# Fidelity Lower Bounds for Stabilizer and CSS Quantum Codes

## TL;DR

This paper provides bounds on the fidelity of stabilizer and CSS quantum codes, analyzing their performance and asymptotic behavior to guide code design for desired fidelity levels.

## Contribution

It introduces new lower bounds on fidelity using quantum enumerators, compares stabilizer and CSS codes, and explores asymptotic performance limits.

## Key findings

- Stabilizer codes outperform CSS codes in fidelity and rate in symmetric depolarizing channels.
- Expurgated ensembles improve fidelity bounds over basic ensembles.
- Asymptotic analysis shows CSS codes have fundamental performance limitations compared to stabilizer codes.

## Abstract

In this paper we estimate the fidelity of stabilizer and CSS codes. First, we derive a lower bound on the fidelity of a stabilizer code via its quantum enumerator. Next, we find the average quantum enumerators of the ensembles of finite length stabilizer and CSS codes. We use the average quantum enumerators for obtaining lower bounds on the average fidelity of these ensembles. We further improve the fidelity bounds by estimating the quantum enumerators of expurgated ensembles of stabilizer and CSS codes. Finally, we derive fidelity bounds in the asymptotic regime when the code length tends to infinity. These results tell us which code rate we can afford for achieving a target fidelity with codes of a given length. The results also show that in symmetric depolarizing channel a typical stabilizer code has better performance, in terms of fidelity and code rate, compared with a typical CSS codes, and that balanced CSS codes significantly outperform other CSS codes. Asymptotic results demonstrate that CSS codes have a fundamental performance loss compared to stabilizer codes.