Engineering $XY$ and $ZZ$ interactions between two remote superconducting qubits via a multimode waveguide

Waveguides provide a flexible platform for long-range interactions between superconducting qubits. Here, we study how to engineer the effective $XY$ and $ZZ$ interactions between two qubits via qubit frequency assignments and multimode interference determined by the mode configuration and standing-wave parities of a finite multimode waveguide. For a single-mode waveguide, we find that $XY$ and $ZZ$ interactions cannot be simultaneously suppressed to zero and the residual qubit-qubit coupling always exists. For example, the $XY$ interaction may be suppressed to a negligible level when two qubits and the waveguide mode satisfy the two-photon resonant condition, but the $ZZ$ interaction cannot be switched off. For multimode configurations within the central dispersive window, the engineered interaction types are very different for odd and even numbers of modes. Even-mode configurations can readily produce strong $XY$ interactions, whereas odd-mode configurations can strongly suppress the $XY$ interaction while retaining a finite $ZZ$ interaction. Full-Hamiltonian simulations of minimal two- and three-mode configurations confirm $XY$-dominated SWAP dynamics and $ZZ$-dominated CZ-type dynamics, respectively. These results establish a frequency assignment criterion of qubits and waveguide modes for selectively engineering long-range qubit-qubit interaction types induced by the waveguide with finite multimode.