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

TL;DR

The paper tackles the high computational cost of solving large sparse linear systems by reducing fill-in via graph partitioning. It introduces a hybrid quantum-classical approach using Variational Quantum Imaginary Time Evolution (VarQITE) to solve the Graph Partitioning Problem (GPP) formulated as a QUBO/Hamiltonian energy minimization, and integrates this into LS-DYNA workflows for finite element analyses. Through noiseless simulations and experiments on IonQ Aria and Forte, the authors demonstrate that VarQITE can produce partitions with competitive merit factors and, in several cases, reduce total wall-clock time for linear solves, especially when combined with a classical refinement step inspired by Fiduccia-Mattheyses. The work provides empirical evidence of potential quantum utility in large-scale FEA within the NISQ era, outlines a scalable ansatz (HeavyNeighborsAnsatz) tailored to graph structure, and proposes a practical refinement loop to mitigate hardware noise. These results point to a promising near-term pathway for quantum-accelerated preconditioning and reordering in industrial simulations, with clear directions for scaling and further optimization.

Abstract

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.

Figures (13)

• Figure 1: The execution flow chart of LS-DYNA including calls to the quantum computer. Starting from the upper right is a car which is modeled with a finite element mesh, together with an impact. The time evolution is given as a linear system of equations to be solved $Ax = b$. Here $A$ is a sparse matrix which when decomposed using $LDL^T$ decomposition leads to the dense matrices shown in the figure. A graph partitioning problem is formulated from the adjacency graph of $A$ and is solved on the quantum computer, indicated by an ion trap on the lower right. This yields $P$, a permutation matrix to reorder $A$ obtained by solving the graph partitioning problem recursively. The solution is used to partition the mesh as illustrated by the blue line of vertices. The reordered matrix $A$ retains a sparse structure after $LDL^T$ decomposition. The linear system is now solved using the the reordered matrix $A$ and the reordered vector $b$ to obtain the required deformation simulation of the original model. The process can be iterated.
• Figure 2: VarQITE ansatz: (A) An example of a $5$ node graph with the nodes and edge weights labeled, (B) The nodes sorted according to $W^*$, the total edge weight of their induced ego graph of radius $1$. (C) The VarQITE ansatz generated using the ego graphs. In this illustration, the ansatz is constructed with $4$ gates per layer (this number can be adjusted as required). The included gates are shown in dark blue while the excluded gates are shown in white (since there are only $4$ gates per layer, some of the gates will need to be excluded). The first layer (Layer 0) has entangling gates between qubits connected by the graph edges in the descending order of edge weights. The second layer (Layer 1) has entangling gates between qubits sorted by their induced ego graph of radius $1$. More layers may be added, generated by ego graphs of radii equal to $2, 3, 4$, etc.
• Figure 3: Integration of VarQITE into LS-DYNA for the evaluation of the factor of merit in a level-1 nested dissection process. LS-DYNA (orange arrows and boxes) executes two times in this integration framework. The VarQITE algorithm (blue arrows and boxes), which computes optimal partitions, operates externally between the two executions of LS-DYNA.
• Figure 4: Convergence of VarQITE on a Noiseless Simulator: Plots for (A) RoofCrush 30 nodes, (B) RoofCrush 32 nodes, (C) BloodPump 30 nodes, (D) BloodPump 32 nodes. The y-axis shows the value of 1-AR where AR is the Approximation Ratio of the solution. The solid blue and purple lines correspond to the average and best value from the sampled distribution, respectively. The shaded blue region corresponds to the values between the 10th and 90th percentile of the sampled distribution. All results were obtained using 2,000 shots to sample the quantum circuit.
• Figure 5: Comparison of merit factors resulting from the VarQITE algorithm and LS-GPart. The two panels shown are (A) RoofCrush and (B) BloodPump, and each problem uses a Level 1 nested dissection. Each panel includes (top) the number of non-zeros ("fill-in"), and (bottom) the number of operations estimated from symbolic factorization by LS-DYNA. The VarQITE data is restricted to graph partitions that maintain a nodal weight balance within a 5% tolerance. The QPU data is from graphs with the same number of nodes as the corresponding simulation data, the plotting is offset for clarity. The plot also shows results from applying the modified FM algorithm to the QPU data. Shown on the horizontal axis are different numbers of nodes for the coarsened graph, corresponding to the number of qubits. Note that the largest VarQITE experiments reported on have 32 nodes/qubits, however, the horizontal scale is extended logarithmically to much larger sizes of coarsening in order to capture the regime in which LS-GPart operates. The 4 best solutions are plotted for each number of nodes/qubits in the coarsening, shown as the grey dots in the charts. As can be seen in the lower panel, for the BloodPump problem, solutions obtained by coarsening to 28 nodes lead to better merit factors when compared to LS-GPart (orange dots and lines) in terms of non-zeroes and in terms of factorization operations, even when compared to numbers of nodes that are as high as 50,000.