Quantum codes excel at correcting local noise but fail to correct leakage faults that excite qubits to states outside the computational space. Aliferis and Terhal \cite{aliferis07} have shown that an accuracy threshold exists for leakage faults using gadgets called leakage reduction units (LRUs). However, these gadgets reduce the accuracy threshold and increase overhead and experimental complexity, and these costs have not been thoroughly understood. We explore a variety of techniques for leakage-resilient, fault-tolerant error correction in topological codes. Our contributions are threefold. First, we develop a leakage model that is physically motivated and efficient to simulate. Second, we use Monte-Carlo simulations to survey several syndrome extraction circuits. Third, given the capability to perform 3-outcome measurements, we present a dramatically improved syndrome processing algorithm. Our simulations show that simple circuits with one extra CNOT per check operator and no additional ancillas reduce the accuracy threshold by less than a factor of $4$ when leakage and depolarizing noise rates are comparable. This becomes a factor of $2$ when the decoder uses 3-outcome measurements. Finally, when the physical error rate is less than $2\times 10^{-4}$, placing LRUs after every gate may achieve the lowest logical error rates of all of the circuits we considered. We anticipate that the closely related planar codes might exhibit the same accuracy thresholds and that the ideas may generalize naturally to other topological codes.
Fault-tolerant weighted union-find decoding on the toric code