Achievable rates for concatenated square Gottesman-Kitaev-Preskill codes

TL;DR

This paper develops a new concatenated coding strategy for GKP codes that achieves optimal communication rates over all noise strengths in displacement noise and pure loss channels, surpassing previous limitations.

Contribution

It introduces a concatenation-based approach combining GKP codes with quantum polar codes and random coding, enabling capacity achievement across all noise levels.

Findings

Achieves GKP code capacity over all noise strengths.

Constructs capacity-achieving GKP codes via concatenation and random coding.

Demonstrates the effectiveness of concatenation for optimal quantum communication.

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 strength 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 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.