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Performance and Achievable Rates of the Gottesman-Kitaev-Preskill Code for Pure-Loss and Amplification Channels

Guo Zheng1,*,†, Wenhao He2,*, Gideon Lee1, Kyungjoo Noh3, and Liang Jiang1,‡

  • *These authors contributed equally.
  • †Contact author: guozheng@uchicago.edu
  • ‡Contact author: liang.jiang@uchicago.edu
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PRX Quantum 6, 030314 – Published 28 July, 2025

DOI: https://doi.org/10.1103/gh1c-xyn1

Abstract

Quantum error-correction codes protect information from realistic noisy channels and lie at the heart of quantum computation and communication tasks. Understanding the optimal performance and other information-theoretic properties, such as the achievable rates, of a given code is crucial, as these factors determine the fundamental limits imposed by the encoding in conjunction with the noise channel. Here, we use the transpose channel to analytically obtain the near-optimal performance of any Gottesman-Kitaev-Preskill (GKP) code under pure loss and pure amplification. We present rigorous connections between GKP code’s near-optimal performance and its dual lattice geometry and average input energy. With no energy constraint, we show that when |τ/(1−τ)| is an integer, specific families of GKP codes simultaneously achieve the loss and amplification capacity. τ is the transmissivity (gain) for loss (amplification). Our results establish GKP code as the first structured bosonic code family that achieves the capacity of loss and amplification.

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Physics Subject Headings (PhySH)

Popular Summary

Quantum communication systems must overcome the challenge of noise to transmit information reliably. A central goal in quantum error correction is to design codes that protect information effectively under practical noise conditions. In this work, we study the Gottesman-Kitaev-Preskill (GKP) code—a leading candidate for error correction in continuous-variable systems—and show that it can achieve the maximum possible rate of information transmission (the channel capacity) under two important and practical noise models: energy loss and amplification.

Our approach builds on a recently developed performance metric called near-optimal fidelity, which allows us to evaluate the performance of GKP codes analytically rather than numerically. By combining this tool with insights from lattice geometry, we find a precise connection between the lattice structure of the GKP code and its ability to correct errors. We show that under certain conditions, GKP codes built from specific lattices can match the ultimate theoretical limits of quantum communication, even without placing restrictions on energy. Remarkably, our analysis establishes GKP codes as the first structured code family with proven optimality under such a practical noise model.

Looking ahead, these results pave the way for better design and evaluation of quantum codes in both theory and experiment. The framework we introduce could be used to benchmark other continuous-variable codes and potentially guide the development of hardware-efficient quantum error correction for next-generation quantum networks and processors.

Article Text

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PRX Quantum 6, 030314– Published 28 July, 2025
  • Received 11 February 2025
  • Revised 15 May 2025
  • Accepted 1 July 2025
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