Performance and achievable rates of the Gottesman-Kitaev-Preskill code for pure-loss and amplification channels

TL;DR

This paper analytically evaluates the near-optimal performance and achievable rates of Gottesman-Kitaev-Preskill (GKP) codes for pure-loss and amplification channels, establishing them as the first structured bosonic codes to reach these capacities.

Contribution

It provides a rigorous analytical framework linking GKP code performance to lattice geometry and input energy, demonstrating capacity achievement under certain conditions.

Findings

GKP codes achieve near-optimal performance under pure loss and amplification.

Specific GKP code families reach capacity when transmissivity-related parameter is an integer.

GKP codes are the first structured bosonic codes to attain loss and amplification channel capacities.

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 $\vert\frac{\tau}{1 - \tau}\vert$ is an integer, specific families of GKP codes simultaneously achieve the loss and amplification capacity. $\tau$ 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.