Effective Hamiltonian and parametric tuning for the cross-cross-resonance gate in the transmon model

Abstract

The two-qubit entangler is one of the most important components of quantum computers. The cross-cross resonance (CCR) gate has recently been proposed and shown to operate faster than the cross-resonance (CR) gate as an iSWAP gate. However, unlike the CR gate, the CCR gate requires frequency modifications of the input microwave pulses induced by a Stark shift. We present a theoretical study on the CCR gate in the two-transmon system. Using the Schrieffer-Wolff transformation and the rotating-wave approximation, we obtain both the effective Hamiltonian and the input microwave pulse frequencies of 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, whose strength increases as the product of the two pulse amplitudes ${\mathrm{\ensuremath{\Omega}}}_{0}{\mathrm{\ensuremath{\Omega}}}_{1}$. We perform numerical simulations using the set of parameters for the IBM Hanoi quantum processor. We show that the CCR gate has a phase tunability and that varying the relative phase of the input microwave pulses can significantly improve the gate performance. This demonstrates that the static $ZZ$ term in the CCR gate can be effectively reduced by optimizing the relative phase of the input microwave pulses.