We study fundamental limitations of the generic Quantum Approximate Optimization Algorithm (QAOA) on constrained problems where valid solutions form a low dimensional manifold inside the Boolean hypercube, and we present a provable route to exponential improvements via constraint embedding. Focusing on permutation constrained objectives, we show that the standard generic QAOA ansatz, with a transverse field mixer and diagonal r local cost, faces an intrinsic feasibility bottleneck: even after angle optimization, circuits whose depth grows at most sublinearly with n cannot raise the total probability mass on the feasible manifold much above the uniform baseline suppressed by the size of the full Hilber space. Against this envelope we introduce a minimal constraint enhanced kernel (CE QAOA) that operates directly inside a product one hot subspace and mixes with a block local XY Hamiltonian. For permutation constrained problems, we prove an angle robust, depth matched exponential enhancement where the ratio between the feasible mass from CE QAOA and generic QAOA grows exponentially in $n^2$ for all depths up to a linear fraction of n, under a mild polynomial growth condition on the interaction hypergraph. Thanks to the problem algorithm co design in the kernel construction, the techniques and guarantees extend beyond permutations to a broad class of NP-Hard constrained optimization problems.

翻译：我们研究了通用量子近似优化算法（QAOA）在约束问题上的根本局限性，其中有效解形成布尔超立方体内的低维流形，并提出了一种通过约束嵌入实现指数级提升的可证明路径。聚焦于置换约束目标函数，我们证明标准的通用QAOA拟设（采用横向场混合器和对角r局域代价函数）存在固有的可行性瓶颈：即使经过角度优化，深度随n次线性增长的电路也无法将可行流形上的总概率质量提升到超越均匀基线的水平（该基线受完整希尔伯特空间规模抑制）。针对这一限制，我们引入了一种最小化约束增强核（CE-QAOA），该算法直接在乘积单热子空间内运行，并与块局域XY哈密顿量进行混合。对于置换约束问题，我们证明在相互作用超图满足温和多项式增长条件下，CE-QAOA与通用QAOA的可行质量比在深度达到n的线性分数时随$n^2$呈指数增长，且该增强具有角度鲁棒性和深度匹配特性。得益于核构造中的问题-算法协同设计，该技术与保证可超越置换问题，推广至一大类NP-难约束优化问题。