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arXiv:2411.04277 · Physical Review A

Exploring the quantum capacity of a Gaussian random-displacement channel using Gottesman-Kitaev-Preskill codes and maximum-likelihood decoding

Mao Lin, Kyungjoo Noh

Published May 27, 2025• 2 citations
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Abstract

Determining the quantum capacity of a noisy quantum channel is an important problem in the field of quantum communication theory. In this work, we consider the Gaussian random-displacement channel ${\mathcal{N}}_{\ensuremath{\sigma}}$, a type of bosonic Gaussian channels relevant in various bosonic quantum information processing systems. In particular, we attempt to make progress on the problem of determining the quantum capacity of a Gaussian random-displacement channel by analyzing the error-correction performance of several families of multimode Gottesman-Kitaev-Preskill (GKP) codes. In doing so we analyze the surface-square GKP codes using an efficient and exact maximum-likelihood decoder (MLD) up to a large code distance of $d=39$. We find that the error threshold of the surface-square GKP code is remarkably close to $\ensuremath{\sigma}=1/\sqrt{e}\ensuremath{\simeq}0.6065$ at which the best-known lower bound of the quantum capacity of ${\mathcal{N}}_{\ensuremath{\sigma}}$ vanishes. We also analyze the performance of color-hexagonal GKP codes up to a code distance of $d=13$ using a tensor-network decoder serving as an approximate MLD. By focusing on multimode GKP codes that encode just one logical qubit over multiple bosonic modes, we show that GKP codes can achieve nonzero quantum state transmission rates for a Gaussian random-displacement channel ${\mathcal{N}}_{\ensuremath{\sigma}}$ at larger values of $\ensuremath{\sigma}$ than previously demonstrated. Thus, our work reduces the gap between the quantum communication theoretic bounds and the performance of explicit bosonic quantum error-correcting codes in regards to the quantum capacity of a Gaussian random-displacement channel.

DOI
10.1103/physreva.111.052445

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Determining the quantum capacity of a noisy quantum channel is an important problem in the field of quantum communication theory. In this work, we consider the Gaussian random-displacement channel N\ensuremathσ{\mathcal{N}}_{\ensuremath{\sigma}}N\ensuremathσ​, a type of bosonic Gaussian channels relevant in various bosonic quantum information processing systems. In particular, we attempt to make progress on the problem of determining the quantum capacity of a Gaussian random-displacement channel by analyzing the error-correction performance of several families of multimode Gottesman-Kitaev-Preskill (GKP) codes. In doing so we analyze the surface-square GKP codes using an efficient and exact maximum-likelihood decoder (MLD) up to a large code distance of d=39d=39d=39. We find that the error threshold of the surface-square GKP code is remarkably close to \ensuremathσ=1/e\ensuremath≃0.6065\ensuremath{\sigma}=1/\sqrt{e}\ensuremath{\simeq}0.6065\ensuremathσ=1/e​\ensuremath≃0.6065 at which the best-known lower bound of the quantum capacity of N\ensuremathσ{\mathcal{N}}_{\ensuremath{\sigma}}N\ensuremathσ​ vanishes. We also analyze the performance of color-hexagonal GKP codes up to a code distance of d=13d=13d=13 using a tensor-network decoder serving as an approximate MLD. By focusing on multimode GKP codes that encode just one logical qubit over multiple bosonic modes, we show that GKP codes can achieve nonzero quantum state transmission rates for a Gaussian random-displacement channel N\ensuremathσ{\mathcal{N}}_{\ensuremath{\sigma}}N\ensuremathσ​ at larger values of \ensuremathσ\ensuremath{\sigma}\ensuremathσ than previously demonstrated. Thus, our work reduces the gap between the quantum communication theoretic bounds and the performance of explicit bosonic quantum error-correcting codes in regards to the quantum capacity of a Gaussian random-displacement channel.
The Gottesman-Kitaev-Preskill (GKP) code is a promising bosonic quantum error-correcting code, encoding logical qubits into a bosonic mode in such a way that many physically relevant noise types can be corrected effectively. A particularly relevant noise channel is the pure loss channel, which the GKP code is known to protect against. In particular, it is commonly pointed out that losses can be corrected by the GKP code by transforming the losses into random Gaussian displacements through a quantum-limited amplification channel. However, implementing such amplification in practice is not ideal and could easily introduce an additional overhead of noise from associated experimental imperfections. Here, we analyze the performance of teleportation-based GKP error correction against loss in the absence of an amplification channel. We show that amplification is not required to perform GKP error correction and that performing amplification actually worsens the performance for practically relevant parameter regimes.
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.
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.
Abstract Boson quantum error correction is an important means to realize quantum error correction information processing. In this paper, we consider the connection of a single-mode Gottesman-Kitaev-Preskill (GKP) code with a two-dimensional (2D) surface (surface-GKP code) on a triangular quadrilateral lattice. On the one hand, we use a Steane-type scheme with maximum likelihood estimation for surface-GKP code error correction. On the other hand, the minimum-weight perfect matching (MWPM) algorithm is used to decode surface-GKP codes. In the case where only the data GKP qubits are noisy, the threshold reaches σ ≈ 0.5 ( pˉ≈12.3%\bar{p}\approx 12.3 \%pˉ​≈12.3% p ¯ ≈ 12.3 % ). If the measurement is also noisy, the threshold is reached σ ≈ 0.25 ( pˉ≈10.02%\bar{p}\approx 10.02 \%pˉ​≈10.02% p ¯ ≈ 10.02 % ). More importantly, we introduce a neural network decoder. When the measurements in GKP error correction are noise-free, the threshold reaches σ ≈ 0.78 ( pˉ≈15.12%\bar{p}\approx 15.12 \%pˉ​≈15.12% p ¯ ≈ 15.12 % ). The threshold reaches σ ≈ 0.34 ( pˉ≈11.37%\bar{p}\approx 11.37 \%pˉ​≈11.37% p ¯ ≈ 11.37 % ) when all measurements are noisy. Through the above optimization method, multi-party quantum error correction will achieve a better guarantee effect in fault-tolerant quantum computing.
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.
Quantum error correction (QEC) plays an essential role in fault-tolerantly realizing quantum algorithms of practical interest.Among different approaches to QEC, encoding logical quantum information in harmonic oscillator modes has been shown to be promising and hardware efficient.In this work, we study multimode Gottesman-Kitaev-Preskill (GKP) codes, encoding a qubit in many oscillators, through a lattice perspective.In particular, we implement a closest point decoding strategy for correcting random Gaussian shift errors.For decoding a generic multimode GKP code, we first identify its corresponding lattice followed by finding the closest lattice point in its symplectic dual lattice to a candidate shift error compatible with the error syndrome.We use this method to characterize the error-correction capabilities of several known multimode GKP codes, including their code distances and fidelities.We also perform numerical optimization of multimode GKP codes up to ten modes and find three instances (with three, seven, and nine modes) with better code distances and fidelities compared to the known GKP codes with the same number of modes.While exact closest point decoding incurs exponential time cost in the number of modes for general unstructured GKP codes, we give several examples of structured GKP codes (i.e., of the repetition-rectangular GKP code types) where the closest point decoding can be performed exactly in linear time.For the surface-GKP code, we show that the closest point decoding can be performed exactly in polynomial time with the help of a minimum-weight-perfect-matching algorithm (MWPM).We show that this MWPM closest point decoder improves both the fidelity and the noise threshold of the surface-GKP code to 0.602 compared to the previously studied MWPM decoder assisted by log-likelihood analog information, which yields a noise threshold of 0.599.