Continuous-Variable Quantum MacWilliams Identities

TL;DR

This paper introduces a continuous-variable quantum MacWilliams identities and derives bounds on quantum error correcting codes against displacement noise, showing GKP codes based on certain lattices achieve optimal distances.

Contribution

It presents a novel continuous-variable quantum MacWilliams identities and applies them to establish bounds and optimality results for quantum error correction codes.

Findings

Derived bounds on quantum error correcting codes against displacement noise

Introduced a continuous-variable quantum MacWilliams identities

Showed GKP codes based on E8 and Leech lattices are optimal

Abstract

We derive bounds on general quantum error correcting codes against the displacement noise channel. The bounds limit the distances attainable by codes and also apply in an approximate setting. Our main result is a quantum analogue of the classical Cohn-Elkies bound on sphere packing densities attainable in Euclidean space. We further derive a quantum version of Levenshtein's sphere packing bound and argue that Gottesman--Kitaev--Preskill (GKP) codes based on the $E_8$ and Leech lattices achieve optimal distances. The main technical tool is a continuous-variable version of the quantum MacWilliams identities, which we introduce. The identities relate a pair of weight distributions which can be obtained for any two trace-class operators. General properties of these weight distributions are discussed, along with several examples.