Leakage Suppression in the Toric Code

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 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 can increase overhead and experimental complexity, and these costs have not been thoroughly understood. Our work explores a variety of techniques for leakage-resilient, fault-tolerant error correction in the context of topological codes. Our contributions are threefold. First, we develop a leakage model that differs in critical details from earlier models. Second, we use Monte-Carlo simulations to survey several syndrome extraction circuits. Third, given the capability to perform three-outcome measurements, we present a dramatically improved syndrome processing algorithm. Our simulation results show that simple circuits with one extra CNOT per qubit 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 x 10^-4, placing LRUs after every gate may achieve the lowest logical error rates of all of the circuits we considered. We expect the closely related planar and rotated codes to exhibit the same accuracy thresholds and that the ideas may generalize naturally to other topological codes.