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.
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.
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.
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.
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.
A promising route towards fault-tolerant quantum error correction is the concatenation of a Gottesman-Kitaev-Preskill (GKP) code with a qubit code. Development of such concatenated codes requires simulation tools which realistically model noise, while being able to simulate the dynamics of many modes. However, so far, large-scale simulation tools for concatenated GKP codes have been limited to idealized noise models and GKP code implementations. Here, we introduce the Bosonic Pauli+ model (BP+), which can be simulated efficiently for a large number of modes, while capturing the rich dynamics in the bosonic multi-mode Hilbert space. We demonstrate the method by simulating a hybrid surface code, where the data qubits are finite-energy GKP qubits stabilized using the small-Big-small (sBs) protocol, and the syndrome qubits are standard two-level systems. Using BP+, we present logical error rates of such an implementation. Confidence in the accuracy of the method is gained by comparing its predictions with full time evolution simulations for several relevant quantum circuits. While developed specifically for GKP qubits stabilized using the sBs protocol, the mathematical structure of BP+ is generic and may be applicable also to the simulation of concatenations using other bosonic codes.
The Gottesman-Kitaev-Preskill (GKP) code is a promising bosonic candidate for realizing fault-tolerant quantum computation. Among existing error-correction protocols for GKP code, the Steane-type scheme is a canonical and widely adopted paradigm, yet its intrinsic noise propagation pattern limits further performance improvement. In this work, we propose a preprocessing-based Steane-type (P-Steane) scheme, which introduces a tunable preprocessing stage with squeezing parameters a and b to actively reshape noise propagation, thereby constituting a parameter framework. This framework spans a spectrum of protocols beyond existing methods, reproducing the performance of both the ME-Steane scheme (a=1, b=1) and the teleportation-based scheme (a=1/2, b=2) as special cases. Crucially, in the small-noise regime and when the data qubit is noisier than the ancilla qubits, P-Steane scheme achieves the minimum product of position- and momentum-quadrature output noise variances when 2a=b, and consistently outperforms the ME-Steane scheme within a specific squeezing-parameter range under this condition.
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.