ABSTRACT

The advent of quantum computers, operating on entirely different physical principles and abstractions from those of classical digital computers, sets forth a new computing paradigm that can potentially result in game-changing efficiencies and computational performance. Specifically, the ability to simultaneously evolve the state of an entire quantum system leads to quantum parallelism and interference. Despite these prospects, opportunities to bring quantum computing to bear on problems of computational mechanics remain largely unexplored. In this work, we demonstrate how quantum computing can indeed be used to solve representative volume element (RVE) problems in computational homogenisation with polylogarithmic complexity of \( \mathcal{O}((\log N)^c) \), compared to \( \mathcal{O}(N^c) \) in classical computing. Thus, our quantum RVE solver attains exponential acceleration with respect to classical solvers, bringing concurrent multiscale computing closer to practicality. The proposed quantum RVE solver combines conventional algorithms such as a fixed-point iteration for a homogeneous reference material and the Fast Fourier Transform (FFT). However, the quantum computing reformulation of these algorithms requires a fundamental paradigm shift and a complete rethinking and overhaul of the classical implementation. We employ or develop several techniques, including the Quantum Fourier Transform (QFT), quantum encoding of polynomials, classical piecewise Chebyshev approximation of functions and an auxiliary algorithm for implementing the fixed-point iteration and show that, indeed, an efficient implementation of RVE solvers on quantum computers is possible. We additionally provide theoretical proofs and numerical evidence confirming the anticipated \( \mathcal{O} \left ((\log N)^c \right) \) complexity of the proposed solver.

SPEAKER CV

Prof. Miguel (Michael) Ortiz  is the Frank and Ora Lee Marble Professor Emeritus of Aeronautics and Mechanical Engineering at Caltech. Originally from Spain, he earned his B.S. in Civil Engineering from the Polytechnic University of Madrid, followed by an M.S. and Ph.D. in Civil Engineering from UC Berkeley. His research spans mechanics of materials and computational solid mechanics. Ortiz has held prestigious positions at Brown University, Caltech, and Stanford University, and has received numerous accolades, including the Timoshenko Medal and the John von Neumann Medal. He continues to contribute to the field as a Bonn Research Chair at Bonn University.