Grover's search with local and total depolarizing channel errors

In this article the effect of noise on Grover's algorithm is analyzed, modeled as a total depolarizing channel (TDCh) and a local depolarizing channel in each qubit (LDCh). The focus was not in error correction (e.g. by the fault-tolerant method), but to provide an insight to the kind of error, or degradation, that needs to be corrected. In the last years analytical results regarding mainly the TDCh model have been obtained. In this paper we extend these previous results to the local case, concluding that the degradation of Grover's algorithm with the latter is worse than the former. It has been shown that for both cases with an \(N\)-dependent small enough error-width, smaller than \(1/\sqrt{N}\) for total error and \(1/(\sqrt{N}\log_2{N})\) for the local case, correction is not needed.