Thresholds for Correcting Errors, Erasures, and Faulty Syndrome Measurements in Degenerate Quantum Codes

We suggest a technique for constructing lower (existence) bounds for the fault-tolerant threshold to scalable quantum computation applicable to degenerate quantum codes with sublinear distance scaling. We give explicit analytic expressions combining probabilities of erasures, depolarizing errors, and phenomenological syndrome measurement errors for quantum low-density parity-check codes with logarithmic or larger distances. These threshold estimates are parametrically better than the existing analytical bound based on percolation.

Figure 1

(a) Toric code generators. Plaquette $A_{\square }$ (shaded rounded square) and vertex $B_{+}$ (shaded diamonds) operators constructed as products of four Pauli $X$ and Pauli $Z$ operators, respectively. (b) A reducible cluster is counted as one or two clusters, depending on the order in which the numbered qubits are chosen.