Errors in quantum computers are of two kinds: sudden perturbations to isolated qubits, and slow random drifts of all the qubits. The latter may be reduced, but not eliminated, by means of symmetrization, namely by using many replicas of the computer, and forcing their joint quantum state to be completely symmetric. On the other hand, isolated errors can be corrected by quantum codewords that represent a logical qubit in a redundant way, by several physical qubits. If one of the physical qubits is perturbed, for example if it gets entangled with an unknown environment, there still is enough information encoded in the other physical qubits to restore the logical qubit, and disentangle it from the environment. The recovery procedure may consist of unitary operations, without the need of actually identifying the error.
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 $\mathcal{C}\ensuremath{\subset}\mathcal{H}$ 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.