Computation on an error-avoiding quantum code and symmetrization

Let H be the state space of a quantum computer coupled with the environment by a set of error operators spanning a Lie algebra L. Suppose L admits an error-avoiding quantum code, i.e., a subspace C⊂H annihilated by L. We show that a universal set of gates over C is obtained by any generic pair of L-invariant gates. Such gates—if not available from the outset—can be obtained by resorting to a symmetrization with respect to the group generated by L. Any computation can then be performed completely within the coding decoherence-free subspace.