Characterization and critical faults of leakage errors on the surface code

Leakage is a particularly damaging error that occurs when a qubit leaves the defined computational subspace. Leakage errors leave qubits ill-defined and limit the effectiveness of quantum error correcting codes. The effects of leakage errors on the surface code has been studied in various contexts. However, the effects of a leaked data qubit versus a leaked ancilla qubit can be quite different.
In this paper, we study the effects of data leakage and ancilla leakage separately. We show that data leakage is much less damaging. If leakage errors can be confined to data qubits, and efficiently removed with a minimum overhead leakage reducing circuit (LRC), then the surface code maintains the code distance in the presence of leakage. We also show that the distance damaging fault in the surface code comes from ancilla leakage at a particular point in the syndrome extraction circuit. Because of this critical fault location, if leakage errors are eliminated in this particular part of the circuit, the surface code can maintain its effective distance, regardless of other leakage errors that occur on either ancilla or data qubits. We simulate two physical realizations of these toy models that can be applied to both superconducting and ion trapped architectures.