Voxel-based block variational quantum linear solver: a hybrid quantum-classical method for static analysis of solids

2026-09-24

AI Summary

This study addresses the computational bottleneck of solving sparse linear systems in large-scale solid static analysis, alongside challenges inherent to quantum finite element methods such as matrix decomposition overhead, barren plateaus, and measurement costs. To overcome these limitations, this work proposes a voxel-block variational quantum linear solver. The method constructs a problem-size-independent Linear Combination of Unitaries (LCU) decomposition and leverages the principle of minimum potential energy as the objective function to mitigate barren plateaus. Furthermore, a block Hadamard test is introduced to substantially reduce the number of required circuit configurations. Noise-inclusive simulations demonstrate that the proposed approach significantly decreases both the LCU term count and convergence iterations. By preserving solution accuracy within finite precision, this framework offers an efficient paradigm for quantum finite element analysis on regular grids.

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.

Problem

• variational quantum linear solver
• finite element analysis
• sparse linear systems
• barren plateaus
• solid mechanics

Innovation

• Variational Quantum Linear Solver
• Finite Element Analysis
• Minimum Potential Energy
• Block-Hadamard Test
• LCU Decomposition