Recently, it has been demonstrated that quantum implementations of Fourier-based spectral methods can achieve O(polylog(N)) complexity in computational mechanics [1, 2]. This represents an exponential speed-up over equivalent classical approaches. In both works, a truncated Fourier basis is used to discretise all spatial fields, and a fixed-point iteration is employed to solve a sequence of problems with a homogeneous material distribution. The mapping between the physical and Fourier spaces is carried out using the quantum Fourier transform (QFT) for periodic fields or the quantum sine transform (QST) for fields with zero Dirichlet data. We extend these works towards a framework for quantum–classical topology optimisation while introducing new approaches to account for material heterogeneity [3]. In the proposed topology optimisation framework, the cost function and its derivatives are determined using the quantum solver, while the new material distribution is determined classically. Our formulation provides a pathway to extend quantum computational strategies to broader classes of mechanical problems, including anisotropic materials and complex structural layouts. We outline the algorithmic structure, discuss circuit-level implementation aspects, and present numerical demonstrations illustrating the computational scaling behaviour. References [1] Liu, B., Ortiz, M., and Cirak, F., Towards quantum computational mechanics, Computer Methods in Applied Mechanics and Engineering, Vol. 432, 117403, 2024. [2] Febrianto, E., Wang, Y., Liu, B., Ortiz, M., and Cirak, F., A quantum spectral method for non- periodic boundary value problems, arXiv preprint arXiv:2511.11494, 2025. [3] Martyn, J. M., Rossi, Z. M., Tan, A. K., and Chuang, I. L., Grand Unification of Quantum Algorithms, PRX Quantum 2(4):040203, 2021.