  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.
