# Leakage mitigation for quantum error correction using a mixed qubit scheme

## TL;DR

This paper investigates leakage errors in ion trap qubits within quantum error correction, comparing hyperfine and Zeeman qubits under realistic noise models, and proposes a mixed qubit scheme to improve leakage mitigation.

## Contribution

It introduces a mixed qubit scheme combining hyperfine and Zeeman qubits to effectively mitigate leakage errors without extra circuits under low magnetic field conditions.

## Key findings

- Zeeman qubits outperform hyperfine qubits at magnetic field noise below 32 μG.
- Hyperfine qubits perform better at magnetic field noise below 10 μG with Molmer-Sorenson gates.
- The mixed qubit scheme reduces leakage errors without additional leakage reduction circuits.

## Abstract

Leakage errors take qubits out of the computational subspace and will accumulate if not addressed. A leaked qubit will reduce the effectiveness of quantum error correction protocols due to the cost of implementing leakage reduction circuits and the harm caused by interacting leaked states with qubit states. Ion trap qubits driven by Raman gates have a natural choice between qubits encoded in magnetically insensitive hyperfine states that can leak and qubits encoded in magnetically sensitive Zeeman states of the electron spin that cannot leak. In our previous work, we compared these two qubits in the context of the toric code with a depolarizing leakage error model and found that for magnetic field noise with a standard deviation less than 32 $\mu$G that the $^{174}$Yb$^+$ Zeeman qubit outperforms the $^{171}$Yb$^+$ hyperfine qubit. Here we examine a physically motivated leakage error model based on ions interacting via the Molmer-Sorenson gate. We find that this greatly improves the performance of hyperfine qubits but the Zeeman qubits are more effective for magnetic field noise with a standard deviation less than 10 $\mu$G. At these low magnetic fields, we find that the best choice is a mixed qubit scheme where the hyperfine qubits are the ancilla and the leakage is handled without the need of an additional leakage reduction circuit.