Coherence Limits in Interference-Based cos(2$φ$) Qubits

We investigate the coherence properties of parity-protected $\cos(2φ)$ qubits based on interferences between two Josephson elements in a superconducting loop. We show that qubit implementations of a $\cos(2φ)$ potential using a single loop, such as those employing semiconducting junctions, rhombus circuits, flowermon and KITE structures, can be described by the same Hamiltonian as two multi-harmonic Josephson junctions in a SQUID geometry. We find that, despite the parity protection arising from the suppression of single Cooper pair tunneling, there exists a fundamental trade-off between charge and flux noise dephasing channels. Using numerical simulations, we examine how relaxation and dephasing rates depend on external flux and circuit parameters, and we identify the best compromise for maximum coherence. With currently existing circuit parameters, the qubit lifetime $T_1$ can exceed milliseconds while the dephasing time $T_φ$ remains limited to only a few microseconds due to either flux or charge noise. Our findings establish practical limits on the coherence of this class of qubits and raise questions about the long-term potential of this approach.