Gaussian conversion protocols for cubic phase state generation: Part 1

Universal quantum computing with continuous variables requires non-Gaussian resources, in addition to a Gaussian set of operations. A known resource enabling universal quantum computation is the cubic phase state, a non-Gaussian state whose experimental implementation has so far remained elusive.
We introduce a deterministic Gaussian conversion protocol that allows for the conversion of a non-Gaussian state that has been achieved experimentally, namely the trisqueezed state [Sandbo Chang et al, Phys. Rev. X 10, 011011 (2020)], to a cubic phase state.
First, we derive an upper bound on the conversion fidelity that is obtainable with the most general Gaussian completely positive trace-preserving (CPTP) map. To do so, we analyse how the characteristic function associated with the trisqueezed state is transformed by these general Gaussian maps. Then, we analyse the performances of symplectic protocols and displacements, and show that they saturate the bound found for general Gaussian maps. Finally, we show that exact state conversion is possible asymptotically, in the limit of infinite squeezing in the target cubic phase state.