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Superconducting microwave resonator couplers will likely become an essentialcomponent in modular semiconductor quantum dot (QD) spin qubit processors, asthey help alleviate cross-talk and wiring issues as the number of qubitsincreases. Here, we focus on a three-qubit system composed of two modules: atwo-electron triple QD resonator-coupled to a single-electron double QD. Usinga combination of analytical techniques and numerical results, we derive aneffective Hamiltonian that describes the three-qubit logical subspace and showthat it accurately captures the dynamics of the system. We examine theperformance of short-range and long-range entangling gates, revealing theeffect of a spectator qubit in reducing the gate fidelities in both cases. Wefurther study the competition between non-adiabatic errors andspectator-associated errors in short-range operations and quantify theirrelative importance across practical parameter ranges for short and long gatetimes. We also analyze the impact of charge noise together with residualcoupling to the spectator qubit on inter-module entangling gates and find thatfor current experimental settings, leakage errors are the main source ofinfidelities in these operations. Our results help pave the way towardidentifying optimal modular QD architectures for quantum information processingon semiconductor chips.

Long-distance photon-mediated and short-distance entangling gates in three-qubit quantum dot spin systems