Accelerating large-scale linear algebra using variational quantum imaginary time evolution

The solution of large sparse linear systems via factorization methods such as LU or Cholesky decomposition, can be computationally expensive due to the introduction of non-zero elements, or ``fill-in.'' Graph partitioning can be used to reduce the ``fill-in,'' thereby speeding up the solution of the linear system. We introduce a quantum approach to the graph partitioning problem based on variational quantum imaginary time evolution (VarQITE). We develop a hybrid quantum/classical method to accelerate Finite Element Analysis (FEA) by using VarQITE in Ansys's LS-DYNA multiphysics simulation software. This allows us to study different types of FEA problems, from mechanical engineering to computational fluid dynamics in simulations and on quantum hardware (IonQ Aria and IonQ Forte). We demonstrate that VarQITE has the potential to impact LS-DYNA workflows by measuring the wall-clock time to solution of FEA problems. We report performance results for our hybrid quantum/classical workflow on selected FEA problem instances, including simulation of blood pumps, automotive roof crush, and vibration analysis of car bodies on meshes of up to six million elements. We find that the LS-DYNA wall clock time can be improved by up to 12\% for some problems. Finally, we introduce a classical heuristic inspired by Fiduccia-Mattheyses to improve the quality of VarQITE solutions obtained from hardware runs. Our results highlight the potential impact of quantum computing on large-scale FEA problems in the NISQ era.