoising, detection, and estimation, developing efficient methods for computing the GFT over graphs and networks is a crucial task. This paper presents a quantum computation method for calculating the GFT of a graph signal, executed through five main steps: (1) The graph signal is first normalized to a unit norm during preprocessing. (2) A quantum state preparation circuit loads this unit-norm data into a quantum register, generating a superposition state. (3) The quantum circuit representing the unitary GFT matrix is obtained using the IBM Qiskit transpiler, which is then applied to transform the vertex-domain data into the spectral domain. (4) Repeated quantum measurements are performed to generate a histogram that encodes the spectral-domain data. (5) Finally, the histogram's probability data is scaled by the original input signal's norm to yield the complete GFT result. The effectiveness of this proposed quantum method is demonstrated through the GFT computation of temperature data from a sensor network.
signals into frequency-domain forms is the Fourier Transform (FT). It is extensively utilized in several fields, such as spectrum analysis, audio and video processing, and 5G networks. Effective frequency analysis is essential for channel optimization, signal reliability, and noise reduction in 5G networks. The FFT is appropriate for real-time applications since it greatly increases processing speed. However, the Quantum Fourier Transform makes use of quantum entanglement and superposition, providing exciting new developments for secure data transfer and high-speed communication in the future. Furthermore, both the Inverse Fast Fourier Transform (IFFT) and the Inverse Quantum Fourier Transform are used to correctly recreate the signal. Implementation of Quantum's real valued computing of quantum-based signal processing is also covered in the survey.
Fourier Transform (FT) is an important signal processing technique that translates time-domain signals to frequency-domain forms. It is widely used in several applications, including 5G networks, image processing, & spectrum analysis. In 5G networks, efficient frequency analysis is critical for noise reduction, channel optimisation, and signal dependability.This study explores the implementation of Classical Fourier Transform, Fast Fourier Transform, and Quantum Fourier Transform using Python (NumPy, Pandas, Matplotlib) and Qiskit. The FFT significantly improves processing speed, making it suitable for real-time applications. On the other hand, QFT leverages quantum superposition and entanglement, offering promising advancements for future high-speed communication and secure data transmission. Additionally, the signal is successfully reconstructed using both the Inverse Fast Fourier Transform (IFFT) and the Inverse Quantum Fourier Transform (IQFT). While IFFT efficiently restores the original signal with minimal numerical artifacts, IQFT ensures exact reconstruction without errors. The study also includes execution on IBM Quantum’s real hardware, showcasing the practical feasibility of quantum-based signal processing.
Graph kernels are essential for learning the properties of graph data, a fundamental task in the analysis of such data. However, a major limitation of current graph kernel methods is their high computational cost, which becomes prohibitive when dealing with large-scale graphs. In order to address this bottleneck, we propose a quantum algorithm designed to analyze the intrinsic structural information within graph data through an efficient quantum kernel method. The core idea of our kernel method involves encoding pertinent information from graphs into quantum states, utilizing quantum kernel functions to represent the similarities among various graphs. This process is facilitated through the application of quantum walks and quantum signal processing techniques. Building upon this framework, we propose a quantum kernel ridge regression algorithm that consists of two key steps. First, we employ our quantum kernel method to prepare a quantum state, in which the amplitudes encode the kernel functions. Second, we implement quantum feature mapping in parallel to encode the corresponding kernel matrix into the quantum circuit. Crucially, we demonstrate that the computational complexity of our regression algorithm scales primarily as O(M3/2) with respect to the size of the graph set M. This suggests that our quantum kernel method could process graph data with resource requirements scaling approximately as O(polylog(N)), where N denotes the number of nodes in the graph. Accordingly, this underscores the advantage of our approach over classical methods in the context of large-scale graph analysis. To validate our approach, we conduct numerical experiments on chemical molecular datasets, demonstrating the method's effectiveness and advantages. In summary, this work represents a significant advancement in quantum graph learning, enhancing its potential for analyzing large-scale graph data.
Abstract In this paper, we describe a parameterized quantum circuit that can be considered as convolutional and pooling layers for graph neural networks. The circuit incorporates the parameterized quantum Fourier circuit where the qubit connections for the controlled gates derived from the Laplacian operator. Specifically, we show that the eigenspace of the Laplacian operator of a graph can be approximated by using QFT based circuit whose connections are determined from the adjacency matrix. For an N × N Laplacian, this approach yields an approximate polynomial-depth circuit requiring only n = log(N) qubits. These types of circuits can eliminate the expensive classical computations for approximating the learnable functions of the Laplacian through Chebyshev polynomial or Taylor expansions. Using this circuit as a convolutional layer provides an n− dimensional probability vector that can be considered as the filtered and compressed graph signal. Therefore, the circuit along with the measurement can be considered a very efficient convolution plus pooling layer that transforms an N-dimensional signal input into n−dimensional signal with an exponential compression. We then apply a classical neural network prediction head to the output of the circuit to construct a complete graph neural network. Since the circuit incorporates geometric structure through its graph connection-based approach, we present graph classification results for the benchmark datasets listed in TUDataset library (AIDS, Letter-high, Letter-med, Letter-low, MUTAG, ENZYMES, PROTEINS, COX2, BZR, DHFR, MSRC-9). Using only [1-100] learnable parameters for the quantum circuit and minimal classical layers (1000-5000 parameters) in a generic setting, the obtained results are comparable to and in some cases better than many of the baseline results, particularly for the cases when geometric structure plays a significant role.
In digital signal processing, the finite impulse response (FIR) filter is a fundamental tool for processing discrete-time signals. This paper explores the implementation of FIR filters using quantum computation methods. In this study, a quantum circuit for the FIR filter is designed using a normalized filter coefficient vector, orthogonal triangular decomposition commonly known as QR decomposition, and the transpilation tools provided by IBM’s software Qiskit SDK V2.3. Then, each block of the input signal is normalized to a unit-norm vector, loaded into a quantum register, and processed by the FIR filter quantum circuit to produce an output state. Quantum measurement is then performed on the output state to obtain a histogram, from which the first-bin data are scaled to compute the output sample of the filter. Finally, signal filtering experiments using FIR mean filters are conducted to demonstrate the effectiveness of the proposed quantum computation approach.
Quantum methods for image analysis and processing in the frequency domain represent an innovative approach to image processing using quantum computing. One of the key tools in this field is the quantum Fourier transform (QFT), which can be applied to quantum images to analyze and process data in the frequency domain. The process involves several steps, including quantum image preparation, QFT application, frequency domain data processing, and inverse Fourier transform to reconstruct the spatial domain image. Quantum image processing techniques offer new opportunities for solving complex problems in the field of image processing, especially in the context of the use of quantum computing. However, this area is still under active research and requires further research and technology development for practical application.