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

Abstract

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 simple method of using a single application of the cubic phase gate to perform the logical non-Clifford T gate. Here we show that this cubic phase gate approach performs extraordinarily poorly, even for arbitrarily large amounts of squeezing in the GKP state. Thus, contrary to common belief, the cubic phase gate is not suitable for achieving universal fault-tolerant quantum computation with GKP states.

Non-Clifford gate on optical qubits by nonlinear feedforward

“…We analyze the performance of our scheme and find that it can work as an almost ideal T gate with high fidelity 1. In contrast, when using the cubic phase gate as the T gate based on GKP's original proposal, the fidelity saturates at 0.78 [23]. We also find that this fidelity can be improved to 0.95 by optimizing the gains of the cubic phase gate and other Gaussian gates.…”

“…In the GKP's original paper, the case of c 0 = 1 2 , c 1 = 1 4 , c 2 = − 1 2 was proposed [15]. The fidelity of this case is plotted as a blue line and it saturates ∼0.78 [23]. We find that the fidelity can be improved by optimizing the gains (details about the optimization are given in supplementary material…”

“…which was pointed out not to be suitable for the T gate [23]. However, we can think of other physical implementations of the T gate which has better performance even if only one cubic phase gate is used.…”

“…One method is to use the cubic phase gate combined with Gaussian operations. Recently, however, it was pointed out that the cubic phase gate is not the most suitable for the T gate on GKP qubits [23]. This can be somewhat expected as the cubic phase gate is a gate intended for universal processing of CV systems [24] and not tailored for non-Clifford operations on GKP qubits.…”

“…3 is drawn by the orange line (the case the gaussian ancilla in mode "B" is infinitely squeezed) and by the red dots (the case it is finitely squeezed state with the same squeezing level as the input GKP state). The blue and green lines show the case where cubic phase gate is used as [15,23] and the case where it is used with optimization, respectively. [33]), and when c 0 = − 1 6 , c 1 = 1 4 , c 2 = 1 6 , the fidelity goes up to ∼ 0.95 (plotted as a green line).…”