Quantum error correction with the toric Gottesman-Kitaev-Preskill code

We examine the performance of the single-mode Gottesman-Kitaev-Preskill (GKP) code and its concatenation with the toric code for a noise model of Gaussian shifts, or displacement errors. We show how one can optimize the tracking of errors in repeated noisy error correction for the GKP code. We do this by examining the maximum-likelihood problem for this setting and its mapping onto a 1D Euclidean path-integral modeling a particle in a random cosine potential. We demonstrate the efficiency of a minimum-energy decoding strategy as a proxy for the path integral evaluation. In the second part of this paper, we analyze and numerically assess the concatenation of the GKP code with the toric code. When toric code measurements and GKP error correction measurements are perfect, we find that by using GKP error information the toric code threshold improves from $10\%$ to $14\%$. When only the GKP error correction measurements are perfect we observe a threshold at $6\%$. In the more realistic setting when all error information is noisy, we show how to represent the maximum likelihood decoding problem for the toric-GKP code as a 3D compact QED model in the presence of a quenched random gauge field, an extension of the random-plaquette gauge model for the toric code. We present a decoder for this problem which shows the existence of a noise threshold at shift-error standard deviation ${\sigma }_0\approx 0.243$ for toric code measurements, data errors and GKP ancilla errors. If the errors only come from having imperfect GKP states, then this corresponds to states with just four photons or more. Our last result is a no-go result for linear oscillator codes, encoding oscillators into oscillators. For the Gaussian displacement error model, we prove that encoding corresponds to squeezing the shift errors. This shows that linear oscillator codes are useless for quantum information protection against Gaussian shift errors.

Figure 1

A single round of fault-tolerant GKP syndrome measurement for both $q$ and $p$ shifts. Here $|\overline{+}\rangle$ is the $+1$ eigenstate of $S_q$ and $X$, and $|\overline{0}\rangle$ is the $+1$ eigenstate of $S_p$ and $Z$. The cnot gate is the logical cnot for the GKP code which induces the transformation $q_{\mathrm{target}}\to q_{\mathrm{control}}+q_{\mathrm{target}}$ (while $p_{\mathrm{control}}\to p_{\mathrm{control}}-p_{\mathrm{target}},q_{\mathrm{control}}\to q_{\mathrm{control}},p_{\mathrm{target}}\to p_{\mathrm{target}})$. Each measurement is a perfect homodyne measurement of $\widehat{q}$ or $\widehat{p}$. $\mathcal{N}$ are ${\mathcal{N}}_{\mathrm{M}}$ are Gaussian displacement channels in Eq. (5) which model shift errors on the encoded state in each round of error correction, respectively, shift errors in the homodyne measurement.

Figure 2

Repeated rounds of error correction for the GKP code to detect and keep track of error shifts in $\widehat{q}$ followed by a final destructive measurement of the data (modeled as $\mathcal{N}$ followed by a perfect measurement of $\widehat{q})$. No explicit corrections based on the measured values are shown.

Figure 3

Sketch of GKP decoding problem and its solution strategy. For each round of error correction $t\in \left[M\right]$ one has a Villain potential $V_{{\sigma }_{\mathrm{M}}}(q_t-{\varphi }_t)$, depicted in blue, with the minimum centered at the measured value $q_t$. Green and red intervals along the horizontal axis denote the sets $I_0$ and $I_1$ in Eq. (16), they correspond to realizations leaving no error or an $X$ error, respectively. In a memoryless decoding strategy one decides how to shift the code state after each measurement based on the value of $q_t$. In a maximum-likelihood decoding procedure one evaluates the path integral in Eq. (22). In a minimum-energy decoder one determines an optimal path for the sequence ${\varphi }_t,t\in \left[M\right]$, given the random potential and the quadratic “kinetic energy” term proportional to ${({\varphi }_t-{\varphi }_{t-1})}^2$, see Eq. (26). Red and black lines show two decoding trajectories which start at the same point but have different winding numbers. Upon a final decoding step, choosing the black trajectory leads to deciding that no logical $X$ error has taken place, since the final value ${\varphi }_M$ lies in the green region. Choosing the red trajectory leads to deciding that a logical $X$ has taken place since the final value ${\varphi }_M$ lies in the red region. Examples of actual trajectories are shown in Fig. 13 in the Appendix.

Figure 4

Some of the numerical results for the forward-minimization and maximum-likelihood decoders (with cutoff $K=2)$. To observe the exponential decay towards $1/2$ we plot $1-2P_{\mathrm{err}}$ on a log scale for different number of rounds and different bare standard deviation ${\sigma }_0$. Low hundred thousands of trials are performed for each data point and the confidence intervals at $95\%$ are shown. For each ${\sigma }_0$ we fit an exponential decay, the slope gives us an effective logical error rate per round. On both plots the value for ${\sigma }_0$ varies between 0.1 and 0.9, on the left by 0.05 increments, on the right by 0.1 increments. All effective logical error rates are plotted in Fig. 5.

Figure 5

Plots of the observed effective logical error rate per round for different decoding techniques. In the left plot we have taken the same standard deviation for measurement and data errors. On the right we have taken the measurement standard deviation equal to half that of the data errors. The horizontal axis shows the bare standard deviation ${\sigma }_0$.

Figure 6

The two-dimensional layout of oscillators, e.g., high-Q cavities, for the toric-GKP code. Shown is a fragment of a surface or toric code lattice. The different $\pm$ signs are defined by the orientations as explained in the main text.

Figure 7

A single round of error correction for a $Z$-check for the toric-GKP code in Fig. 6, on oscillators numbered 1 to 4. The GKP error correction unit is given in Fig. 1, $|\overline{0}\rangle$ is a $+1$ eigenstate of $Z$ and $S_p$. The inverse cnot which induces the transformation $q_{\mathrm{target}}\to q_{\mathrm{target}}-q_{\mathrm{control}}$ (while $p_{\mathrm{control}}\to p_{\mathrm{control}}+p_{\mathrm{target}})$ is denoted using a $\ominus$ at the target qubit instead of a $\oplus$. A parallel execution of the cnots for the $X$-checks is possible in the toric code.

Figure 8

Threshold comparison between decoding with or without GKP error information. On the left, the simulation only takes the average error rate into account and one obtains a threshold between ${\sigma }_0\approx 0.54$ and ${\sigma }_0\approx 0.55$ corresponding to $P\left(X\right)\approx 10\%$ and $P\left(X\right)\approx 10.7\%$, respectively. On the right, the simulation takes the GKP error information into account. In this case, the crossing point is around ${\sigma }_0^c\approx 0.6$ corresponding to $P\left(X\right)\approx 14\%$. The data are labeled by the distance of the toric code. “Bare GKP” is the logical error rate for a single GKP qubit whose errors are processed perfectly without measurement errors (green line in Fig. 5).

Figure 9

Notations for repeated toric-GKP errors. Data oscillators are located on the bonds of the square lattice. A GKP measurement error for the bond $b\equiv (ij,t)$ in the measurement cycle $t$ is denoted as ${\delta }_{ij}\left(t\right)\equiv {\delta }_b$, while the corresponding measurement error for the toric code generator on horizontal plaquette $p\equiv (h,t)$ is denoted as ${\xi }_h\left(t\right)\equiv {\xi }_p$. A data qubit error ${\epsilon }_{kl}\left(t\right)\equiv {\epsilon }_p$ is associated with the vertical plaquette $p$ directly below the bond $kl$ in the layer $t$.

Figure 10

Schematic phase diagram of the 3D anisotropic $q=2$ Villain gauge model Eq. (43) with ${\sigma }_{\mathrm{T}}=\sigma$. Solid line bounds the Meissner phase (light orange shading) in the clean isotropic limit, $\eta ={\sigma }_{\mathrm{M}}$, which corresponds to the Villain version of the model Eq. (44). With $\eta \to 0$, decodability condition for the toric-GKP code with $\sigma ={\sigma }_{\mathrm{T}}$ can not be satisfied in the region to the right from the dashed line, ${\sigma }^2>4/{\lambda }_c\left({\mathbb{Z}}_2\right)$, and along the upper boundary, ${\sigma }_{\mathrm{M}}\to \infty$, and it is satisfied along the left boundary, $\sigma \to 0$. We therefore expect the boundary of the Meissner phase in the clean $\eta \to 0$ limit as shown with dash-dotted line; this region includes the entire Meissner phase of the isotropic model. The green-hatched region represents the expected location of the ML decodable phase for the toric-GKP code; the sign of the curvature matches the bound in Ref. [24].

Figure 11

Numerical results for perfect GKP measurements, with $\sigma ={\sigma }_{\mathrm{T}}$ and ${\sigma }_{\mathrm{M}}=0$. The logical error rate as a function of the bare standard deviation ${\sigma }_0$, see Eq. (14), is shown for toric code distances $d=L$ as indicated in the caption. The vertical dotted line indicates the position of the crossing point in the data, a threshold at ${\sigma }_0\approx 0.47$, which corresponds to a logical error probability $p=6\%$ for single-qubit GKP code with ${\sigma }_{\mathrm{M}}=0$. The latter value is recovered as the vertical position on the dash-dotted line which reproduces the green line ${\sigma }_{\mathrm{M}}=0$ from Fig. 5 (left). The observed threshold is well above ${\sigma }_0=0.41$ which can be superficially expected from the $p=2.9\%$ threshold for the 2D toric code under phenomenological error model.

Figure 12

For both plots the dash-dotted line shows the logical error rate per round for a single GKP qubit corrected with the forward-minimization decoder, with $\sigma ={\sigma }_{\mathrm{T}}={\sigma }_{\mathrm{M}}$, as a function of the bare standard deviation ${\sigma }_0=\sigma /2\sqrt{\pi }$. (Left) Numerical results for Algorithm 1 without preprocessing of the GKP syndrome information (see text). We observe a crossing point at ${\sigma }_0\approx 0.243$ which can be translated to $p=1.3\%$. (Right) Numerical results for the Algorithm 2 with preprocessing of GKP syndrome information (see text). We observe that this decoder improves the logical error rates. However, the improvement being greater for smaller distances, the crossing point moves left to ${\sigma }_0\approx 0.235$ which corresponds to an error rate $p=1\%$.

Figure 13

Two examples of 11 rounds of error correction with data and measurement errors sampled from Gaussian distributions with ${\sigma }_0=0.4$. In the example on the left, forward-minimization and dynamic programming reach the same conclusion, whereas on the right forward-minimization reaches a wrong conclusion. One can compare this data with the sketched trajectories in Fig. 3.

Figure 14

In both figures periodic boundary conditions are assumed. (Left) Example of stabilizer checks and logical operators for the distance-5 continuous-variable toric-code. The stabilizer checks are also shown in Fig. 7. The support of the logical ${\mathit{p}}_{\mathit{1}}$ (blue/dark gray) and ${\mathit{q}}_1$ (purple/light gray) of one of the encoded oscillators is depicted. Shifts of strength $v$ on the support of ${\mathit{p}}_1$ is a logical $\mathit{p}$-shift of strength $v$ of the first oscillator. (Right) The support of the spread-out version of the logical operators ${\mathit{p}}_1$ and ${\mathit{q}}_1$. Both operators have the same support (blue/dark gray and purple/light gray) and one can verify that these logical operators are orthogonal to the stabilizer checks as well as commuting with them. A logical $\mathit{p}$-shift of strength $v$ on the first oscillator is now realized by applying $v/5$ on the support of the spread-out ${\mathit{p}}_1$. In general, if the torus had dimension $L_x\times L_y$, the spread-out ${\mathit{p}}_1$ would have a shift rescaling of $\frac{1}{L_x}$ over the whole lattice while ${\mathit{q}}_1$ would a shift rescaling of $\frac{1}{L_y}$. At the same time ${\mathit{p}}_2$ (respectively, ${\mathit{q}}_2)$ would have rescaling $\frac{1}{L_y}$ (respectively, $\frac{1}{L_x})$.