LX-mixers for QAOA: Optimal mixers restricted to subspaces and the stabilizer formalism

TL;DR

This paper introduces LX-mixers for QAOA, utilizing stabilizer formalism to efficiently construct mixers that preserve specific subspaces, significantly reducing resource requirements for constrained optimization problems.

Contribution

It presents a systematic method to design resource-efficient mixers for QAOA based on stabilizer codes, generalizing known mixers and reducing gate complexity.

Findings

Significant reduction in CX gates compared to previous methods

Systematic construction of mixers for arbitrary subspaces

Enhanced understanding of mixer design via stabilizer formalism

Abstract

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.