  We present a method for implementing a weak optical Kerr interaction
(single-mode Kerr Hamiltonian) in a measurement-based fashion using the common
set of universal elementary interactions for continuous-variable quantum
computation. Our scheme is a conceptually distinct alternative to the use of
naturally occurring, weak Kerr nonlinearities or specially designed nonlinear
media. Instead, we propose to exploit suitable offline prepared quartic ancilla
states together with beam splitters, squeezers, and homodyne detectors. For
perfect ancilla states and ideal operations, our decompositions for obtaining
the measurement-based Kerr Hamiltonian lead to a realization with near-unit
fidelity. Nonetheless, even by using only approximate ancilla states in the
form of superposition states of up to four photons, high fidelities are still
attainable. Our scheme requires four elementary operations and its
deterministic implementation corresponds to about ten ancilla- based gate
teleportations. We test our measurement-based Kerr interaction against an ideal
Kerr Hamiltonian by applying them both to weak coherent states and
single-photon superposition states.
