  If suitable quantum optical interactions were available, transforming optical
field mode operators in a nonlinear fashion, the all-photonics platform could
be one of the strongest contenders for realizing a quantum computer. Unlike
other, matter-based (solid-state or atomic) platforms, photonic qubits can be
operated at room temperature and high clock rates (GHz or, in principle, even
THz). In addition, recent continuous-variable time-domain approaches are
extremely well scalable. Moreover, while single-photon qubits may be processed
directly, "brighter" logical qubits may be embedded in individual oscillator
modes, using so-called bosonic codes, for an in-principle fault-tolerant
processing. In this paper, we show how elements of all-optical, universal, and
fault-tolerant quantum computation can be implemented using only beam splitters
together with single-mode cubic phase gates in reasonable numbers, and possibly
off-line squeezed-state or single-photon resources. Our approach is based on a
decomposition technique combining exact gate decompositions and approximate
Trotterization. This allows for efficient decompositions of certain nonlinear
continuous-variable multimode gates into the elementary gates, where the few
cubic gates needed may even be weak or all identical, thus facilitating
potential experiments. The final gate operations include two-mode controlled
phase rotation and three-mode Rabi-type Hamiltonian gates, which are shown to
be employable for realizing high-fidelity single-photon two-qubit entangling
gates or, as a bosonic-code example, creating high-quality
Gottesman-Kitaev-Preskill states. We expect our method of general use with
various applications, including those that rely on quartic Kerr-type
interactions.
