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

Abstract

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 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 $\lambda$ 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 $\sigma \approx 0.67$ when $\lambda =2.1$, compared with the threshold $\sigma \approx 0.60$ 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 ($\kappa /g\approx 0.71\%$) to encode a logical qubit with the error rate $2.53\times 10^{-7}$, compared with the qubit-based surface code using 3041 qubits to achieve almost the same logical error rate.

arXiv comment17 pages, 10 figures

References in corpus (48)

• Mixed State Entanglement and Quantum Error Correction
• Quantum information with continuous variables
• Gaussian Quantum Information
• Surface codes: Towards practical large-scale quantum computation
• Topological quantum memory
• Encoding a qubit in an oscillator
• Quantum Error Correction for Quantum Memories
• Multiple Particle Interference and Quantum Error Correction
• Quantum Computing with Very Noisy Devices
• Reliable Quantum Computers
• Continuous variable quantum information: Gaussian states and beyond
• Topological Quantum Distillation
• Demonstration of Entanglement of Electrostatically Coupled Singlet-Triplet Qubits
• High threshold universal quantum computation on the surface code
• Quantum error correction of a qubit encoded in grid states of an oscillator
• Real-time quantum error correction beyond break-even
• New class of quantum error-correcting codes for a bosonic mode
• Experimental Quantum Computations on a Topologically Encoded Qubit
• Quantum computing with nearest neighbor interactions and error rates over 1%
• Stabilizer Formalism for Operator Quantum Error Correction
• Performance and structure of single-mode bosonic codes
• Hardware-efficient autonomous quantum error correction
• Demonstration of quantum error correction and universal gate set on a binomial bosonic logical qubit
• Simulation of topological field theories by quantum computers
• The XZZX Surface Code
• Efficient Algorithms for Maximum Likelihood Decoding in the Surface Code
• Ultrahigh Error Threshold for Surface Codes with Biased Noise
• Bosonic quantum error correction codes in superconducting quantum circuits
• Bias-preserving gates with stabilized cat qubits
• Beating the break-even point with a discrete-variable-encoded logical qubit
• Optimal Resources for Topological 2D Stabilizer Codes: Comparative Study
• Fault-tolerant bosonic quantum error correction with the surface-GKP code
• Repetition Cat Qubits for Fault-Tolerant Quantum Computation
• Fault-tolerant detection of a quantum error
• Improved quantum capacity bounds of Gaussian loss channels and achievable rates with Gottesman-Kitaev-Preskill codes
• Quantum Error Correction with the Toric-GKP Code
• Quantum Error Correction with the Gottesman-Kitaev-Preskill Code
• Fault-tolerant thresholds for quantum error correction with the surface code
• Fault-tolerant thresholds for the surface code in excess of 5% under biased noise
• Fault-Tolerant Computing With Biased-Noise Superconducting Qubits
• Programmable interference between two microwave quantum memories
• Continuous-variable gate teleportation and bosonic-code error correction
• Low overhead fault-tolerant quantum error correction with the surface-GKP code
• Engineering bilinear mode coupling in circuit QED: Theory and experiment
• 2-D color code quantum computation
• Fault-Tolerant Quantum Computing in the Pauli or Clifford Frame with Slow Error Diagnostics
• Enhanced noise resilience of the surface-GKP code via designed bias
• Quantum error correction with the color-Gottesman-Kitaev-Preskill code

Cited by in corpus (10)

• Advances in Bosonic Quantum Error Correction with Gottesman-Kitaev-Preskill Codes: Theory, Engineering and Applications
• Quantum spherical codes
• Biased Gottesman-Kitaev-Preskill repetition code
• Weight Reduced Stabilizer Codes with Lower Overhead
• Bosonic Pauli+: Efficient Simulation of Concatenated Gottesman-Kitaev-Preskill Codes
• Q-Pandora Unboxed: Characterizing Noise Resilience of Quantum Error Correction Codes
• Linear optical quantum computing with a hybrid squeezed cat code
• Fault-Tolerant Non-Clifford GKP Gates using Polynomial Phase Gates and On-Demand Noise Biasing
• Exploring the quantum capacity of a Gaussian random displacement channel using Gottesman-Kitaev-Preskill codes and maximum likelihood decoding
• Multimode rotationally symmetric bosonic codes from group-theoretic construction