Optimizing a parameterized controlled gate using free quaternion selection

Abstract In variational quantum algorithms (VQAs), parameterization is typically applied to single-qubit gates. In this study, we instead parameterize a generalized controlled gate and propose an algorithm to locally minimize the cost function by maximally optimizing these parameters. This method extends the free quaternion selection technique, which was originally developed for single-qubit gate optimization. To evaluate its performance, we apply the proposed method to a variety of quantum optimization tasks, including the variational quantum eigensolver for both Ising and molecular Hamiltonians, fidelity maximization in general VQAs, and unitary compilation of time evolution operators. Across these applications, our method demonstrates efficient optimization, enhanced expressibility, and the ability to construct shallower circuits compared to existing techniques. Moreover, the method can be generalized to optimize particle-number-conserving gates, which are particularly relevant for quantum chemistry. Leveraging this capability, we further demonstrate that the method achieves superior quantum compilation of molecular time-evolution operators by approximating them with shallower circuits than standard Trotter decomposition.