Effective Hamiltonian in superconducting qubits

Qubits with more than two energy levels, such as superconducting transmons, usually are externally driven in order to engineer one and two qubit gates. However due to the presence of higher excited levels the fidelity of the gates requires improvement. Such a system carries a large Hilbert space and recognizing effective qubits requires to use perturbation theory. This puts a lage limitation on the system parameters and interactions. We discuss a method that allows to go beyond regular perturbative limitations and separates classical effects from quantum fluctuations in the Hamiltonian of a weakly-anharmonic qubits. We compare results taken from applying Schrieffer-Wolff transformation, Least action principle, and our method. Our results will become practical tools for experimental efforts in circuit QED.