Abstract
Concatenating bosonic error-correcting codes with qubit codes can substantially boost the error-correcting power of the original qubit codes. It is not clear how to concatenate optimally, given that there are several bosonic codes and concatenation schemes to choose from, including the recently discovered Gottesman-Kitaev-Preskill (GKP) – stabilizer codes [Phys. Rev. Lett. 125, 080503 (2020)] that allow protection of a logical bosonic mode from fluctuations of the conjugate variables of the mode. We develop efficient maximum-likelihood decoders for and analyze the performance of three different concatenations of codes taken from the following set: qubit stabilizer codes, analog or Gaussian stabilizer codes, GKP codes, and GKP-stabilizer codes. We benchmark decoder performance against additive Gaussian white noise, corroborating our numerics with analytical calculations. We observe that the concatenation involving GKP-stabilizer codes outperforms the more conventional concatenation of a qubit stabilizer code with a GKP code in some cases. We also propose a GKP-stabilizer code that suppresses fluctuations in both conjugate variables without extra quadrature squeezing and formulate qudit versions of GKP-stabilizer codes.
- Received 7 October 2022
- Revised 9 April 2023
- Accepted 8 May 2023
DOI:https://doi.org/10.1103/PRXQuantum.4.020342
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Popular Summary
Protecting quantum information against noise from the environment is the main challenge in realizing large-scale quantum computers. Quantum error-correcting codes protect logical quantum information by embedding it into a larger Hilbert space so that different code words are sufficiently far apart and a small amount of noise will not mix them.
Depending on the underlying physical system, there are two mainstream code designs: discrete-variable (DV) codes make use of finite-dimensional subsystems such as qubits or qudits, while continuous-variable (CV) codes make use of photons or, more generally, quantum harmonic oscillators. Concatenating DV with CV codes combines the unique features of both DV and CV codes. However, due to the rapid development of new CV codes, performance studies of hybrid DV-CV codes are far from exhaustive.
In this work, we provide a unified encoding and decoding framework for several DV-CV concatenation schemes and numerically study their performance. Our results will guide the architecture design of large-scale DV-CV concatenated codes.
