Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2611.00694

Photon-loss tolerance of the surface-GKP code with anti-photon-loss decoding

Guan-Ju Yan, Pei-Lin Du, Ke-Min Hu, Jun-Jie He, Yu-Ze Zhu, Xing-Yu Chen, Lin Yang, An-Bang Wu, Hao Tang, Xian-Min Jin

quant-phSubmitted Oct 31, 2026• Announced Nov 3, 2026
arXivHTMLPDF

Abstract

Photon loss somewhat limits fault-tolerant photonic quantum computing. We investigate its effect on finite-energy Gottesman--Kitaev--Preskill (GKP) states by modeling the pure-loss channel directly. We obtain photon-loss threshold estimates for the surface-GKP code using nearest-lattice-point inner GKP decoding and minimum-weight perfect matching for the outer surface code. For GKP states with $ε=0.01$ and $ε=0.05$, corresponding to GKP squeezing of $20$ and $13\,\mathrm{dB}$, the estimated thresholds are $γ^\ast=5.5\%$ ($0.24\,\mathrm{dB}$) and $γ^\ast=5.9\%$ ($0.26\,\mathrm{dB}$), respectively. We then introduce an anti-photon-loss inner decoder that compensates for the contraction of the measured quadrature. It increases the corresponding threshold estimates to $γ^\ast=9.8\%$ ($0.45\,\mathrm{dB}$) and $γ^\ast=6.3\%$ ($0.28\,\mathrm{dB}$). Our results highlight the importance of accounting for loss-induced lattice contraction in inner decoding and demonstrate that quadrature compensation enhances the photon-loss tolerance of surface-GKP codes for both high- and moderate-quality GKP states, providing a route toward more photon-loss-robust optical quantum computing.

Comments
9 pages,5 figures

Related papers

Low-Overhead Fault-Tolerant Quantum Error Correction with the Surface-GKP Code

Kyungjoo Noh, Christopher Chamberland, Fernando G. S. L. Brandão

Fault-tolerant quantum error correction is essential for implementing quantum algorithms of significant practical importance.In this work, we propose a highly effective use of the surface Gottesman-Kitaev-Preskill (GKP) code, i.e., the surface code consisting of bosonic GKP qubits instead of bare two-level qubits.In our proposal, we use error-corrected two-qubit gates between GKP qubits and introduce a maximum-likelihood decoding strategy for correcting shift errors in the two-GKP-qubit gates.Our proposed decoding reduces the total CNOT failure rate of the GKP qubits, e.g., from 0.87% to 0.36% at a GKP squeezing of 12 dB, compared to the case where the simple closest-integer decoding is used.Then, by concatenating the GKP code with the surface code, we find that the threshold GKP squeezing is given by 9.9 dB under the the assumption that finite squeezing of the GKP states is the dominant noise source.More importantly, we show that a low logical failure rate p L < 10 -7 can be achieved with moderate hardware requirements, e.g., 291 modes and 97 qubits at a GKP squeezing of 12 dB as opposed to 1457 bare qubits for the standard rotated surface code at an equivalent noise level (i.e., p = 0.36%).Such a low failure rate of our surface-GKP code is possible through the use of space-time correlated edges in the matching graphs of the surface-code decoder.Further, all edge weights in the matching graphs are computed dynamically based on analog information from the GKP error correction using the full history of all syndrome measurement rounds.We also show that a highly squeezed GKP state of GKP squeezing 12 dB can be experimentally realized by using a dissipative stabilization method, namely, the big-small-big method, with fairly conservative experimental parameters.Lastly, we introduce a three-level ancilla scheme to mitigate ancilla decay errors during a GKP state preparation.

arXiv:2103.06994•PRX Quantum•Jan 28, 2022•79 citations

Fault-tolerant bosonic quantum error correction with the surface–Gottesman-Kitaev-Preskill code

Kyungjoo Noh, Christopher Chamberland

The performance of the circuit-based surface-GKP bosonic quantum error correction code under noise due to finite squeezing of the GKP states and photon losses is investigated. The authors show under what conditions fault-tolerant quantum error correction is possible with the surface-GKP code.

arXiv:1908.03579•Physical Review A•Jan 13, 2020•161 citations

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

Lisa Hänggli, Margret Heinze, Robert König

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.

arXiv:2004.00541•Physical Review A•Nov 10, 2020•41 citations

Very low overhead fault-tolerant quantum error correction with the surface-GKP code

Kyungjoo Noh, Christopher Chamberland, Fernando G. S. L. Brandão

Fault-tolerant quantum error correction is essential for implementing quantum algorithms of significant practical importance. In this work, we propose a highly effective use of the surface-GKP code, i.e., the surface code consisting of bosonic GKP qubits instead of bare two-dimensional qubits. In our proposal, we use error-corrected two-qubit gates between GKP qubits and introduce a maximum likelihood decoding strategy for correcting shift errors in the two-GKP-qubit gates. Our proposed decoding reduces the total CNOT failure rate of the GKP qubits, e.g., from $0.87\%$ to $0.36\%$ at a GKP squeezing of $12$dB, compared to the case where the simple closest-integer decoding is used. Then, by concatenating the GKP code with the surface code, we find that the threshold GKP squeezing is given by $9.9$dB under the the assumption that finite-squeezing of the GKP states is the dominant noise source. More importantly, we show that a low logical failure rate $p_{L} < 10^{-7}$ can be achieved with moderate hardware requirements, e.g., $291$ modes and $97$ qubits at a GKP squeezing of $12$dB as opposed to $1457$ bare qubits for the standard rotated surface code at an equivalent noise level (i.e., $p=0.36\%$). Such a low failure rate of our surface-GKP code is possible through the use of space-time correlated edges in the matching graphs of the surface code decoder. Further, all edge weights in the matching graphs are computed dynamically based on analog information from the GKP error correction using the full history of all syndrome measurement rounds. We also show that a highly-squeezed GKP state of GKP squeezing $\gtrsim 12$dB can be experimentally realized by using a dissipative stabilization method, namely, the Big-small-Big method, with fairly conservative experimental parameters. Lastly, we introduce a three-level ancilla scheme to mitigate ancilla decay errors during a GKP state preparation.

•Mar 11, 2021•9 citations

Multidimensional Bose quantum error correction based on neural network decoder

Haowen Wang, Yun-Jia Xue, Yingjie Qu +2 more

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 ( $$\bar{p}\approx 12.3 \%$$ p ¯ ≈ 12.3 % ). If the measurement is also noisy, the threshold is reached σ ≈ 0.25 ( $$\bar{p}\approx 10.02 \%$$ 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 ( $$\bar{p}\approx 15.12 \%$$ p ¯ ≈ 15.12 % ). The threshold reaches σ ≈ 0.34 ( $$\bar{p}\approx 11.37 \%$$ 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.

npj Quantum Information•Nov 18, 2022•26 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.

arXiv:2112.14447•Physical Review A•Dec 20, 2021•15 citations
Photon loss somewhat limits fault-tolerant photonic quantum computing. We investigate its effect on finite-energy Gottesman--Kitaev--Preskill (GKP) states by modeling the pure-loss channel directly. We obtain photon-loss threshold estimates for the surface-GKP code using nearest-lattice-point inner GKP decoding and minimum-weight perfect matching for the outer surface code. For GKP states with ε=0.01ε=0.01ε=0.01 and ε=0.05ε=0.05ε=0.05, corresponding to GKP squeezing of 202020 and 13 dB13\,\mathrm{dB}13dB, the estimated thresholds are γ∗=5.5%γ^\ast=5.5\%γ∗=5.5% (0.24 dB0.24\,\mathrm{dB}0.24dB) and γ∗=5.9%γ^\ast=5.9\%γ∗=5.9% (0.26 dB0.26\,\mathrm{dB}0.26dB), respectively. We then introduce an anti-photon-loss inner decoder that compensates for the contraction of the measured quadrature. It increases the corresponding threshold estimates to γ∗=9.8%γ^\ast=9.8\%γ∗=9.8% (0.45 dB0.45\,\mathrm{dB}0.45dB) and γ∗=6.3%γ^\ast=6.3\%γ∗=6.3% (0.28 dB0.28\,\mathrm{dB}0.28dB). Our results highlight the importance of accounting for loss-induced lattice contraction in inner decoding and demonstrate that quadrature compensation enhances the photon-loss tolerance of surface-GKP codes for both high- and moderate-quality GKP states, providing a route toward more photon-loss-robust optical quantum computing.
Fault-tolerant quantum error correction is essential for implementing quantum algorithms of significant practical importance.In this work, we propose a highly effective use of the surface Gottesman-Kitaev-Preskill (GKP) code, i.e., the surface code consisting of bosonic GKP qubits instead of bare two-level qubits.In our proposal, we use error-corrected two-qubit gates between GKP qubits and introduce a maximum-likelihood decoding strategy for correcting shift errors in the two-GKP-qubit gates.Our proposed decoding reduces the total CNOT failure rate of the GKP qubits, e.g., from 0.87% to 0.36% at a GKP squeezing of 12 dB, compared to the case where the simple closest-integer decoding is used.Then, by concatenating the GKP code with the surface code, we find that the threshold GKP squeezing is given by 9.9 dB under the the assumption that finite squeezing of the GKP states is the dominant noise source.More importantly, we show that a low logical failure rate p L < 10 -7 can be achieved with moderate hardware requirements, e.g., 291 modes and 97 qubits at a GKP squeezing of 12 dB as opposed to 1457 bare qubits for the standard rotated surface code at an equivalent noise level (i.e., p = 0.36%).Such a low failure rate of our surface-GKP code is possible through the use of space-time correlated edges in the matching graphs of the surface-code decoder.Further, all edge weights in the matching graphs are computed dynamically based on analog information from the GKP error correction using the full history of all syndrome measurement rounds.We also show that a highly squeezed GKP state of GKP squeezing 12 dB can be experimentally realized by using a dissipative stabilization method, namely, the big-small-big method, with fairly conservative experimental parameters.Lastly, we introduce a three-level ancilla scheme to mitigate ancilla decay errors during a GKP state preparation.
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.
Fault-tolerant quantum error correction is essential for implementing quantum algorithms of significant practical importance. In this work, we propose a highly effective use of the surface-GKP code, i.e., the surface code consisting of bosonic GKP qubits instead of bare two-dimensional qubits. In our proposal, we use error-corrected two-qubit gates between GKP qubits and introduce a maximum likelihood decoding strategy for correcting shift errors in the two-GKP-qubit gates. Our proposed decoding reduces the total CNOT failure rate of the GKP qubits, e.g., from 0.87%0.87\%0.87% to 0.36%0.36\%0.36% at a GKP squeezing of 121212dB, compared to the case where the simple closest-integer decoding is used. Then, by concatenating the GKP code with the surface code, we find that the threshold GKP squeezing is given by 9.99.99.9dB under the the assumption that finite-squeezing of the GKP states is the dominant noise source. More importantly, we show that a low logical failure rate pL<10−7p_{L} < 10^{-7}pL​<10−7 can be achieved with moderate hardware requirements, e.g., 291291291 modes and 979797 qubits at a GKP squeezing of 121212dB as opposed to 145714571457 bare qubits for the standard rotated surface code at an equivalent noise level (i.e., p=0.36%p=0.36\%p=0.36%). Such a low failure rate of our surface-GKP code is possible through the use of space-time correlated edges in the matching graphs of the surface code decoder. Further, all edge weights in the matching graphs are computed dynamically based on analog information from the GKP error correction using the full history of all syndrome measurement rounds. We also show that a highly-squeezed GKP state of GKP squeezing ≳12\gtrsim 12≳12dB can be experimentally realized by using a dissipative stabilization method, namely, the Big-small-Big method, with fairly conservative experimental parameters. Lastly, we introduce a three-level ancilla scheme to mitigate ancilla decay errors during a GKP state preparation.
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.
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.