<p>We investigate a class of complex quadratic programming problems characterized by unit-modulus and discrete argument constraints. This problem can be reformulated as a mixed-integer quadratic programming problem, which could be addressed using a commercial solver such as Gurobi. However, the solver’s efficiency is often unsatisfying if the problem formulation is inadequately designed. In this paper, we introduce several quadratic convex reformulations aimed at enhancing the solver’s performance. We extend the classical diagonal perturbation-based reformulation technique to this problem. Additionally, by leveraging the unique structure of the problem, we derive a new quadratic convex reformulation that provides a tighter continuous relaxation compared to the diagonal perturbation-based approach. The numerical tests on random instances and the unimodular code design problem demonstrate the superiority of the newly proposed reformulation.</p>