Effective Hamiltonian for Cross-Cross Resonance gate and Frequency tuning in Transmon qubits

Two-qubit entangler is one of the most important components for quantum computer. For a quantum computer based on transmon qubit, the Cross-Cross Resonance(CCR) gate has been recently proposed, which seems to operate faster than Cross-Resonance(CR) gate as an iSWAP gate. However, unlike the CR gate, the CCR gate needs a frequency change induced by Stark shift. Here, we present the theoretical study of CCR gate in transmon qubits. Using the Schrieffer–Wolff transformation and rotating-wave approximation, we obtain both the effective Hamiltonian for CCR gate and the frequency condition of CCR gate. We also perform numerical simulation using Quantum Toolbox in Python(QuTip). Our numerical simulation result shows an excellent agreement with the analytical one.