Noiseless Subsystems and the Structure of the Commutant in Quantum Error Correction

John A. Holbrook, David W. Kribs and Raymond Laflamme Department of Mathematics and Statistics, University of Guelph, Guelph, Ontario, Canada N1G 2W1. Institute for Quantum Computing, University of Waterloo, Waterloo, ON, CANADA N2L 3G1. Perimeter Institute for Theoretical Physics, 35 King St. North, Waterloo, ON, CANADA N2J 2W9.∗ Abstract The effect of noise on a quantum system can be described by a set of operators obtained from the interaction Hamiltonian. Recently it has been shown that generalized quantum error correcting codes can be derived by studying the algebra of this set of operators. This led to the discovery of noiseless subsystems. They are described by a set of operators obtained from the commutant of the noise generators. In this paper we derive a general method to compute the structure of this commutant in the case of unital noise.