Characterization of Single- and Two-qubit Gates between Transmons and Capacitively Shunted Flux Qubits: Part 2, Theory

We theoretically model an experiment on a superconducting circuit made of a capacitively shunted flux qubit (CSFQ) and a Transmon qubit both capacitively coupled to a bus resonator in dispersive regime. To model this circuit we take into account the contribution of higher excited states in qubits and block-diagonalize the Hamiltonian perturbatively in the regime of small interaction couplings compared to frequency detuning. We apply external driving microwave pulses over all energy levels and consider the transitions they impose effectively within the computational subspace. More specifically we study single qubit gates and cross-resonance gate, and analyze the fidelity of their operations. Our theoretical results are in agreement with experiment, showing a promising approach to controllably improve single- and two-qubit gate operations in such circuits due to the relatively large and positive anharmonicity of CSFQ.