# Handling Leakage with Subsystem Codes

## TL;DR

This paper demonstrates that subsystem codes, especially subsystem surface codes and Bacon-Shor codes, are effective at handling leakage errors in quantum computing, outperforming traditional codes under certain low-error conditions.

## Contribution

The study provides a comparative analysis showing subsystem codes' superior performance against leakage errors, highlighting their intrinsic locality advantage in quantum error correction.

## Key findings

- Subsystem surface codes outperform subspace surface codes below certain error thresholds.
- Bacon-Shor codes offer better leakage protection at low to intermediate distances.
- Relaxing extraction methods extends the advantage of subsystem codes to higher distances.

## Abstract

Leakage is a particularly damaging error that occurs when a qubit state falls out of its two-level computational subspace. Compared to independent depolarizing noise, leaked qubits may produce many more configurations of harmful correlated errors during error-correction. In this work, we investigate different local codes in the low-error regime of a leakage gate error model. When restricting to bare-ancilla extraction, we observe that subsystem codes are good candidates for handling leakage, as their locality can limit damaging correlated errors. As a case study, we compare subspace surface codes to the subsystem surface codes introduced by Bravyi et al. In contrast to depolarizing noise, subsystem surface codes outperform same-distance subspace surface codes below error rates as high as $\lessapprox 7.5 \times 10^{-4}$ while offering better per-qubit distance protection. Furthermore, we show that at low to intermediate distances, Bacon-Shor codes offer better per-qubit error protection against leakage in an ion-trap motivated error model below error rates as high as $\lessapprox 1.2 \times 10^{-3}$. For restricted leakage models, this advantage can be extended to higher distances by relaxing to unverified two-qubit cat state extraction in the surface code. These results highlight an intrinsic benefit of subsystem code locality to error-corrective performance.