We propose a framework to solve non-linear and history-dependent mechanical problems based on a hybrid classical computer-quantum annealer approach. Quantum Computers are anticipated to solve particular operations exponentially faster. The available possible operations are however not as versatile as with a classical computer. However, quantum annealers (QAs) is well suited to evaluate the minimum state of a Hamiltonian quadratic potential. Therefore, we reformulate the elasto-plastic finite element problem as a double minimisation process framed at the structural scale using the variational updates formulation. In order to comply with the expected quadratic nature of the Hamiltonian, the resulting non-linear minimisation problems are iteratively solved with the suggested Quantum Annealing-assisted Sequential Quadratic Programming (QA-SQP): a sequence of minimising quadratic problems is performed by approximating the objective function by a quadratic Taylor's series. Each quadratic minimisation problem of continuous variables is then transformed into a binary quadratic problem. This binary quadratic minimisation problem can be solved on quantum annealing hardware such as the D-Wave system. The applicability of the proposed framework is demonstrated with one and two-dimensional elasto-plastic numerical benchmarks. The current work provides a pathway of performing general non-linear finite element simulations assisted by quantum computing.

翻译：本文提出一种基于经典计算机与量子退火器混合架构的框架，用于求解非线性及历史依赖的力学问题。量子计算机有望以指数级速度加速特定运算，但其可用操作不如经典计算机灵活。然而，量子退火器特别适合评估哈密顿二次势能的基态。为此，我们采用变分更新公式，将弹塑性有限元问题重构为结构尺度上的双重最小化过程。为符合哈密顿量预期的二次形式，所生成的非线性最小化问题通过提出的量子退火辅助序列二次规划（QA-SQP）迭代求解：通过目标函数的二次泰勒级数近似，依次求解一系列二次最小化问题。每个连续变量的二次最小化问题进一步转化为二元二次问题，该问题可在D-Wave系统等量子退火硬件上求解。通过一维和二维弹塑性数值基准算例验证了所提框架的适用性。本研究为量子计算辅助下的一般非线性有限元模拟提供了可行路径。