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.

中文翻译：

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用于 QAOA 的 LX 混频器：仅限于子空间和稳定器形式的最佳混频器


我们提出了一种新的形式主义来理解和构建保留给定子空间的混合器。该方法连接并利用纠错码中使用的稳定器形式。当量子近似优化算法 （QAOA）（一种用于解决组合优化问题的常用元启发式方法）应用于问题的约束导致较大但易于指定的可行子空间的设置时，这可能很有用。所提出的方法提供了一种系统的方法来构建混合器，这些混合器在受控非门的数量上是资源高效的，可以理解为众所周知的 X 和 XY 混合器的泛化和 Grover 混合器的松弛：给定任何子空间的基础，可以构建一个资源高效的混合器来保留子空间。提供的数值示例显示，与以前的结果相比，CX 门数显著减少。我们称我们的方法为逻辑 X-Mixer 或逻辑 X QAOA （$\textbf{LX-QAOA}$），因为它可以理解为将子空间划分为稳定器 S 的代码空间，并连续应用与这些代码空间相关的逻辑旋转 X 门。总的来说，我们希望这个新视角可以导致对量子算法的发展有更深入的了解。