Quantum Fourier Transform Methods for Exponential Speedup in Computational Mechanics and Signal Processing

Description

This research proposes the development of novel quantum algorithms that integrate Fast Fourier Transform (FFT) techniques to achieve exponential speedup in solving computational mechanics problems and signal processing tasks. The core challenge lies in the fact that while the Quantum Fourier Transform (QFT) serves as the quantum analogue to the classical FFT and leverages quantum parallelism to reduce time complexity , its practical implementation in areas like computational homogenization and multiscale analysis remains largely unexplored . The central aim is to design and validate quantum solvers that combine conventional FFT-based algorithms with quantum computing principles to reduce computational complexity from classical (O(N^c)) to quantum polylogarithmic (O((log N)^c)) . The methodology will employ the Quantum Fourier Transform alongside techniques such as quantum encoding of polynomials and fixed-point iterations, reimagining classical FFT-based approaches within a quantum circuit framework . The expected contribution is the demonstration of exponential acceleration for representative volume element (RVE) problems, bringing concurrent multiscale computing closer to practicality . Application domain one focuses on computational mechanics and materials science, where the quantum RVE solver will enable efficient simulation of complex microstructures that are currently intractable with classical methods . Application domain two targets signal processing and autocorrelation computation, where QFT-based methods can leverage quantum parallelism for denoising, system identification, and telecommunications applications with reduced computational complexity . The significance of this work lies in establishing a foundational bridge between classical FFT-dominated fields and quantum computing, potentially revolutionizing how we approach large-scale scientific computing problems. The feasibility is supported by recent theoretical proofs and numerical evidence confirming the anticipated polylogarithmic complexity of quantum FFT-based solvers, as well as advances in quantum signal processing that enable efficient polynomial approximations.