  Superconducting circuits with coupler architecture receive considerable
attention due to their advantages in tunability and scalability. Although
single-qubit gates with low error have been achieved, high-fidelity two-qubit
gates in coupler architecture are still challenging. This paper pays special
attention to examining the gate error sources and primarily concentrates on the
related physical mechanism of ZZ parasitic couplings using a systematic
effective Hamiltonian approach. Benefiting from the effective Hamiltonian, we
provide simple and straightforward insight into the ZZ parasitic couplings that
were investigated previously from numerical and experimental perspectives. The
analytical results obtained provide exact quantitative conditions for
eliminating ZZ parasitic couplings, and trigger four novel realizable parameter
regions in which higher fidelity two-qubit gates are expected. Beyond the
numerical simulation, we also successfully drive a simple analytical result of
the two-qubit gate error from which the trade-off effect between qubit energy
relaxation effects and ZZ parasitic couplings is understood, and the resulting
two-qubit gate error can be estimated straightforwardly. Our study opens up new
opportunities to implement high-fidelity two-qubit gates in superconducting
coupler architecture.
