Quantum Weight Enumerators for Real Codes with $X$ and $Z$ Exactly Transversal

TL;DR

This paper establishes that real quantum error-correcting codes with exact transversal $X$ and $Z$ operations have specific weight enumerator identities, linking transversality to error correction capabilities and code distance.

Contribution

It introduces new identities for weight enumerators of such codes, revealing that error detection implies error correction, thus connecting transversality with code robustness.

Findings

Error detecting codes are automatically error correcting.

Weight enumerator identities constrain code properties.

Transversality relates directly to code distance.

Abstract

In this note we show that the weight enumerators of a real quantum error correcting code with $ X $ and $ Z $ exactly transversal must satisfy certain identities. One consequence of these identities is that if the code is error detecting then it is automatically error correcting for free; implying a relationship between transversality and code distance.