Variational Quantum Latent Encoding for Topology Optimization

Abstract: A variational framework for structural topology optimization is developed, integrating quantum and classical latent encoding strategies within a coordinate-based neural decoding architecture. In this approach, a low-dimensional latent vector, generated either by a variational quantum circuit or sampled from a Gaussian distribution, is mapped to a higher-dimensional latent space via a learnable projection layer. This enriched representation is then decoded into a high-resolution material distribution using a neural network that takes both the latent vector and Fourier-mapped spatial coordinates as input. The optimization is performed directly on the latent parameters, guided solely by physics-based objectives such as compliance minimization and volume constraints evaluated through finite element analysis, without requiring any precomputed datasets or supervised training. Quantum latent vectors are constructed from the expectation values of Pauli observables measured on parameterized quantum circuits, providing a structured and entangled encoding of information. The classical baseline uses Gaussian-sampled latent vectors projected in the same manner. The proposed variational formulation enables the generation of diverse and physically valid topologies by exploring the latent space through sampling or perturbation, in contrast to traditional optimization methods that yield a single deterministic solution. Numerical experiments show that both classical and quantum encodings produce high-quality structural designs. However, quantum encodings demonstrate advantages in several benchmark cases in terms of compliance and design diversity. These results highlight the potential of quantum circuits as an effective and scalable tool for physics-constrained topology optimization and suggest promising directions for applying near-term quantum hardware in structural design.