Concatenated Quantum Codes
  One of the main problems for the future of practical quantum computing is to stabilize the computation against unwanted interactions with the environment and imperfections in the applied operations. Existing proposals for quantum memories and quantum channels require gates with asymptotically zero error to store or transmit an input quantum state for arbitrarily long times or distances with fixed error. In this report a method is given which has the property that to store or transmit a qubit with maximum error $\epsilon$ requires gates with error at most $c\epsilon$ and storage or channel elements with error at most $\epsilon$, independent of how long we wish to store the state or how far we wish to transmit it. The method relies on using concatenated quantum codes with hierarchically implemented recovery operations. The overhead of the method is polynomial in the time of storage or the distance of the transmission. Rigorous and heuristic lower bounds for the constant $c$ are given. 
Emanuel Knill and Raymond Laflamme