# Optimal quantum subsystem codes in 2-dimensions

## TL;DR

This paper presents a method to construct optimal 2D quantum subsystem codes from classical codes, achieving parameters that saturate known bounds and enabling efficient decoding, with potential for threshold behavior through gauge fixing.

## Contribution

It introduces a construction of 2D quantum subsystem codes from classical codes that saturate Bravyi's bound and can be efficiently decoded using classical algorithms.

## Key findings

- Constructed quantum codes with parameters $[[N,K,D]]$ satisfying $N o 2n_1n_2$, $K=k$, $D= ext{min}(d_1,d_2)$.
- Utilized classical expander codes to build quantum codes with optimal parameters.
- Provided a method to gauge fix subsystem codes to hypergraph product codes with asymptotic thresholds.

## Abstract

Given any two classical codes with parameters $[n_1,k,d_1]$ and $[n_2,k,d_2]$, we show how to construct a quantum subsystem code in 2-dimensions with parameters $[[N,K,D]]$ satisfying $N\le 2n_1n_2$, $K=k$, and $D=\min(d_1,d_2)$. These quantum codes are in the class of generalized Bacon-Shor codes introduced by Bravyi. We note that constructions of good classical codes can be used to construct quantum codes that saturate Bravyi's bound $KD=O(N)$ on the code parameters of 2-dimensional subsystem codes. One of these good constructions uses classical expander codes. This construction has the additional advantage of a linear time quantum decoder based on the classical Sipser-Spielman flip decoder. Finally, while the subsystem codes we create do not have asymptotic thresholds, we show how they can be gauge-fixed to certain hypergraph product codes that do.