  Strong nonlinear coupling of superconducting qubits and/or photons is a
critical building block for quantum information processing. Due to the
perturbative nature of the Josephson nonlinearity, linear coupling is often
used in the dispersive regime to approximate nonlinear coupling. However, this
dispersive coupling is weak and the underlying linear coupling mixes the local
modes which, for example, distributes unwanted self-Kerr to photon modes. Here,
we use the quarton to yield purely nonlinear coupling between two linearly
decoupled transmon qubits. The quarton's zero $\phi^2$ potential enables a
giant gigahertz-level cross-Kerr which is an order of magnitude stronger
compared to existing schemes, and the quarton's positive $\phi^4$ potential can
cancel the negative self-Kerr of qubits to linearize them into resonators. This
giant cross-Kerr between bare modes of qubit-qubit, qubit-photon, and even
photon-photon is ideal for applications such as single microwave photon
detection and implementation of bosonic codes.
