  We present a concept of non-Gaussian measurement composed of a non-Gaussian
ancillary state, linear optics and adaptive heterodyne measurement, and on the
basis of this we also propose a simple scheme of implementing a quantum cubic
gate on a traveling light beam. In analysis of the cubic gate in the Heisenberg
representation, we find that nonlinearity of the gate is independent from
nonclassicality; the nonlinearity is generated solely by a classical nonlinear
adaptive control in a measurement-and-feedforward process while the
nonclassicality is attached by the non-Gaussian ancilla that suppresses excess
noise in the output. By exploiting the noise term as a figure of merit, we
consider the optimum non-Gaussian ancilla that can be prepared within reach of
current technologies and discuss performance of the gate. It is a crucial step
towards experimental implementation of the quantum cubic gate.
