  We present a novel formalism to both understand and construct mixers that
preserve a given subspace. The method connects and utilizes the stabilizer
formalism that is used in error correcting codes. This can be useful in the
setting when the quantum approximate optimization algorithm (QAOA), a popular
meta-heuristic for solving combinatorial optimization problems, is applied in
the setting where the constraints of the problem lead to a feasible subspace
that is large but easy to specify. The proposed method gives a systematic way
to construct mixers that are resource efficient in the number of controlled not
gates and can be understood as a generalization of the well-known X and XY
mixers and a relaxation of the Grover mixer: Given a basis of any subspace, a
resource efficient mixer can be constructed that preserves the subspace. The
numerical examples provided show a dramatic reduction of CX gates when compared
to previous results. We call our approach logical X-Mixer or logical X QAOA
($\textbf{LX-QAOA}$), since it can be understood as dividing the subspace into
code spaces of stabilizers S and consecutively applying logical rotational X
gates associated with these code spaces. Overall, we hope that this new
perspective can lead to further insight into the development of quantum
algorithms.
