Magic State Injection with Erasure Qubits

Erasure qubits constitute a promising approach for tackling the daunting resources required for fault-tolerant quantum computing. By heralding erasure errors, both the error-correction threshold and the sub-threshold scaling of the logical error rate are significantly improved. While previous research has focused primarily on fault-tolerant quantum memories, we extend this investigation to magic state injection--a critical yet resource-intensive component of fault-tolerant quantum computation. We show that, after postselecting on erasures, the logical error rate of the injected magic state is set by the residual Pauli error, while the space-time overhead is only marginally increased as compared to non-erasure qubits with a similar noise strength. These conclusions hold both for injection into the surface code, and for injection and cultivation on the color code. For the former, we show that most of the gains can be achieved by using just three strategically placed erasure qubits in the surface code patch, independent of the patch size. For the latter, in contrast, it is beneficial to have all the qubits in the cultivation patch be erasure qubits. Our results for cultivation suggest that algorithmically relevant logical error rates may be within reach without magic state distillation for erasure rates $\lesssim 4\times 10^{-3}$ and residual Pauli error rates $\sim 10^{-4}$.