Unsuitability of cubic phase gates for non-Clifford operations on Gottesman-Kitaev-Preskill states

With the Gottesman-Kitaev-Preskill (GKP) encoding, Clifford gates and error correction can be carried out using simple Gaussian operations. Still, non-Clifford gates, required for universality, require non-Gaussian elements. In their original proposal, GKP suggested a particularly method of single application cubic phase gate to perform logical $T$ gate. Here we show that this approach performs extraordinarily poorly, even arbitrarily large amounts squeezing in state. Thus, contrary common belief, is not suitable achieving universal fault-tolerant quantum computation with states.