We examine general Gottesman-Kitaev-Preskill (GKP) codes for continuous-variable quantum error correction, including concatenated GKP codes, through the lens of lattice theory, in order to better understand the structure of this class of stabilizer codes. We derive formal bounds on code parameters, show how different decoding strategies are precisely related, propose new ways to obtain GKP codes by means of glued lattices and the tensor product of lattices and point to natural resource savings that have remained hidden in recent approaches. We present general results that we illustrate through examples taken from different classes of codes, including scaled self-dual GKP codes and the concatenated surface-GKP code.We examine general Gottesman-Kitaev-Preskill (GKP) codes for continuous-variable quantum error correction, including concatenated GKP codes, through the lens of lattice theory, in order to better understand the structure of this class of stabilizer codes. We derive formal bounds on code parameters, show how different decoding strategies are precisely related, propose new ways to obtain GKP codes by means of glued lattices and the tensor product of lattices and point to natural resource savings that have remained hidden in recent approaches. We present general results that we illustrate through examples taken from different classes of codes, including scaled self-dual GKP codes and the concatenated surface-GKP code.

Jan 1, 202241 citations

Quantum low-density lattice codes

Timo Hillmann, Jens Eisert, Francesco Arzani

Gottesman-Kitaev-Preskill (GKP) codes provide a family of promising schemes for encoding discrete quantum information (qudits) into infinite-dimensional bosonic modes based on mathematical lattices. While such codes, when concatenated with discrete-variable codes, are relatively well studied, the construction and decoding of native GKP codes has largely remained open due to the computationally hard problems encountered. To address this challenge, we advocate a strategy of co-designing the decoder and the quantum error-correcting code itself by constructing lattices for which decoding is feasible: The requirement of efficient decoding effectively determines the quantum error-correcting code. This construction is built on classical low-density lattice codes (LDLCs), a lattice analogue of low-density parity-check codes, here lifted to families of GKP codes. Concretely, we introduce quantum versions of classical, randomly constructed LDLCs. We show that after suitable dimensionality reduction these codes have code properties comparable to or better than concatenated GKP-surface codes of equal number of modes. However, the GKP-LDLCs constructed here do not have a strictly sparse parity check matrix, which motivates our study of the performance of natively analog message-passing decoders originally developed for LDLCs when applied to concatenated GKP-LDPC codes. We show that the fully analog, linear-time decoder achieves performances close to state-of-the-art hybrid qubit-analog decoders. To facilitate future research on the structure and performance of general GKP codes, the relevant source code will be released in open-source Julia packages LatticeDecoder.jl and SymplecticGKP.jl.Gottesman-Kitaev-Preskill (GKP) codes provide a family of promising schemes for encoding discrete quantum information (qudits) into infinite-dimensional bosonic modes based on mathematical lattices. While such codes, when concatenated with discrete-variable codes, are relatively well studied, the construction and decoding of native GKP codes has largely remained open due to the computationally hard problems encountered. To address this challenge, we advocate a strategy of co-designing the decoder and the quantum error-correcting code itself by constructing lattices for which decoding is feasible: The requirement of efficient decoding effectively determines the quantum error-correcting code. This construction is built on classical low-density lattice codes (LDLCs), a lattice analogue of low-density parity-check codes, here lifted to families of GKP codes. Concretely, we introduce quantum versions of classical, randomly constructed LDLCs. We show that after suitable dimensionality reduction these codes have code properties comparable to or better than concatenated GKP-surface codes of equal number of modes. However, the GKP-LDLCs constructed here do not have a strictly sparse parity check matrix, which motivates our study of the performance of natively analog message-passing decoders originally developed for LDLCs when applied to concatenated GKP-LDPC codes. We show that the fully analog, linear-time decoder achieves performances close to state-of-the-art hybrid qubit-analog decoders. To facilitate future research on the structure and performance of general GKP codes, the relevant source code will be released in open-source Julia packages LatticeDecoder.jl and SymplecticGKP.jl.

quant-pharXiv:2609.03021Sep 4, 20260 citations

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

Jiaxuan Zhang, Jian Zhao, Yu-Chun Wu +1 more

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.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.

arXiv:2112.14447Physical Review ADec 20, 202115 citations

Achievable Rates for Concatenated Square Gottesman-Kitaev-Preskill Codes

Mahadevan Subramanian, Guo Zheng, Liang Jiang

The Gottesman-Kitaev-Preskill (GKP) codes are known to achieve optimal rates under displacement noise and pure-loss channels, which establishes theoretical foundations for its optimality. However, such optimal rates are only known to be achieved at a discrete set of noise strengths with the current self-dual symplectic lattice construction. In this work, we develop a new coding strategy using concatenated continuous variable-discrete variable encodings to go beyond past results and establish GKP’s optimal rate over all noise strengths. In particular, for displacement noise, the rate is obtained through a constructive approach by concatenating GKP codes with a quantum polar code and analog decoding. For a pure-loss channel, we prove the existence of capacity-achieving GKP codes through a random coding approach. These results highlight the capability of concatenation-based GKP codes and provides new methods for constructing good GKP lattices.The Gottesman-Kitaev-Preskill (GKP) codes are known to achieve optimal rates under displacement noise and pure-loss channels, which establishes theoretical foundations for its optimality. However, such optimal rates are only known to be achieved at a discrete set of noise strengths with the current self-dual symplectic lattice construction. In this work, we develop a new coding strategy using concatenated continuous variable-discrete variable encodings to go beyond past results and establish GKP’s optimal rate over all noise strengths. In particular, for displacement noise, the rate is obtained through a constructive approach by concatenating GKP codes with a quantum polar code and analog decoding. For a pure-loss channel, we prove the existence of capacity-achieving GKP codes through a random coding approach. These results highlight the capability of concatenation-based GKP codes and provides new methods for constructing good GKP lattices.

arXiv:2505.10499PRX QuantumNov 21, 20251 citation

Concatenation of the Gottesman-Kitaev-Preskill code with the XZZX surface code

Jiaxuan Zhang, Yu-Chun Wu, Guo‐Ping Guo

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.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.

arXiv:2207.04383Physical Review AJun 9, 202316 citations

Probabilistic Error Cancellation for Single-Mode Gottesman-Kitaev-Preskill Codes

Victoria Wadewitz, Alessandro Ciani

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.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.

quant-pharXiv:2609.17095Sep 16, 20260 citations

\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.