约束QAOA的表达能力：你可能错过的内容

中文摘要 AI 辅助

保持可行性的混合器哈密顿量为约束量子优化提供了一种比惩罚编码更具吸引力的替代方案，然而，强制约束如何改变量子近似优化算法（QAOA）可达到的状态空间，在很大程度上仍未得到探索。我们在最大独立集（MIS）问题的背景下解决这个问题。除了标准的受控比特翻转混合器，我们还分析了备选的 \\(\mathit{Flip\text{-}or\text{-}Stay}\\) 混合器，其对可行子空间 \\(W_{\mathcal{F}}\\) 的限制恰好对应于独立集重构图的移位负拉普拉斯算子。尽管两种混合器在独立集之间诱导相同的跃迁，但对角修改改变了谱结构，并能显著扩大可达量子态集合。对于每个至少有两个顶点的连通输入图，我们证明了与任一混合器相关的独立参数化局部控制都能在可行子空间上生成完整的酉李代数。标准QAOA表现出更微妙的代数结构。我们证明了采用传统混合器的MIS-QAOA相关的动力学李代数嵌入到其Flip-or-Stay对应物中。此外，我们构造了一个无限图族，对于该图族，相应的真空态动力学群轨道表现出严格不同的维度，从而确立了两种架构之间的定性表达能力分离。对于任一混合器，我们证明了在空集初始化的标准QAOA可以在有限深度制备一个完全支持在最大独立集上的态。对于自由架构，我们还推导了酉设计极限下的精确损失方差公式，该公式通过可行独立集的数量及其基数的方差来表达。

英文摘要

Feasibility-preserving mixer Hamiltonians offer an attractive alternative to penalty encodings for constrained quantum optimization, yet how enforcing constraints alters the state space reachable by the Quantum Approximate Optimization Algorithm (QAOA) remains largely unexplored. We address this question within the context of the Maximum Independent Set (MIS) problem. Alongside the standard controlled bit-flip mixer, we analyze the alternative $\mathit{Flip\text{-}or\text{-}Stay}$ mixer, whose restriction to the feasible subspace $W_{\mathcal{F}}$ corresponds precisely to the shifted negative Laplacian of the independent-set reconfiguration graph. Although both mixers induce identical transitions between independent sets, the diagonal modification changes the spectral structure and can substantially enlarge the set of reachable quantum states. For every connected input graph with at least two vertices, we prove that independently parameterized local controls associated with either mixer generate the full unitary Lie algebra on the feasible subspace. Standard QAOA exhibits a more nuanced algebraic structure. We prove that the dynamical Lie algebra associated with MIS-QAOA employing the conventional mixer embeds into its Flip-or-Stay counterpart. Furthermore, we construct an infinite family of graphs for which the corresponding vacuum-state dynamical group orbits exhibit strictly distinct dimensions, establishing a qualitative expressivity separation between the two architectures. For either mixer, we show that standard QAOA initialized in the empty set can prepare a state supported entirely on maximum independent sets at finite depth. For the free architectures, we also derive an exact loss-variance formula in the unitary-design limit, expressed through the number of feasible independent sets and the variance of their cardinalities.

发表机构

• University of California, Riverside（加州大学河滨分校）
• University of Delaware（特拉华大学）

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