Exploring the quantum capacity of a Gaussian random displacement channel using Gottesman-Kitaev-Preskill codes and maximum likelihood decoding

TL;DR

This paper investigates the quantum capacity of Gaussian random displacement channels by analyzing multi-mode GKP codes with maximum likelihood decoding, showing they can transmit quantum information at higher noise levels than previously known.

Contribution

It demonstrates that GKP codes with advanced decoding techniques can achieve non-zero quantum capacity at noise levels where capacity bounds vanish, narrowing the gap between theory and practical codes.

Findings

Surface-square GKP codes have an error threshold near σ=0.6065.

Color-hexagonal GKP codes perform well up to code distance 13.

GKP codes can transmit quantum information at higher noise levels than prior results.

Abstract

Determining the quantum capacity of a noisy quantum channel is an important problem in the field of quantum communication theory. In this work, we consider the Gaussian random displacement channel $N_{\sigma}$, a type of bosonic Gaussian channels relevant in various bosonic quantum information processing systems. In particular, we attempt to make progress on the problem of determining the quantum capacity of a Gaussian random displacement channel by analyzing the error-correction performance of several families of multi-mode Gottesman-Kitaev-Preskill (GKP) codes. In doing so we analyze the surface-square GKP codes using an efficient and exact maximum likelihood decoder (MLD) up to a large code distance of $d=39$. We find that the error threshold of the surface-square GKP code is remarkably close to $\sigma=1/\sqrt{e}\simeq 0.6065$ at which the best-known lower bound of the quantum capacity of $N_{\sigma}$ vanishes. We also analyze the performance of color-hexagonal GKP codes up to a code distance of $d=13$ using a tensor-network decoder serving as an approximate MLD. By focusing on multi-mode GKP codes that encode just one logical qubit over multiple bosonic modes, we show that GKP codes can achieve non-zero quantum state transmission rates for a Gaussian random displacement channel $N_{\sigma}$ at larger values of $\sigma$ than previously demonstrated. Thus our work reduces the gap between the quantum communication theoretic bounds and the performance of explicit bosonic quantum error-correcting codes in regards to the quantum capacity of a Gaussian random displacement channel.