Fault-tolerant bosonic quantum error correction with the surface–Gottesman-Kitaev-Preskill code

Bosonic quantum error correction is a viable option for realizing error-corrected quantum information processing in continuous-variable bosonic systems. Various single-mode bosonic quantum error-correcting codes such as cat, binomial, and Gottesman-Kitaev-Preskill (GKP) codes have been implemented experimentally in circuit QED and trapped-ion systems. Moreover, there have been many theoretical proposals to scale up such single-mode bosonic codes to realize large-scale fault-tolerant quantum computation. Here, we consider the concatenation of the single-mode GKP code with the surface code, namely, the surface-GKP code. In particular, we thoroughly investigate the performance of the surface-GKP code by assuming realistic GKP states with a finite squeezing and noisy circuit elements due to photon losses. By using a minimum-weight perfect matching decoding algorithm on a three-dimensional spacetime graph, we show that fault-tolerant quantum error correction is possible with the surface-GKP code if the squeezing of the GKP states is higher than 11.2 dB in the case where the GKP states are the only noisy elements. We also show that the squeezing threshold changes to 18.6 dB when both the GKP states and circuit elements are comparably noisy. At this threshold, each circuit component fails with probability $0.69\%$. Finally, if the GKP states are noiseless, fault-tolerant quantum error correction with the surface-GKP code is possible if each circuit element fails with probability less than $0.81\%$. We stress that our decoding scheme uses the additional information from GKP-stabilizer measurements and we provide a simple method to compute renormalized edge weights of the matching graphs. Furthermore, our noise model is general because it includes full circuit-level noise.

Figure 1

(a) Computational basis states ($|0_{\mathrm{gkp}}\rangle$, $|1_{\mathrm{gkp}}\rangle$) and complementary basis states ($|{+}_{\mathrm{gkp}}\rangle$, $|{-}_{\mathrm{gkp}}\rangle$) of an approximate GKP qubit with an average photon number $\overline{n}=5$. (b) Circuits for measuring the ${\widehat{S}}_q$ and ${\widehat{S}}_p$ stabilizers. ${\mathcal{M}}_q$ and ${\mathcal{M}}_p$ represent the homodyne measurement of the position and momentum operators, respectively. Also, the controlled-$\oplus$ symbol represents the sum gate and similarly the controlled-$\ominus$ symbol represents the inverse-sum gate [see Eq. (6)]. Note that the size of the correction shifts $exp\left[i{\widehat{p}}_a{z}_q^{\left(b\right)}\right]$ and $exp[-i{\widehat{q}}_a{z}_p^{\left(b\right)}]$ in the ${\widehat{S}}_q$ and ${\widehat{S}}_p$ stabilizer measurements are determined by the homodyne measurement outcomes $z_q^{\left(b\right)}$ and $z_p^{\left(b\right)}$.

Figure 2

$p\left[\sigma \right]\left(z\right)$ for $\sigma =0.2$, 0.5, and 1. $p\left[\sigma \right]\left(z\right)$ is defined in Eq. (11) and represents the conditional probability of having a Pauli $X$ (or $Z$) error, given the measurement outcome ${\xi }_q=z+n\sqrt{\pi }$ (or ${\xi }_p=z+n\sqrt{\pi }$) for some integer $n$.

Figure 3

The surface-GKP codes with $d=3$ and $d=5$. White circles represent the data GKP qubits and gray circles represent the ancilla GKP qubits that are used to measure GKP stabilizers of each data GKP qubit. Green and orange circles represent the syndrome GKP qubits that are used to measure the $Z$-type and $X$-type surface code stabilizers of the data GKP qubits, respectively. In general, there are $d^2$ data GKP qubits and $(d^2-1)/2Z$-type and $X$-type syndrome GKP qubits. See also Fig. 5 for the reason behind our choice of inverse-sum gates in the $X$-type stabilizer measurements.

Figure 4

Circuits for surface code stabilizer measurements.

Figure 5

Noise propagation from the $X4$ qubit to the $Z2$ qubit during surface code stabilizer measurements. The red lightening symbol represents the initial position of a shift error on the qubit $X4$. During the propagation of the shift error to the qubit $Z2$, the sign of the shift error is flipped by the inverse-sum gate $su{m}_{X4\to D5}^{†}$. This sign flip then results in cancellations of the propagated shift errors on the qubit $Z2$ (empty lightening symbol).

Figure 6

The logical $X$ error rate of the surface-GKP code for various $d$ when (a) $\sigma =0$ (Case I), (b) ${\sigma }_{\mathrm{gkp}}=0$ (Case II), and (c) $\sigma ={\sigma }_{\mathrm{gkp}}$ (Case III), which is the same as the logical $Z$ error rate. The solid lines represent logical error rates when information from the GKP-stabilizer measurements is used to renormalize edge weights in the matching graphs. The dotted lines correspond to the case when information from GKP-stabilizer measurements is ignored. In all cases, given that ${\sigma }_{\mathrm{gkp}}$ and $\sigma$ are below certain fault-tolerance thresholds, the logical $X$ or $Z$ error rates are suppressed to an arbitrarily small value as we increase the code distance $d$.

Figure 7

Visualization of the function $p_{\mathrm{err}}\left(\sigma \right)$ (blue). The asymptotic expression $p_{\mathrm{asy}}\left(\sigma \right)=(\sqrt{8}{\sigma }^2/\pi )exp[-\pi /\left(8{\sigma }^2\right)]$ is represented by the yellow dashed line. $p_{\mathrm{err}}\left(\sigma \right)$ and $p_{\mathrm{asy}}\left(\sigma \right)$ agree well with each other in the $\sigma \ll 1$ limit.

Figure 8

Measurement of the GKP stabilizers for $d=3$. See also Fig. 1 for the graphical notation.

Figure 9

Measurement of the surface code stabilizers for $d=3$.

Figure 10

$Z$-type and $X$-type 2D space graphs for the surface-GKP code with $d=5$. These 2D graphs will be stacked up to construct $Z$-type and $X$-type 3D spacetime graphs.