  The engineering of Kerr interactions has great potential for quantum
information processing applications in multipartite quantum systems and for
investigation of many-body physics in a complex cavity-qubit network. We study
how coupling multiple different types of superconducting qubits to the same
cavity modes can be used to modify the self- and cross-Kerr effects acting on
the cavities and demonstrate that this type of architecture could be of
significant benefit for quantum technologies.
  Using both analytical perturbation theory results and numerical simulations,
we first show that coupling two superconducting qubits with opposite
anharmonicities to a single cavity enables the effective self-Kerr interaction
to be diminished, while retaining the number splitting effect that enables
control and measurement of the cavity field. We demonstrate that this reduction
of the self-Kerr effect can maintain the fidelity of coherent states and
generalised Schr\"{o}dinger cat states for much longer than typical coherence
times in realistic devices. Next, we find that the cross-Kerr interaction
between two cavities can be modified by coupling them both to the same pair of
qubit devices. When one of the qubits is tunable in frequency, the strength of
entangling interactions between the cavities can be varied on demand, forming
the basis for logic operations on the two modes. Finally, we discuss the
feasibility of producing an array of cavities and qubits where intermediary and
on-site qubits can tune the strength of self- and cross-Kerr interactions
across the whole system. This architecture could provide a way to engineer
interesting many-body Hamiltonians and a useful platform for quantum simulation
in circuit quantum electrodynamics.
