Direct calculation of the ZZ-interaction rates in the multi-mode circuit-QED

Hamiltonians of the superconducting qubits of Transmon type involve non-zero ZZ-interaction terms due to their finite and small anharmonicities. These terms might lead to the unwanted accumulation of spurious phases during the execution of the two-qubit gates. Exact calculation of the ZZ-interaction rates requires the full diagonalization of the circuit Hamiltonians which very quickly becomes computationally demanding as the number of the modes in the coupler circuit increases. Here we propose a direct analytical method for the accurate estimation of the ZZ-interaction rates between low-anharmonicity qubits in the dispersive limit of the multi-mode circuit-QED. We observe very good agreement between the predictions of our method and the measurement data collected from the multi-qubit devices. Our method being an extension of our previous work in [1] is a new addition to the toolbox of the quantum microwave engineers as it relates the ZZ-interaction rates directly to the entries of the impedance matrix defined between the qubit ports.