Quantum remeshing and efficient encoding for fracture mechanics

TL;DR

This work introduces a variational quantum framework to address 2D fracture mechanics by encoding nodal displacements as quantum amplitudes and solving the elastic energy minimization problem via a tensor-product decomposition of the stiffness matrix, enabling polylogarithmic measurements. A cascaded warm-start remeshing strategy leverages coarse solutions to initialize finer meshes, mitigating barren plateaus and enabling scalable simulations up to tens of qubits. The method is validated through numerical simulations and an experimental demonstration on Quandela Ascella, achieving energy estimates within ~96% of the noiseless optimum. The approach yields efficient extraction of key fracture observables such as the Stress Intensity Factor (SIF) and Crack Opening Displacement (COD), and outlines a path toward extending to 3D and more complex boundary conditions for practical quantum-accelerated structural simulations.

Abstract

We present a variational quantum algorithm for structural mechanical problems, specifically addressing crack opening simulations that traditionally require extensive computational resources. Our approach provides an alternative solution for a relevant 2D case by implementing a parametrized quantum circuit that stores nodal displacements as quantum amplitudes and efficiently extracts critical observables. The algorithm achieves optimal nodal displacements by minimizing the elastic energy obtained from finite element method. The energy is computed with only a polylogarithmic number of measurements. Extracting relevant scalar observables such as the stress intensity factor is then done efficiently on the converged solution. To validate the scalability of our approach, we develop a warm start strategy based on a remeshing technique that uses coarse solutions to circumvent barren plateaus in the optimization landscape of the more refined problems. Our method has been experimentally validated on Quandela's photonic quantum processor Ascella and comprehensive numerical simulations demonstrate its scalability across increasingly complex quantum systems.

Figures (22)

• Figure 1: Geometry of the pre-cracked 2D plate under external loading, showing the crack tip and nodal grid. This mesh underlies the FEM formulation (Eq. 3) and the operator decomposition (Eq. 5).
• Figure 2: We first optimize a 5-qubit state, and duplicate it to use it as a warm start for the 7-qubit ansatz. This scheme is applied iteratively until the desired system size is reached as seen in Fig. \ref{['fig:c5719']}.
• Figure 3: Cascaded simulations from 5 to 19 qubit mesh. In (b) the iterative remeshing, performing 48h optimization at each step, allows to reach 19 qubit simulation with $50\%$ precision of observable COD and SIF, far beyond the capacities of any tested cold start strategies. (c) illustrates a single step remeshing capabilities to bypass the initial barren plateau with 50 Layers ansatz. The $n+2$-qubit state is optimized for 2 hours starting from an ideal state before duplication, obtained from $n$-qubit discretization.
• Figure 4: The results from 10 runs on Quandela’s linear optical QPU. The blue line shows the mean energy estimate at each iteration, while the shaded region is bounded by the minimum and maximum values. Run 8, which produced the minimum energy estimate overall, is shown as a pink line.
• Figure 5: $\mathbb{K}_{aa}$ component contribution.