Towards Quantum Computational Mechanics

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.

References in corpus (23)

• Elementary gates for quantum computation
• Quantum algorithm for solving linear systems of equations
• A numerical method for computing the overall response of nonlinear composites with complex microstructure
• On the role of entanglement in quantum computational speed-up
• Quantum algorithm for systems of linear equations with exponentially improved dependence on precision
• Perspectives of quantum annealing: Methods and implementations
• An approximate Fourier transform useful in quantum factoring
• Variational quantum algorithms for nonlinear problems
• Quantum Risk Analysis
• Concrete Categorical Model of a Quantum Circuit Description Language with Measurement
• Option Pricing using Quantum Computers
• Quantum algorithm and circuit design solving the Poisson equation
• A divide-and-conquer algorithm for quantum state preparation
• High-precision quantum algorithms for partial differential equations
• Quantum Algorithm for Simulating the Wave Equation
• Computational Homogenization of Polycrystals
• A Constrained Sequential-Lamination Algorithm for the Simulation of Sub-Grid Microstructure in Martensitic Materials
• Enhancing the Quantum Linear Systems Algorithm using Richardson Extrapolation
• Hybrid quantum algorithms for flow problems
• Box algorithm for the solution of differential equations on a quantum annealer
• Lecture Notes on Quantum Algorithms for Scientific Computation
• FEqa: Finite Element Computations on Quantum Annealers
• Computational modelling and data-driven homogenisation of knitted membranes

Cited by in corpus (6)

• A novel design update framework for topology optimization with quantum annealing: Application to truss and continuum structures
• Quantum Realization of the Finite Element Method
• Divergence-free algorithms for solving nonlinear differential equations on quantum computers
• QAFE${}^2$: Quantum accelerated multiscale finite element analysis
• Vortex Detection from Quantum Data
• Nonlinear path-following via the asymptotic numerical method on a quantum processor