Quantum Weight Enumerators

TL;DR

This paper introduces two new quantum error-correcting code enumerators connected by a simple duality, providing clearer insights into code properties and extending enumerator theory to larger block sizes.

Contribution

The paper presents novel quantum code enumerators with a simpler duality transform, enhancing understanding of code properties and extending the theory beyond block size two.

Findings

New enumerators connected by a simple duality transform

Simpler conditions for quantum code minimum distance

Extension of enumerator theory to larger block sizes

Abstract

In a recent paper ([quant-ph/9610040]), Shor and Laflamme define two ``weight enumerators'' for quantum error correcting codes, connected by a MacWilliams transform, and use them to give a linear-programming bound for quantum codes. We introduce two new enumerators which, while much less powerful at producing bounds, are useful tools nonetheless. The new enumerators are connected by a much simpler duality transform, clarifying the duality between Shor and Laflamme's enumerators. We also use the new enumerators to give a simpler condition for a quantum code to have specified minimum distance, and to extend the enumerator theory to codes with block-size greater than 2.