Error correction for mutually interacting qubits

For a simple model of mutually interacting qubits it is shown how the errors induced by mutual interactions can be eliminated using concatenated coding. The model is solved exactly for arbitrary interaction strength, for two well-known codes, one and two levels deep: this allows one to see under which circumstances error amplitudes add coherently or incoherently. For deeper concatenation, approximate results are derived that make it possible to calculate an approximate “threshold” value for the product of interaction strength and free evolution time, below which the failure probability for an encoded qubit decreases exponentially with the depth of the encoding. The results suggest that concatenated coding could fully handle the errors arising from mutual interactions, at no extra cost, in terms of resources needed, than what would be required to deal with random environmental errors.