Three-qubit direct dispersive parity measurement with Tunable Coupling Qubits

We consider the direct three-qubit parity measurement scheme with two measurement resonators, using circuit quantum electrodynamics to analyze its functioning for several different types of superconducting qubits. We find that for the most common, transmon-like qubit, the presence of additional qubit-state dependent coupling terms of the two resonators hinders the possibility of performing the direct parity measurement. We show how this problem can be solved by employing the Tunable Coupling Qubit (TCQ) in a particular designed configuration. In this case, we effectively engineer the original model Hamiltonian by cancelling the harmful terms. We further develop an analysis of the measurement in terms of information gains and provide some estimates of the typical parameters for optimal operation with TCQs.