Non-Clifford gate on optical qubits by nonlinear feedforward

In a continuous-variable optical system, the Gottesman-Kitaev-Preskill (GKP) qubit is a promising candidate for fault-tolerant quantum computation. To implement non-Clifford operations on GKP qubits, non-Gaussian operations are required. In this context, the implementation of a cubic phase gate by combining nonlinear feedforward with ancillary states has been widely researched. Recently, however, it was pointed out that the cubic phase gate is not the most suitable for non-Clifford operations on GKP qubits. In this paper, we show that we can achieve linear optical implementation of non-Clifford operations on GKP qubits with high fidelity by applying the nonlinear feedforward originally developed for the cubic phase gate and using a GKP-encoded ancillary state. Our paper shows the versatility of the nonlinear feedforward technique—important for optical implementation of the fault-tolerant continuous-variable quantum computation.

Figure 1

Quantum circuits to implement the $T$ gate. (a) Magic state injection suggested in the original GKP's paper [15]. (b) The linear optical system to implement the $T$ gate. We replace CNOT gate with a 50:50 beam splitter (BS) instead of QND gate. The measurement and the phase gate $\widehat{S}$ in Fig. 1 are replaced by homodyne detection (HD) and shear operation $\widehat{P}\left(\kappa \right)$, respectively. We add extra squeezing operation ${\widehat{U}}_{\mathrm{sq}}$ and displacement operators to feedforward to fully achieve the $T$ gate at Fig. 1.

Figure 2

The setup of the dynamic squeezing gate. We need only Gaussian ancillary state ${|x=0\rangle }_{\mathrm{B}}$ and linear feedforward of measured value from homodyne detection HD controlling the displacement [11].

Figure 3

The whole setup of the $T$ gate. We need only two ancillary states, ${|T_{\mathrm{L}}\rangle }_{\mathrm{A}}$ and ${|x=0\rangle }_{\mathrm{B}}$. The feedforward is nonlinear about the measured value from the first homodyne detection HD1. For Fig. 4, we substitute ideal ${|T_{\mathrm{L}}\rangle }_{\mathrm{A}}$, ${|{\psi }_{\mathrm{L}}\rangle }_{\mathrm{in}}$, and ${|x=0\rangle }_{\mathrm{B}}$ by realistic ${|T_{\mathrm{\Delta }}\rangle }_{\mathrm{A}}$ [Eq. (13)], ${|{\psi }_{\mathrm{\Delta }}\rangle }_{\mathrm{in}}$ [Eq. (12)], and ${|{\mathrm{Sq}}_{\sigma }\rangle }_{\mathrm{B}}$ [Eq. (14)].

Figure 4

The numerical evaluation of the $T$ gate. The approximate GKP state ${|{\psi }_{\mathrm{\Delta }}\rangle }_{\mathrm{in}}=\frac{1}{\sqrt{2}}({|0_{\mathrm{\Delta }}\rangle }_{\mathrm{in}}+{|1_{\mathrm{\Delta }}\rangle }_{\mathrm{in}})$ is used as the input state, and its squeezing level is taken as the horizontal axis. The logical density matrix is obtained from the output state of the $T$ gate, and we calculate the logical fidelity to the target state ${|T\rangle }_{\mathcal{L}}=\frac{1}{\sqrt{2}}({|0\rangle }_{\mathcal{L}}+{\mathrm{e}}^{i\frac{\pi }{4}}{|1\rangle }_{\mathcal{L}})$. The fidelity when using our $T$ gate setup shown in Fig. 3 is drawn by the solid orange line (the Gaussian ancilla case in mode B is infinitely squeezed) and by the red circles (the case it is finitely squeezed state with the same squeezing level as the input GKP state). The dotted blue line and dashed green line show the case where cubic phase gate is used as in Refs. [15, 27] and the case where it is used with optimization, respectively.

Figure 5

The logical fidelity as the $T$ gate when the gains of cubic phase gate and other Gaussian gates are adjusted.