  The standard stabilizer formalism provides a setting to show that quantum
computation restricted to operations within the Clifford group are classically
efficiently simulable: this is the content of the well-known Gottesman-Knill
theorem. This work analyzes the mathematical structure behind this theorem to
find possible generalizations and derivation of constraints required for
constructing a non-trivial generalized Clifford group. We prove that if the
closure of the stabilizing set is dense in the set of $SU(d)$ transformations,
then the associated Clifford group is trivial, consisting only of local gates
and permutations of subsystems. This result demonstrates the close relationship
between the density of the stabilizing set and the simplicity of the
corresponding Clifford group. We apply the analysis to investigate
stabilization with binary observables for qubits and find that the formalism is
equivalent to the standard stabilization for a low number of qubits. Based on
the observed patterns, we conjecture that a large class of generalized
stabilizer states are equivalent to the standard ones. Our results can be used
to construct novel Gottesman-Knill-type results and consequently draw a sharper
line between quantum and classical computation.
