Effective Suppression of Static ZZ term by Parametric Tuning for the Cross-Cross Resonance Gate in the Transmon Model

The two-qubit entangler is one of the most important components of quantum computers. The cross-cross resonance (CCR) gate has recently been proposed as a faster alternative to the cross-resonance (CR) gate, operating as an iSWAP gate. However, unlike the CR gate, the CCR gate requires frequency modifications of the input microwave pulses due to a Stark shift. We presented theoretical analysis of the CCR gate in a two-transmon system. By employing the Schrieffer-Wolff transformation and the rotating-wave approximation, we derive both the effective Hamiltonian and the optimal input microwave pulse frequencies for the CCR gate. It is well known that the static ZZ term degrades the performance of the CR gate in the transmon system.

For the CCR gate, an additional dynamic ZZ term arises from the combined coupling to the qubits by the input microwave pulses, with its strength increasing as the product of the two pulse amplitudes Ω₀Ω₁. We performed the numerical simulations using the set of parameters for the IBM Hanoi quantum processor. We showed that the CCR gate exhibits phase tunability, where varying the relative phase of the input microwave pulses can significantly enhance gate performance, especially near θ=3π/2. This indicates that the static ZZ term in the CCR gate can be effectively mitigated by optimizing the relative phase of the input microwave pulses.