Summary
Large solid objects, like bridges or buildings, need to be analyzed by breaking them into many small pieces to see how forces affect them. Solving these problems usually takes a lot of time and computer memory. The paper introduces a new way to use quantum computers together with classical computers to solve the equations faster and more efficiently for regular grid structures. Their method breaks down the problem in a special way, uses a better way to adjust the quantum computer’s settings, and reduces the number of measurements needed, making the overall process quicker. They tested this approach on a simulated computer and showed it works with good accuracy while needing fewer steps.
What this means in practice
- •For structural engineers: Speed up static analysis of solid structures modeled on regular grids by using quantum-classical hybrid methods to reduce computation time.
- •For quantum software developers: Implement improved quantum algorithms for solving large sparse linear systems arising from finite element discretization with fewer quantum resources and faster convergence.
Authors
Feng Wu, Chen Li, Li Zhu, Yuxiang Yang, Xu Guo
Abstract
In solid mechanics, finite element discretization of large-scale static problems produces large sparse linear systems whose solution requires substantial computation time and memory. The variational quantum linear solver (VQLS) offers a hybrid quantum-classical route, but its use in quantum finite element analysis is limited by the decomposition of nonunitary matrices, barren plateaus, and the measurement cost of expectation values. We propose a voxel-based block variational quantum linear solver (Voxel-BVQLS) that combines structured matrix decomposition, the principle of minimum potential energy, and batched quantum tests. First, we construct an LCU decomposition of the stiffness matrix from the recursive block-banded structure of voxel-grid finite element matrices, with the number of unitary terms bounded independently of the problem size. Second, we optimize the ansatz parameters using a minimum-potential-energy objective in place of a conventional VQLS loss function, thereby mitigating barren plateaus in the studied problems. Third, we introduce a block-Hadamard test whose circuit directly estimates weighted sums of multiple inner products, reducing the number of circuit configurations required per iteration. We assessed the proposed method in noiseless classical simulations using three examples. These examples show that the method reduces both the number of unitary terms in the LCU decomposition and the number of iterations required to converge, while still yielding solutions of finite accuracy. Voxel-BVQLS thus provides a structured hybrid quantum-classical framework for quantum finite element analysis on regular grids.