  The Pauli stabilizer formalism is perhaps the most thoroughly studied means
of procuring quantum error-correcting codes, whereby the code is obtained
through commutative Pauli operators and ``stabilized'' by them. In this work we
will show that every quantum error-correcting code, including Pauli stabilizer
codes and subsystem codes, has a similar structure, in that the code can be
stabilized by commutative ``Paulian'' operators which share many features with
Pauli operators and which form a \textbf{Paulian stabilizer group}. By
facilitating a controlled gate we can measure these Paulian operators to
acquire the error syndrome. Examples concerning codeword stabilized codes and
bosonic codes will be presented; specifically, one of the examples has been
demonstrated experimentally and the observable for detecting the error turns
out to be Paulian, thereby showing the potential utility of this approach. This
work provides a possible approach to implement error-correcting codes and to
find new codes.
