  We propose a deterministic, measurement-free implementation of a cubic phase
gate for continuous-variable quantum information processing. In our scheme, the
applications of displacement and squeezing operations allow us to engineer the
effective evolution of the quantum state propagating through an optical Kerr
nonlinearity. Under appropriate conditions, we show that the input state
evolves according to a cubic phase Hamiltonian, and we find that the cubic
phase gate error decreases inverse-quartically with the amount of quadrature
squeezing, even in the presence of linear loss. We also show how our scheme can
be adapted to deterministically generate a nonclassical approximate cubic phase
state with high fidelity using a ratio of native nonlinearity to linear loss of
only $10^{-4}$, indicating that our approach may be experimentally viable in
the near term even on all-optical platforms, e.g., using quantum solitons in
pulsed nonlinear nanophotonics.
