Pure and mixed Dicke state ansatz for equality and inequality constraints in variational quantum eigensolver

Combinatorial optimization can be addressed with quantum computing through variational quantum algorithms, but a central challenge in this approach is to design an ansatz expressive enough to explore the feasible subspace of the Hilbert space where the optimal solution lies. Another major challenge is tuning the Lagrange multipliers in penalty terms to enforce feasibility and guarantee solution quality. To address both challenges, we propose the first feasibility-preserving mixed Dicke state ansatz for Hamming weight constrained combinatorial optimization, extending the density matrix formalism to structurally encode equality and inequality constraints directly into the quantum circuit, thereby eliminating the need for penalty terms in the objective function. The proposed framework handles both constraint types, with the pure Dicke state ansatz recovered as a special case corresponding to equality constraints, and generalizes to multiple constraint groups via tensor products of individual pure or mixed Dicke states. We validate the proposed approach in the context of combinatorial portfolio optimization across three experimental scenarios with increasing constraint complexity, using the CMA-ES optimizer and comparing its performance against random search with replacement restricted to the feasible subspace. As the feasible search space grows, the proposed ansatz demonstrates a clear advantage over random search in terms of the number of objective function calls required to identify high-quality solutions. Hardware experiments on IBM NISQ processors confirm that noise mitigation and circuit transpilation optimizations remain open challenges for practical deployment. The framework is general and directly applicable to other combinatorial optimization problems with Hamming weight constraints.