In this paper, we propose a robust protocol for achieving precise parity measurements of $N$ cat-state qubits stabilized in $N$ Kerr nonlinear cavities, leveraging the steady states of an auxiliary cavity mode. The interactions between the cat-state qubits and the auxiliary cavity are mediated by a superconducting artificial atom. By modulating the classical field applied to the artificial atom at suitable frequencies, we derive a parity-dependent effective Hamiltonian for the system. When the artificial atom is coupled to a thermal reservoir, a relatively strong energy relaxation is induced. Under the dissipative dynamics governed by the effective Hamiltonian, this relaxation drives the auxiliary cavity mode toward two distinct steady states corresponding to the parities of the $N$ cat-state qubits. As a result, the parity information of the cat-state qubits can be extracted through the measurement of the auxiliary cavity. Numerical simulations indicate that the protocol is robust to systematic calibration errors (in the drives' Rabi frequencies and coupling strengths), cavity photon loss, and ancilla energy relaxation within realistic parameter regimes. Rather than relying on population transfers, the protocol reads out parity by mapping the global parity to distinct auxiliary-cavity steady states, providing an alternative route to parity measurements of many cat-state qubits.

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• Quantum communication
• Quantum computation
• Quantum information processing