Cost-reduced all-Gaussian universality with the Gottesman-Kitaev-Preskill code: Resource-theoretic approach to cost analysis

The Gottesman-Kitaev-Preskill (GKP) quantum error-correcting code has emerged as a key technique in achieving fault-tolerant quantum computation using photonic systems. Whereas [Baragiola et al., Phys. Rev. Lett. 123, 200502 (2019)] showed that experimentally tractable Gaussian operations combined with preparing a GKP codeword $\left|0\right.〉$ suffice to implement universal quantum computation, this implementation scheme involves a distillation of a logical magic state $\left|H\right.〉$ of the GKP code, which inevitably imposes a trade-off between implementation cost and fidelity. In contrast, we propose a scheme of preparing $\left|H\right.〉$ directly and combining Gaussian operations only with $\left|H\right.〉$ to achieve the universality without this magic state distillation. In addition, we develop an analytical method to obtain bounds of fundamental limit on transformation between $\left|H\right.〉$ and $\left|0\right.〉$, finding an application of quantum resource theories to cost analysis of quantum computation with the GKP code. Our results lead to an essential reduction of required non-Gaussian resources for photonic fault-tolerant quantum computation compared to the previous scheme.

Figure 1

A quantum circuit of state injection for applying the $T$ gate to any one-qubit input state $\left|\psi \right.〉$ by Clifford operations assisted by an auxiliary input qubit prepared in $\left|\frac{\pi }{8}\right.〉$ at the top, and that for converting a two-qubit input state ${\left|\frac{\pi }{8}\right.〉}^{\otimes 2}$ to $\left|0\right.〉$ at the bottom. The latter conversion circuit can be implemented only by adaptive Gaussian operations on GKP qubits, namely, Clifford gates cnot, $S$, $S^{†}$, and $H$) that are implemented with Gaussian unitary operations, and conditioning on a $Z$-basis measurement outcome that is implemented with a homodyne detection.

Figure 2

Wigner functions of ideal GKP states $\left|0\right.〉$ on the left and $\left|H\right.〉$ on the right, where each blue filled circle represents a positive delta function $\delta$, each red circled X represents a negative delta function $-\delta$, each yellow filled circle represents a weighted positive delta function $\frac{1}{\sqrt{2}}\delta$, and each black circled X represents a weighted negative delta function $-\frac{1}{\sqrt{2}}\delta$, e.g., $W_{\left|0\right.〉\left.〈0\right|}(q,p)\propto {\sum }_{s,t\in \mathbb{Z}}{(-1)}^{st}\delta \left(q-\sqrt{\pi }s\right)\delta \left(p-\frac{\sqrt{\pi }}{2}t\right)$ up to normalization. These Wigner functions have periodicity, where the gray region shows a period.

Figure 3

Negativities of the Wigner functions of $\left|0_{{\sigma }^2}\right.〉$ (blue solid line) and $\left|H_{{\sigma }^2}\right.〉\propto \left(cos\left(\frac{\pi }{8}\right)\left|0_{{\sigma }^2}\right.〉+sin\left(\frac{\pi }{8}\right)\left|1_{{\sigma }^2}\right.〉\right)$ (orange dashed line) with respect to the squeezing level $-10{log}_{10}\left(2{\sigma }^2\right)$. Negativity of $\left|0_{{\sigma }^2}\right.〉$ approaches 2, and that of $\left|H_{{\sigma }^2}\right.〉$ to $1+\sqrt{2}=2.41\cdots$, as expected from Eqs. (4) and (5).

Figure 4

A phase-estimation-like protocol for generating approximate GKP codewords proposed in Ref. [40], which is a refinement of the protocol in Ref. [38]. The operations surrounded by the dashed line is recursively performed, while the parameter $\varphi$ of the rotation around Z axis $R_Z\left(\varphi \right)$ is chosen according to the former outcomes of the measurement on the qubits.

Figure 5

A quantum circuit for preparing a GKP $\frac{\pi }{8}$ phase state $\left|\frac{\pi }{8}\right.〉$ from an input GKP codeword $\left|0\right.〉$ using the same setup as that in Ref. [40]. Note that the $T$ gate is a rotation around $Z$ axis in the same way as $R_Z\left(\varphi \right)$ in Fig. 4, and thus this quantum circuit can be implemented using the same experimental setup as that in Fig. 4.