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arXiv:2212.11397 · Physical Review A

Biased Gottesman-Kitaev-Preskill repetition code

Matthew P. Stafford, Nicolas C. Menicucci

Published Nov 27, 2023• 10 citations
arXivHTMLPDF

Abstract

Continuous-variable quantum computing architectures based upon the Gottesman-Kitaev-Preskill (GKP) encoding have emerged as a promising candidate because one can achieve fault tolerance with a probabilistic supply of GKP states and Gaussian operations. Furthermore, by generalizing to rectangular-lattice GKP states, a bias can be introduced and exploited through concatenation with qubit codes that show improved performance under biasing. However, these codes (such as the XZZX surface code) still require weight-four stabilizer measurements and have complex decoding requirements to overcome. In this work, we study the code-capacity behavior of a rectangular-lattice GKP encoding concatenated with a repetition code under an isotropic Gaussian displacement channel. We find a numerical threshold of $\ensuremath{\sigma}=0.599$ for the noise's standard deviation, which outperforms the biased GKP planar surface code with a trade-off of increased biasing at the GKP level. This is all achieved with only weight-two stabilizer operators and simple decoding at the qubit level. Furthermore, with moderate levels of bias (aspect ratio $\ensuremath{\le}2.4$) and nine or fewer data modes, significant reductions in logical error rates can still be achieved for $\ensuremath{\sigma}\ensuremath{\le}0.3$, opening the possibility of using biased GKP repetition codes as a simple low-level qubit encoding for further concatenation.

DOI
10.1103/physreva.108.052428

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The Gottesman-Kitaev-Preskill (GKP) code is an important type of bosonic quantum error-correcting code. Since the GKP code only protects against small shift errors in $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}$ and $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{q}$ quadratures, it is necessary to concatenate the GKP code with a stabilizer code for the larger error correction. In this paper, we consider the concatenation of the single-mode GKP code with the two-dimensional (2D) color code (color-GKP code) on the square-octagon lattice. We use the Steane-type scheme with a maximum-likelihood estimation for GKP error correction and show its advantage for the concatenation. In our main paper, the minimum-weight perfect matching algorithm is applied to decode the color-GKP code. Complemented with the continuous-variable information from the GKP code, the threshold of the 2D color code is improved. If only data GKP qubits are noisy, the threshold reaches $\ensuremath{\sigma}\ensuremath{\approx}0.59(\overline{p}\ensuremath{\approx}13.3%)$ compared with $\overline{p}=10.2%$ of the normal 2D color code. If measurements are also noisy, we introduce the generalized restriction decoder on the three-dimensional space-time graph for decoding. The threshold reaches $\ensuremath{\sigma}\ensuremath{\approx}0.46$ when measurements in the GKP error correction are noiseless, and $\ensuremath{\sigma}\ensuremath{\approx}0.24$ when all measurements are noisy. Lastly, the good performance of the generalized restriction decoder is also shown on the normal 2D color code giving the threshold at $3.1%$ under the phenomenological error model.

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Continuous-variable quantum computing architectures based upon the Gottesman-Kitaev-Preskill (GKP) encoding have emerged as a promising candidate because one can achieve fault tolerance with a probabilistic supply of GKP states and Gaussian operations. Furthermore, by generalizing to rectangular-lattice GKP states, a bias can be introduced and exploited through concatenation with qubit codes that show improved performance under biasing. However, these codes (such as the XZZX surface code) still require weight-four stabilizer measurements and have complex decoding requirements to overcome. In this work, we study the code-capacity behavior of a rectangular-lattice GKP encoding concatenated with a repetition code under an isotropic Gaussian displacement channel. We find a numerical threshold of \ensuremathσ=0.599\ensuremath{\sigma}=0.599\ensuremathσ=0.599 for the noise's standard deviation, which outperforms the biased GKP planar surface code with a trade-off of increased biasing at the GKP level. This is all achieved with only weight-two stabilizer operators and simple decoding at the qubit level. Furthermore, with moderate levels of bias (aspect ratio \ensuremath≤2.4\ensuremath{\le}2.4\ensuremath≤2.4) and nine or fewer data modes, significant reductions in logical error rates can still be achieved for \ensuremathσ\ensuremath≤0.3\ensuremath{\sigma}\ensuremath{\le}0.3\ensuremathσ\ensuremath≤0.3, opening the possibility of using biased GKP repetition codes as a simple low-level qubit encoding for further concatenation.
Bosonic codes provide an alternative option for quantum error correction. An important category of bosonic codes called the Gottesman-Kitaev-Preskill (GKP) code has aroused much interest recently. Theoretically, the error correction ability of the GKP code is limited since it can only correct small shift errors in position and momentum quadratures. A natural approach to promote the GKP error correction for large-scale, fault-tolerant quantum computation is concatenating encoded GKP states with a stabilizer code. The performance of the XZZX surface-GKP code, i.e., the single-mode GKP code concatenated with the XZZX surface code is investigated in this paper under two different noise models. Firstly, in the code-capacity noise model, the asymmetric rectangular GKP code with parameter $\ensuremath{\lambda}$ is introduced. Using the minimum weight perfect matching decoder combined with the continuous-variable GKP information, the optimal threshold of the XZZX-surface GKP code reaches $\ensuremath{\sigma}\ensuremath{\approx}0.67$ when $\ensuremath{\lambda}=2.1$, compared with the threshold $\ensuremath{\sigma}\ensuremath{\approx}0.60$ of the standard surface-GKP code. Secondly, we analyze the shift errors of two-qubit gates in the actual implementation and build the full circuit-level noise model. By setting the appropriate bias parameters, the logical error rate is reduced by several times in some cases. These results indicate the XZZX surface-GKP codes are more suitable for asymmetric concatenation under the general noise models. We also estimate the overhead of the XZZX-surface GKP code which uses about 291 GKP states with the noise parameter 18.5 dB ($\ensuremath{\kappa}/g\ensuremath{\approx}0.71%$) to encode a logical qubit with the error rate $2.53\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}7}$, compared with the qubit-based surface code using 3041 qubits to achieve almost the same logical error rate.
Concatenation of a bosonic code with a qubit code is one of the promising ways to achieve fault-tolerant quantum computation. As one of the most important bosonic codes, the Gottesman-Kitaev-Preskill (GKP) code is proposed to correct small displacement errors in phase space. If the noise in phase space is biased, the square-lattice GKP code can be concatenated with the repetition code that promises a high fault-tolerant threshold to suppress the logical error. In this work, we study the performance of GKP repetition codes with finite-energy ancillary GKP qubits in correcting biased noise. We find that there exists a critical value of noise variance for the ancillary GKP qubit such that the logical Pauli error rate decreases when increasing the code size. Furthermore, one round of GKP error correction has to be performed before concatenating with the repetition code. Our study paves the way for practical implementation of error correction by concatenating the GKP code with low-level qubit codes.
The continuous-variable (CV) Gaussian no-go theorem fundamentally limits the suppression of Gaussian displacement errors using only Gaussian gates and states. Prior studies have employed Gottesman-Kitaev-Preskill (GKP) states as ancillary qumodes to suppress small Gaussian displacement errors, but when the displacement magnitude becomes large, lattice-crossing events arise beyond the correctable range of the GKP state. To address this issue, we concatenate a Gaussian-noise-suppression circuit with an outer analog Steane code that corrects such occasional lattice-crossing events as well as other abrupt displacement errors. Unlike conventional concatenation, which primarily aims to reduce logical error rates, the Steane-GKP duality in encoding provides complementary protection against both large and small displacement errors, enabling CV error correction within the continuous encoding space and contrasting with earlier approaches that concatenate GKP states with repetition codes for discrete qubit or qudit encodings. Analytical results show that, under infinite squeezing, the concatenated code suppresses the variance of Gaussian displacement errors across all qumodes by up to 50 percent while enabling unbiased correction of lattice-crossing events, with a success probability determined by the ratio between the residual Gaussian error standard deviation and the lattice-crossing magnitude. Even with finite squeezing, the proposed architecture continues to provide Gaussian-error suppression together with lattice-crossing correction, and the presence of the outer analog Steane code relaxes the squeezing requirement of the inner GKP states, indicating near-term experimental feasibility. This work establishes a viable route toward fault-tolerant continuous-variable quantum computation and provides new insight into the design of concatenated CV error-correcting architectures.
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 $\ensuremath{\sigma}\ensuremath{\approx}0.58$ 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 $\ensuremath{\sigma}\ensuremath{\approx}0.54$ and $\ensuremath{\sigma}\ensuremath{\approx}0.55$ 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.
In order to solve practical problems on a quantum computer, it is necessary to use fault-tolerant quantum error correction schemes to overcome noise: the errors arising from imperfections in physical components. The Gottesman-Kitaev-Preskill (GKP) code aims at achieving this in a hardware efficient manner by encoding finite-dimensional logical subspaces in the Hilbert space of one or more continuous variable modes. In near term implementations, however, it is not feasible to eliminate errors entirely, so it is natural to also employ alternative error mitigation techniques together with error correction. In this work, we study a quantum error mitigation method known as probabilistic error cancellation in the context of the GKP code. We compare Steane-type and teleportation-based GKP error correction, and calculate the sampling overheads associated with the technique for square and hexagonal GKP codes. We employ the stabilizer subsystem decomposition for GKP codes to obtain an effective logical channel for noisy operations that is used to express the ideal one of a target logical unitary. We consider noise from finite squeezing on the data and the two ancilla modes, needed for error correction, and examine the relationship between the sampling overhead and the noise for different decoding methods. Our results are calculated for single- and two-qubit GKP Clifford gates and one round of error correction, and show the nontrivial combination of error correction and mitigation in continuous variable codes.
The Gottesman-Kitaev-Preskill (GKP) code is an important type of bosonic quantum error-correcting code. Since the GKP code only protects against small shift errors in $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}$ and $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{q}$ quadratures, it is necessary to concatenate the GKP code with a stabilizer code for the larger error correction. In this paper, we consider the concatenation of the single-mode GKP code with the two-dimensional (2D) color code (color-GKP code) on the square-octagon lattice. We use the Steane-type scheme with a maximum-likelihood estimation for GKP error correction and show its advantage for the concatenation. In our main paper, the minimum-weight perfect matching algorithm is applied to decode the color-GKP code. Complemented with the continuous-variable information from the GKP code, the threshold of the 2D color code is improved. If only data GKP qubits are noisy, the threshold reaches $\ensuremath{\sigma}\ensuremath{\approx}0.59(\overline{p}\ensuremath{\approx}13.3%)$ compared with p‾=10.2\overline{p}=10.2%p​=10.2 of the normal 2D color code. If measurements are also noisy, we introduce the generalized restriction decoder on the three-dimensional space-time graph for decoding. The threshold reaches $\ensuremath{\sigma}\ensuremath{\approx}0.46$ when measurements in the GKP error correction are noiseless, and $\ensuremath{\sigma}\ensuremath{\approx}0.24$ when all measurements are noisy. Lastly, the good performance of the generalized restriction decoder is also shown on the normal 2D color code giving the threshold at 3.13.1%3.1 under the phenomenological error model.