Conditional generation of error syndromes in fault-tolerant error correction

In this paper we show how the fault-tolerant quantum error-correction scheme recently proposed by DiVincenzo and Shor Phys. Rev. Lett. 77, 3260 (1996) can be made more efficient. Our scheme, unlike the earlier one, can deal with a single error between the quantum gate operations that are required for the implementation of the quantum error correction without repeated generation of error syndromes. For errors during gate operations we have to produce fewer error syndromes as we generate them conditionally on the previous one. We show how to generalize the method to arbitrary error-correcting codes.