Computation on an error-avoiding quantum code and symmetrization

Let $\mathcal{H}$ be the state space of a quantum computer coupled with the environment by a set of error operators spanning a Lie algebra $\mathcal{L}.$ Suppose $\mathcal{L}$ admits an error-avoiding quantum code, i.e., a subspace $\mathcal{C}\subset \mathcal{H}$ annihilated by $\mathcal{L}.$ We show that a universal set of gates over $\mathcal{C}$ is obtained by any generic pair of $\mathcal{L}$-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 $\mathcal{L}.$ Any computation can then be performed completely within the coding decoherence-free subspace.