Leakage Mitigation Using Quantum Error Correction in a Mixed Ion Scheme

Mixed species surface code layout. Hyperfine (171Yb+) ions are defined as ancilla qubits (white) and Zeeman (174Yb+) ions are defined as data qubits (black). The green (blue) diamond represents an X (Z) stabilizer measurement.
Leakage errors are exceptionally damaging errors that occur when a qubit leaves the defined computational space. If left untreated, leakage errors will accumulate, corrupting data and rendering error syndromes useless. Leakage errors are typically handled by implementing leakage reducing circuits (LRCs) which convert leakage errors into Pauli errors at the cost of additional overhead. 
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. Our previous work compared these two qubits in the context of the toric code and found certain magnetic field regimes in which the Zeeman qubits outperforms the hyperfine qubit. 
In this work, we mixed the two different qubits and analyzed the performance of the toric code using two different leakage models: the standard depolarizing model and a physically motivated leakage error model based on ions interacting via the Mølmer-Sørenson gate. We find that this greatly improves the performance of hyperfine qubits but the Zeeman qubits are more effective for magnetic field noise. 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 LRCs. For more details see our preprint on arXiv:1904.10724.