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) are 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：我们提出了一种基于经典计算机-量子退火器混合框架，用于求解非线性及历史相关的力学问题。量子计算机预计能以指数级速度加速特定运算，然而其可执行操作远不及经典计算机灵活。但量子退火器特别适用于评估哈密顿二次势的最小状态。为此，我们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，并采用变分更新公式进行描述。为满足哈密顿量期望的二次形式特性，所得到的非线性最小化问题通过建议的量子退火辅助序列二次规划（QA-SQP）迭代求解：通过泰勒二次展开近似目标函数，逐步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随后转化为二进制二次问题，该二进制二次最小化问题可在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测试验证了该框架的适用性。当前研究为量子计算辅助下的通用非线性有限元模拟提供了实现路径。