Quantum Approach to Classical Signal Processing

Classical signal processing methods, grounded in linear transformations such as Fourier and Laplace, have long served as cornerstones in engineering and scientific domains. The advent of quantum computing introduces transformative tools, including the Quantum Fourier Transform (QFT) and Quantum Inverse Fourier Transform (Q-IFT), which operate with O(log2 N) complexity, offering exponential al speedups over their classical counterparts O(N*log (N)). This paper delves into the quantum reimagining of signal processing, employing principles of quantum parallelism, superposition, and entanglement to tackle spectral analysis, filtering, and convolution tasks. By encoding signals into quantum states and leveraging unitary operators, we demonstrate superior performance in high-dimensional signal decomposition and noise-tolerant computation. Practical implementations are analyzed through simulations in Qiskit, highlighting the enhancements in spectral resolution and frequency trade-offs. Insights from prior work underscore critical challenges, including gate fidelity and noise-induced decoherence, but also illuminate pathways for hybrid quantum-classical signal processing systems. This study establishes a mathematical and practical framework for transitioning classical signal processing tasks into the quantum paradigm, underscoring the potential for unprecedented advances in efficiency and scalability.