Abstract
Constraint-preserving quantum optimization offers a principled alternative to penalty-based methods by restricting quantum evolution to the feasible subspace of an optimization problem. In this work, we unify and extend recent advances in constrained quantum algorithms by jointly addressing three core components: mixer Hamiltonians, problem encodings, and circuit-level implementations. We develop a general framework for constructing mixers that guarantee feasibility throughout the evolution, including stabilizer-based and projector-defined constructions that apply to a wide range of combinatorial and physical constraints. We analyze encoding strategies for discrete optimization problems, with particular emphasis on non-power-of-two variable domains, and show how subspace encodings and balanced full-space encodings impact both optimization landscapes and hardware resources. Finally, we present compact circuit constructions for implementing constrained evolutions, enabling efficient realization of mixers and excitation operators with reduced entangling-gate and fault-tolerant overhead. Together, these results clarify the trade-offs between expressivity, resource efficiency, and algorithmic performance in constraint-preserving quantum optimization, and provide practical guidelines for designing scalable quantum optimization algorithms on near-term and fault-tolerant quantum hardware.