Computation on a Noiseless Quantum Code and Symmetrization

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