Enhanced noise resilience of the surface–Gottesman-Kitaev-Preskill code via designed bias

We study the code obtained by concatenating the standard single-mode Gottesman-Kitaev-Preskill (GKP) code with the surface code. We show that the noise tolerance of this surface–GKP code with respect to (Gaussian) displacement errors improves when a single-mode squeezing unitary is applied to each mode, assuming that the identification of quadratures with logical Pauli operators is suitably modified. We observe noise-tolerance thresholds of up to $\sigma \approx 0.58$ shift-error standard deviation when the surface code is decoded without using GKP syndrome information. In contrast, prior results by K. Fukui, A. Tomita, A. Okamoto, and K. Fujii, High-Threshold Fault-Tolerant Quantum Computation with Analog Quantum Error Correction, Phys. Rev. X 8, 021054 (2018) and C. Vuillot, H. Asasi, Y. Wang, L. P. Pryadko, and B. M. Terhal, Quantum error correction with the toric Gottesman-Kitaev-Preskill code, Phys. Rev. A 99, 032344 (2019) report a threshold between $\sigma \approx 0.54$ and $\sigma \approx 0.55$ for the standard (toric, respectively) surface–GKP code. The modified surface–GKP code effectively renders the mode-level physical noise asymmetric, biasing the logical-level noise on the GKP qubits. The code can thus benefit from the resilience of the surface code against biased noise. We use the approximate maximum likelihood decoding algorithm of S. Bravyi, M. Suchara, and A. Vargo, Efficient algorithms for maximum likelihood decoding in the surface code, Phys. Rev. A 90, 032326 (2014) to obtain our threshold estimates. Throughout, we consider an idealized scenario where measurements are noiseless and GKP states are ideal. Our paper demonstrates that Gaussian encodings of individual modes can enhance concatenated codes.

Figure 1

Tensor network associated with the BSV decoder. Figure 1 shows the surface code with $d=4$, logical operators $\overline{X}$ and $\overline{Z}$ and two stabilizer generators. Figure 1 gives the tensor network whose contraction gives the value $\pi \left(E\mathcal{S}\right)$ of a coset determined by a Pauli operator $E={\otimes }_{(j,k)}{E}_{j,k}$ acting with the single-qubit Pauli operator $E_{j,k}$ on the qubit at location $(j,k)$. Evaluating this scalar for $E\in \{E_s,E_s\overline{X},E_s\overline{Y},E_s\overline{Z}\}$, where $E_s$ is a representative error giving syndrome $s$, allows us to perform maximum likelihood decoding according to Eq. (7). Local tensors are defined as shown in Fig. 1. (Tensors near the boundaries are defined similarly.) A priori Pauli-error probabilities for a qubit located at $(j,k)$ are denoted ${\pi }_{j,k}\left(Q\right)=p_Q$, where $p_Q$ is the a priori probability of a Pauli error $Q\in \{I,X,Y,Z\}$. (a) Surface code with distance $d$ = 4. (b) Tensor network for coset probability π(ES). (c) Bulk tensors; the first two associated with a qubit at ($j$, $k$).

Figure 2

The lattice ${\mathcal{L}}_r$ (dashed black lines) and the dual lattice ${\mathcal{L}}_r^{\perp }$ (solid blue lines) defining the GKP codes for two different ratio parameters $r=1$ (square lattice) and $r=4$. The Voronoi cell of the dual lattice ${\mathcal{L}}_r^{\perp }$ is marked with blue edges. (a) $r$ = 1, (b) $r$ = 4.

Figure 3

Lattices for $\text{GKP}\left({\mathcal{L}}_r\right)$ for the ratio parameters $r=1,4$. In the phase space, the lattice ${\mathcal{L}}_r$ (dashed black lines) and its dual ${\mathcal{L}}_r^{\perp }$ (solid blue lines) are depicted, respectively. For the square lattice ($r=1$) GKP code undergoing a displacement $D\left(\nu \right)$, $\nu \in {\mathbb{R}}^2$, measuring the stabilizer generators $e^{i2\sqrt{\pi }Q}$ and $e^{i2\sqrt{\pi }P}$ corresponds to measuring the values $\eta =\nu (mod{\mathcal{L}}^{\perp })$, with $\eta ={\left({\eta }_1{\eta }_2\right)}^T$, ${\eta }_1,{\eta }_2\in [-\frac{\sqrt{\pi }}{2},\frac{\sqrt{\pi }}{2}]$. Applying the correction $D\left(\eta \right)$ corresponds to a mapping to the nearest lattice point in ${\mathcal{L}}_r^{\perp }$. The correction is successful if the actual error displacement $\nu ={\left({\nu }_1{\nu }_2\right)}^T$ satisfies ${\nu }_1=2\sqrt{\pi }{n}_1+{\eta }_1$, ${\nu }_2=2\sqrt{\pi }{n}_2+{\eta }_2$, with $n_1,n_2\in \mathbb{Z}$. In other words, the correction is successful if the error vector $\nu$ lies in the areas marked by the blue edges. This correspond to shifts of the dual lattice Voronoi cell by a lattice vector $\xi \in \mathcal{L}$. If the error vector $\nu$ lies in one of the grey shaded areas, the correction results in a logical error $\overline{X},\overline{Y},\overline{Z}$ [cf. table in (30)]. (a) $r$ = 1, (b) $r$ = 4.

Figure 4

Two ways of decoding the surface–GKP code. In both cases the GKP error correction circuit produces syndrome information $s^{j,k}$. When decoding without side information [Fig. 4], this information is ignored and the same prior distribution $\pi$ over logical Pauli errors $\{\overline{I},\overline{X},\overline{Y},\overline{Z}\}$ is used for every site. The surface-code BSV decoder (which also involves syndrome measurement but whose syndromes are not shown here) is run with the associated product (i.i.d.) distribution ${\pi }^n$. Figure 4 shows how to exploit the GKP side information: For each qubit $(j,k)$ a prior distribution ${\pi }_{j,k}={\pi }^{\mathrm{p}}\left(s^{j,k}\right)$ over logical Pauli errors is computed from the GKP syndrome $s^{j,k}$, and this is then used in the BSV decoder as the prior distribution. (a) Decoding without GKP side information. (b) Decoding with GKP side information.

Figure 5

The error probability $P_{\text{err}}(\sigma ,d,r)$ (left) for decoding without GKP side information for different asymmetry ratios $r\in \{1.0,2.0,3.0\}$. Insets give higher-resolution data around the observed threshold estimate ${\sigma }_c$ for the critical noise variance. The latter is indicated by a vertical line. The right-hand side shows a log-plot of the error probability $P_{\text{err}}(\sigma ,d,r)$. For comparison, the error probability of the bare GKP code, i.e., without concatenation with the surface code, is shown in blue with the solid line corresponding to ratio $r=1.0$ and the dashed lines corresponding to the respective ratio of the figure. (a) Ratio $r=1.0$, threshold ${\sigma }_c\approx 0.540.$ (b) Ratio $r=2.0$, threshold ${\sigma }_c\approx 0.562.$ (c) Ratio $r=3.0$, threshold ${\sigma }_c\approx 0.581.$

Figure 6

The dependence of the error probability $P_{\text{err}}(\sigma ,d,r)$ on the noise variance $\sigma$ for decoding without GKP side information. In each figure, the curves $\sigma \mapsto P_{\text{err}}(\sigma ,d,r)$ for asymmetry ratios $r$ between 1.0 and 4.0 are given. The individual figures treat code distances $d\in \{9,13,17,21\}$. (a) Distance $d=9$. (b) Distance $d=13$. (c) Distance $d=17$. (d) Distance $d=21$.

Figure 7

Empirically computed thresholds for different asymmetry ratios $r$. The error bars depict the standard deviation of the fitted threshold value ${\sigma }_c$ (left hand side axis) and range from 0.0006 (for $r=1)$ to 0.0019 (for $r=3$). The values on the right-hand side axis correspond to $q_{\overline{X}}$ [Eq. (46)] for the parameters $\sigma ={\sigma }_c$ and $r=1$. These values are given for comparison with existing estimates for (qubit) surface codes: For example, it is known [35] that the logical $X$-error probability $q_{\overline{X}}$ for minimum matching decoding for independent $X$ and $Z$ noise is around $10.1\%$. We recover this value for ratio $r=1$. The overall logical error probability is given by $p=q_{\overline{X}}^2+2q_{\overline{X}}(1-q_{\overline{X}})$.

Figure 8

Other improvements for ratio $r=2.0$: Here the plots show the error probability $P_{\text{err}}(\sigma ,d,r)$ as a function of $\sigma$ for decoding without side information [Fig. 8] and with side information [Fig. 8] for the rectangular lattice GKP code. Figure 8 shows the error probability for the asymmetric hexagonal lattice GKP code decoded with side information. Observed threshold estimates ${\sigma }_c$ are indicated with vertical lines. (a) Rectangular lattice GKP without side information, threshold ${\sigma }_c\approx 0.562$. (b) Rectangular lattice GKP with side information, threshold ${\sigma }_c\approx 0.6062$ (with standard deviation 0.0007). (c) Asymmetric hexagonal lattice GKP with side information, threshold ${\sigma }_c\approx 0.6045$ (with standard deviation 0.0009).

Figure 9

Subroutine MCWithoutSideInfo simulates one instance of the error-recovery process when GKP side information is ignored. It returns the residual logical error.

Figure 10

Subroutine MCWithSideInfo Monte-Carlo-simulates one instance of the error-recovery process when GKP side information is used. It returns the residual logical error.

Figure 11

Test of the necessary bond dimension for the GKP code (without side information): Error probabilities are estimated for different bond dimensions $\chi$. Heuristically, a bond dimension $\chi$ suffices if $P_{\text{err}}$ does not decrease significantly when the bond dimension is increased further. We observe that the required bond dimension $\chi$ appears to be nonmonotonic as a function of the asymmetry ratio $r$, being maximal in some intermediate regime of $r$ (left hand side). The required bond dimension increases with increasing distance $d$ (right-hand side). (a) Ratio $r=2.0$. (b) Ratio $r=3.0$. (c) Ratio $r=4.0$. (d) Distance $d=9$. (e) Distance $d=13$. (f) Distance $d=17$.