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  • Open Access

Achievable Rates for Concatenated Square Gottesman-Kitaev-Preskill Codes

Mahadevan Subramanian*, Guo Zheng†, and Liang Jiang‡

  • Pritzker School of Molecular Engineering, The University of Chicago, Chicago, Illinois 60637, USA

  • *Contact author: mahadevans@uchicago.edu
  • †Contact author: guozheng@uchicago.edu
  • ‡Contact author: liangjiang@uchicago.edu
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PRX Quantum 6, 040342 – Published 21 November, 2025

DOI: https://doi.org/10.1103/56vj-z7h1

Abstract

The Gottesman-Kitaev-Preskill (GKP) codes are known to achieve optimal rates under displacement noise and pure-loss channels, which establishes theoretical foundations for its optimality. However, such optimal rates are only known to be achieved at a discrete set of noise strengths with the current self-dual symplectic lattice construction. In this work, we develop a new coding strategy using concatenated continuous variable-discrete variable encodings to go beyond past results and establish GKP’s optimal rate over all noise strengths. In particular, for displacement noise, the rate is obtained through a constructive approach by concatenating GKP codes with a quantum polar code and analog decoding. For a pure-loss channel, we prove the existence of capacity-achieving GKP codes through a random coding approach. These results highlight the capability of concatenation-based GKP codes and provides new methods for constructing good GKP lattices.

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Popular Summary

Constructing quantum error-correcting codes that approach information-theoretic limits is essential for minimizing overhead in quantum communication. Gottesman-Kitaev-Preskill (GKP) codes are particularly promising for this purpose owing to their compatibility with bosonic systems. Similar to how random coding arguments were used to establish the existence of capacity-achieving classical codes, previous works have shown that multimode GKP codes can achieve optimal rates for bosonic channels with Gaussian displacement, pure-loss, or amplification noise. However, these results are limited to discrete noise strengths and rely on nonconstructive arguments, offering no explicit code constructions.

In this work, we overcome these limitations by developing explicit and scalable construction methods based on concatenated square GKP codes. Specifically, we show that single-mode square GKP qudits, when concatenated with discrete-variable stabilizer codes, can achieve optimal rates for bosonic channels with Gaussian displacement, pure-loss, or amplification noise over the full range of noise strengths. For Gaussian displacement noise, we provide a fully constructive proof using quantum polar codes, leveraging their classical analogs known for achieving capacity. For pure-loss and amplification channels, we employ random coding arguments but offer more structure by restricting to concatenated square GKP codes.

Our results demonstrate that single-mode GKP codes, when properly concatenated, can achieve theoretically optimal communication rates across a broad class of bosonic noise models, providing both conceptual clarity and practical pathways toward capacity-achieving quantum communication protocols.

Article Text

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PRX Quantum 6, 040342– Published 21 November, 2025
  • Received 28 May 2025
  • Revised 3 September 2025
  • Accepted 20 October 2025
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