  Superconducting microwave resonator couplers will likely become an essential
component in modular semiconductor quantum dot (QD) spin qubit processors, as
they help alleviate cross-talk and wiring issues as the number of qubits
increases. Here, we focus on a three-qubit system composed of two modules: a
two-electron triple QD resonator-coupled to a single-electron double QD. Using
a combination of analytical techniques and numerical results, we derive an
effective Hamiltonian that describes the three-qubit logical subspace and show
that it accurately captures the dynamics of the system. We examine the
performance of short-range and long-range entangling gates, revealing the
effect of a spectator qubit in reducing the gate fidelities in both cases. We
further study the competition between non-adiabatic errors and
spectator-associated errors in short-range operations and quantify their
relative importance across practical parameter ranges for short and long gate
times. We also analyze the impact of charge noise together with residual
coupling to the spectator qubit on inter-module entangling gates and find that
for current experimental settings, leakage errors are the main source of
infidelities in these operations. Our results help pave the way toward
identifying optimal modular QD architectures for quantum information processing
on semiconductor chips.
