Optimizing a parameterized controlled gate with Free Quaternion Selection

In variational algorithms, quantum circuits are conventionally parametrized with respect to single-qubit gates. In this study, we parameterize a generalized controlled gate and propose an algorithm to estimate the optimal parameters for locally minimizing the cost value, where we extend the free quaternion selection method, an optimization method for a single-qubit gate. To benchmark the performance, we apply the proposed method to various optimization problems, including the Variational Quantum Eigensolver (VQE) for Ising and molecular Hamiltonians, Variational Quantum Algorithms (VQA) for fidelity maximization, and unitary compilation of time evolution operators. In these applications, the proposed method shows efficient optimization and greater expressibility with shallower circuits than other methods. Furthermore, this method is also capable of generalizing and fully optimizing particle-number-conserving gates, which are in demand in chemical systems applications. Taking advantage of this property, we have actually approximated time evolution operators of molecular Hamiltonian and simulated the dynamics with shallower circuits in comparison to the standard implementation by Trotter decomposition.
Comment: 21 pages, 15 figures