End-to-End Quantum Algorithm for Topology Optimization in Structural Mechanics

TL;DR

This paper presents a fault-tolerant end-to-end quantum algorithm for topology optimization in structural mechanics, addressing the combinatorial explosion of binary material distributions by encoding designs in a quantum register and applying Grover's search on a space of size $N=2^{n_\text{el}}$. The workflow integrates a FEM-based compliance evaluation with block-encoding of the stiffness matrix, Quantum Singular Value Transform for matrix inversion, Hadamard tests, and Quantum Amplitude Estimation within a Grover oracle to produce a phase encoding of the objective $c(\mathbf{x})$. The authors demonstrate the method on the 2D MBB beam, including volume constraints enforced by Dicke-state initialization, and provide a detailed complexity analysis showing a quadratic speedup relative to classical unstructured search. Numerical simulations validate the subroutines, revealing practical challenges such as the need for high-degree QSVT polynomials when scaling to larger $n_\text{el}$ and the phase-resolution required to distinguish near-optimal designs.

Abstract

Topology optimization is a key methodology in engineering design for finding efficient and robust structures. Due to the enormous size of the design space, evaluating all possible configurations is typically infeasible. In this work, we present an end-to-end, fault-tolerant quantum algorithm for topology optimization that operates on the exponential Hilbert space representing the design space. We demonstrate the algorithm on the two-dimensional Messerschmitt-Bölkow-Blohm (MBB) beam problem. By restricting design variables to binary values, we reformulate the compliance minimization task as a combinatorial satisfiability problem solved using Grover's algorithm. Within Grover's oracle, the compliance is computed through the finite-element method (FEM) using established quantum algorithms, including block-encoding of the stiffness matrix, Quantum Singular Value Transformation (QSVT) for matrix inversion, Hadamard test, and Quantum Amplitude Estimation (QAE). The complete algorithm is implemented and validated using classical quantum-circuit simulations. A detailed complexity analysis shows that the method evaluates the compliance of exponentially many structures in quantum superposition in polynomial time. In the global search, our approach maintains Grover's quadratic speedup compared to classical unstructured search. Overall, the proposed quantum workflow demonstrates how quantum algorithms can advance the field of computational science and engineering.

Figures (22)

• Figure 1: FEM domain discretization using quadrilateral elements. (a) shows a single element with four nodes, each having a horizontal and a vertical degree of freedom (DoF), and (b) depicts a domain consisting of $2\times2$ elements. Either way, we order the DoFs by counting column-wise from top to bottom from left to right. Horizontal DoFs have precedence over vertical DoFs.
• Figure 2: Example solution for an MBB beam problem. The imposed boundary conditions and the square domain are displayed on the left. The horizontal DoFs of all nodes on the left boundary are fixed as well as the vertical DoF at the bottom-right corner. A single force pointing downwards is applied at the top-left corner. The right image shows a rounded optimized structure for $9\times9$ elements and a volume fraction of $V_0=1/2$ using the SIMP method outlined in Ref. andreassen_efficient_2011.
• Figure 3: Quantum circuit for Grover's algorithm. $U_\text{init}$ is used to prepare the superposition of the search space. The following $r$ iterations apply Oracle $O$ and Diffusion $D$ repeatedly, before finally the mesurement is conducted.
• Figure 4: Geometrical illustration of a single Grover iteration.
• Figure 5: Quantum circuit for Grover's Oracle operation $O$.