QUANTUM LOGICAL OPERATIONS ON ENCODED QUBITS

We show how to carry out quantum logical operations (controlled-not an d Toffoli gates) on encoded qubits for several encodings which protect against various 1-bit errors. This improves the reliability of these operations by allowing one to correct for 1-bit errors which either pr eexisted or occurred in the course of operation. The logical operation s we consider allow one to carry out the vast majority of the steps in the quantum factoring algorithm.