End-to-end quantum algorithm for topology optimization in structural mechanics

Quantum Science and Technology, vol. 11, pp. 025029

Abstract

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 an exponentially large Hilbert space representing the design space. We demonstrate the algorithm on the two-dimensional Messerschmitt-Bölkow-Blohm 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 using established quantum algorithms, including block-encoding of the stiffness matrix, quantum singular value transformation for matrix inversion, Hadamard test, and quantum amplitude estimation. 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.

Authors 8

1. Leonhard Hölscher

   BMW (Germany) · Forschungszentrum Jülich · BMW Group (Germany) · Saarland University

   Affiliation as printed

   BMW Group, 80809 Munich, Germany

   Institute for Quantum Computing Analytics (PGI 12), Forschungszentrum Jülich, 52425 Jülich, Germany

   Theoretical Physics, Saarland University, 66123 Saarbrücken, Germany

2. Oliver Ahrend Aachen Chair of Methods for Model‐based Development in Computational Engineering

   RWTH Aachen University

   Affiliation as printed

   Chair of Methods for Model-based Development in Computational Engineering, RWTH Aachen University, 52062 Aachen, Germany

3. Lukas Karch

   BMW (Germany) · BMW Group (Germany)

   Affiliation as printed

   BMW Group, 80809 Munich, Germany

4. Carlotta L'Estocq

   BMW (Germany) · BMW Group (Germany)

   Affiliation as printed

   BMW Group, 80809 Munich, Germany

5. Marc Marfany Andreu

   BMW (Germany) · BMW Group (Germany)

   Affiliation as printed

   BMW Group, 80809 Munich, Germany

6. Tobias Stollenwerk

   Forschungszentrum Jülich

   Affiliation as printed

   Institute for Quantum Computing Analytics (PGI 12), Forschungszentrum Jülich, 52425 Jülich, Germany

7. Frank K. Wilhelm

   Forschungszentrum Jülich · Saarland University

   Affiliation as printed

   Institute for Quantum Computing Analytics (PGI 12), Forschungszentrum Jülich, 52425 Jülich, Germany

   Theoretical Physics, Saarland University, 66123 Saarbrücken, Germany

8. Julia Kowalski Aachen Chair of Methods for Model‐based Development in Computational Engineering

   RWTH Aachen University

   Affiliation as printed

   Chair of Methods for Model-based Development in Computational Engineering, RWTH Aachen University, 52062 Aachen, Germany