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.
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.
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% p ¯ ≈ 12.3 % ). If the measurement is also noisy, the threshold is reached σ ≈ 0.25 ( 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% p ¯ ≈ 15.12 % ). The threshold reaches σ ≈ 0.34 ( 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 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.
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.
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.
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.
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.