Operator Algebraic Formulation of the Stabilizer Formalism for Quantum Error Correction

Abstract

We give an operator algebraic formulation of the stabilizer formalism for error correction in quantum computing. The approach relies on an analysis of commutant structures, and gives a natural extension of the classic stabilizer formalism to the general case of arbitrary (not necessarily abelian) Pauli subgroups and subsystem codes. We show how to identify the largest stabilizer subsystem for every Pauli subgroup and discuss examples.

Quantum state reduction: Generalized bipartitions from algebras of observables

“…Noiseless subsystems, for example, are identified by the generalized bipartition associated with the commutant algebra generated by the errors [35]. Subsystem codes [49,51,52], which generalize the idea of noiseless subsystems, are identified by a generalized bipartition usually associated with a non-abelian group (which also generalizes the construction of stabilizer codes that are associated with abelian groups [53,54]). Similarly, the idea of quantum state compression with respect to a preferred set of observables [55] relies on the generalized bipartition associated with the algebra of preferred observables; it is conceptually equivalent to the notion of quantum state reductions from a restricted algebra of observables that we discussed in Section 3.2.…”