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.

翻译：量子计算机的运行基于与经典数字计算机完全不同的物理原理和抽象机制，其作为一种新型计算范式，有望带来颠覆性的效率提升和计算性能突破。具体而言，量子计算机能够同时演化整个量子系统的状态，从而产生量子并行性和干涉效应。尽管前景广阔，但将量子计算应用于计算力学问题的机遇仍鲜有探索。本文证明，量子计算确实可用于求解计算均匀化中的代表性体积单元（RVE）问题，其复杂度为多对数阶 $\mathcal{O}((\log N)^c)$，而经典计算的复杂度为 $\mathcal{O}(N^c)$。因此，我们的量子RVE求解器相较于经典求解器实现了指数级加速，使并发的多尺度计算更接近实用化。所提出的量子RVE求解器融合了传统算法，如同质参考材料的固定点迭代和快速傅里叶变换（FFT）。然而，这些算法的量子计算重构需要根本性的范式转变，并对经典实现进行彻底的重新思考与重构。我们采用或开发了多种技术，包括量子傅里叶变换（QFT）、多项式量子编码、函数的经典分段切比雪夫近似以及实现固定点迭代的辅助算法，并证明在量子计算机上高效实现RVE求解器是可行的。此外，我们提供了理论证明和数值证据，证实了所提求解器预期的 $\mathcal{O} \left ((\log N)^c \right)$ 复杂度。