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The standard stabilizer formalism provides a setting to show that quantumcomputation restricted to operations within the Clifford group are classicallyefficiently simulable: this is the content of the well-known Gottesman-Knilltheorem. This work analyzes the mathematical structure behind this theorem tofind possible generalizations and derivation of constraints required forconstructing a non-trivial generalized Clifford group. We prove that if theclosure of the stabilizing set is dense in the set of $SU(d)$ transformations,then the associated Clifford group is trivial, consisting only of local gatesand permutations of subsystems. This result demonstrates the close relationshipbetween the density of the stabilizing set and the simplicity of thecorresponding Clifford group. We apply the analysis to investigatestabilization with binary observables for qubits and find that the formalism isequivalent to the standard stabilization for a low number of qubits. Based onthe observed patterns, we conjecture that a large class of generalizedstabilizer states are equivalent to the standard ones. Our results can be usedto construct novel Gottesman-Knill-type results and consequently draw a sharperline between quantum and classical computation.

On The Stabilizer Formalism And Its Generalization