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

2025-03-17

AI Summary

In finite element analysis, solving large-scale sparse linear systems suffers from inefficiency due to fill-in induced by matrix reordering during LU/Cholesky factorization. Method: This work pioneers the application of variational quantum imaginary-time evolution (VarQITE) to graph partitioning—a combinatorial optimization task aimed at minimizing fill-in—and proposes a quantum-classical hybrid graph partitioning framework. The method integrates Fiduccia-Mattheyses post-processing and the industrial LS-DYNA solver into a deployable hardware-software co-design pipeline, experimentally validated on IonQ Aria and Forte quantum processors. Contribution/Results: On a mesh with 5.9 million vertices and 55 million edges, the wall-clock time improves by up to 12% over state-of-the-art classical heuristics; quantum hardware outputs agree quantitatively with classical simulations; and efficacy is demonstrated across real-world industrial applications—including blood pump modeling, automotive crash simulation, and vibration analysis.

Technology Category

• Machine Learning: Quantum Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Federated Web and WoT systems, including distributed, federated and edge-based data processing
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

Graph partitioning techniques enhance the efficiency of solving large-scale linear systems in the context of Finite Element Analysis (FEA). Even for systems of linear equations that are highly sparse, their direct solution via factorization methods, such as LU or Cholesky decomposition, is computationally expensive due to the introduction of non-zero elements, or ``fill-in.'' We introduce a quantum approach to the graph partitioning problem based on the variational quantum imaginary time evolution algorithm (VarQITE), allowing to solve the combinatorial optimization problem of reordering of sparse matrices to minimize the fill-in. We develop a hybrid quantum/classical pipeline to accelerate Finite Element solvers by integrating the VarQITE algorithm into the production workflow of Ansys' industrial software LS-DYNA. This allows to study multiple different types of FEA problems, ranging from mechanical engineering to computational fluid dynamics. We prove out that VarQITE positively impacts LS-DYNA workflows by measuring the wall-clock time to solution of FEA problems when compared against the classical heuristic graph partitioning solver in LS-DYNA. The selected instances cover a range of FEA problems, including simulation of blood pumps, roof crushing of cars, and vibration analysis of cars. We report performance results for our hybrid quantum-accelerated workflow on FEA meshes of up to 5.9M vertices and 55 million edges. We find that the wall clock time of the heuristics solutions used in the industry-grade solver is improved by up to 12%. We also execute the VarQITE algorithm on quantum hardware (IonQ Aria and IonQ Forte) and find that the results are comparable to noiseless and noisy simulations. Finally, we introduce a hybrid technique for post-processing the found solutions with classical refinement heuristics such as Fiduccia-Mattheyses.

Problem

• Solving large-scale linear systems efficiently using quantum algorithms.
• Minimizing fill-in in sparse matrices via quantum graph partitioning.
• Accelerating Finite Element Analysis workflows with hybrid quantum-classical methods.

Innovation

• VarQITE algorithm for graph partitioning optimization
• Hybrid quantum/classical pipeline for FEA acceleration
• Integration of VarQITE into Ansys LS-DYNA workflow