Good quantum error-correcting codes exist.

With the realization that computers that use the interference and superposition principles of quantum mechanics might be able to solve certain problems, including prime factorization, exponentially faster than classical computers @1#, interest has been growing in the feasibility of these quantum computers, and several methods for building quantum gates and quantum computers have been proposed @2,3#. One of the most cogent arguments against the feasibility of quantum computation appears to be the difficulty of eliminating error caused by inaccuracy and decoherence @4#. Whereas the best experimental implementations of quantum gates accomplished so far have less than 90% accuracy @5#, the accuracy required for factorization of numbers large enough to be difficult on conventional computers appears to be closer to one part in billions. We hope that the techniques investigated in this paper can eventually be extended so as to reduce this quantity by several orders of magnitude. In the storage and transmission of digital data, errors can be corrected by using error-correcting codes @6#. In digital computation, errors can be corrected by using redundancy; in fact, it has been shown that fairly unreliable gates could be assembled to form a reliable computer @7#. It has widely been assumed that the quantum no-cloning theorem @8# makes error correction impossible in quantum communication and computation because redundancy cannot be obtained by duplicating quantum bits. This argument was shown to be in error for quantum communication in Ref. @9#, where a code was given that mapped one qubit ~two-state quantum system! into nine qubits so that the original qubit could be recovered perfectly even after arbitrary decoherence of any one of these nine qubits. This gives a quantum code on nine qubits with a rate 1