  A Kerr-nonlinear parametric oscillator (KPO) is one of the promising devices
to realize qubits for universal quantum computing. The KPO can stabilize two
coherent states with opposite phases, yielding a quantum superposition called a
Schr\"{o}dinger cat state. Universal quantum computing with KPOs requires three
kinds of quantum gates: $R_z, R_x$, and $R_{zz}$ gates. We theoretically
propose a two-qubit gate $R_{zz}$ for highly detuned KPOs. In the proposed
scheme, we add another two-photon drive for the first KPO. This leads to the
$R_{zz}$ gate based on the driving of the second KPO depending on the first-KPO
state, which we call "conditional driving." First, we perform simulations using
a conventional KPO Hamiltonian derived from a superconducting-circuit model
under some approximations and evaluate the gate fidelity. Next, we also perform
numerical simulations of the two-qubit gate using the superconducting-circuit
model without the approximations. The simulation results indicate that
two-qubit gates can be implemented with high fidelity ($>99.9\%$) for rotation
angles required for universality.
