CSS-$T$ codes over Binary Extension Fields and their Physical Foundations

TL;DR

This paper explores CSS-$T$ quantum error-correcting codes over binary extension fields, extending their definitions and demonstrating the existence of good LDPC code sequences with transversal $T$-gates.

Contribution

It extends the definition of CSS-$T$ codes over binary extension fields and proves the existence of asymptotically good LDPC CSS-$T$ codes over these fields.

Findings

Existence of asymptotically good LDPC CSS-$T$ codes over binary extension fields

Extension of CSS-$T$ code definitions to $q$-ary fields

Demonstration of transversal $T$-gate compatibility

Abstract

We investigate the class of CSS-$T$ codes, a family of quantum error-correcting codes that allows for a transversal $T$-gate. We extend the definition of a pair of linear codes $(C_1,C_2)$, $C_i\subseteq\mathbb{F}_q^n$, forming a $q$-ary CSS-$T$ code over binary extension fields, and demonstrate the existence of asymptotically good sequences of LDPC CSS-$T$ codes over any such field.