Towards Quantum Computational Mechanics
- URL: http://arxiv.org/abs/2312.03791v2
- Date: Sun, 7 Jan 2024 17:35:54 GMT
- Title: Towards Quantum Computational Mechanics
- Authors: Burigede Liu, Michael Ortiz, Fehmi Cirak
- Abstract summary: We show how quantum computing can be used to solve representative volume element (RVE) problems in computational homogenisation.
Our quantum RVE solver attains exponential acceleration with respect to classical solvers.
- Score: 1.7201069233638664
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- 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.
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