We introduce the Constraint-Enhanced Quantum Approximate Optimization Algorithm (CE-QAOA), a shallow, constraint-aware ansatz that operates inside the one-hot product space of size [n]^m, where m is the number of blocks and each block is initialized with an n-qubit W_n state. We give an ancilla-free, depth-optimal encoder that prepares a W_n state using n-1 two-qubit rotations per block, and a two-local XY mixer restricted to the same block of n qubits with a constant spectral gap. Algorithmically, we wrap constant-depth sampling with a deterministic classical checker to obtain a polynomial-time hybrid quantum-classical solver (PHQC) that returns the best observed feasible solution in O(S n^2) time, where S is the number of shots. We obtain two advantages. First, when CE-QAOA fixes r >= 1 locations different from the start city, we achieve a Theta(n^r) reduction in shot complexity even against a classical sampler that draws uniformly from the feasible set. Second, against a classical baseline restricted to raw bitstring sampling, we show an exp(Theta(n^2)) separation in the minimax sense. In noiseless circuit simulations of TSP instances ranging from 4 to 10 locations from the QOPTLib benchmark library, we recover the global optimum at depth p = 1 using polynomial shot budgets and coarse parameter grids defined by the problem sizes.

翻译：本文提出约束增强量子近似优化算法（CE-QAOA），这是一种浅层、约束感知的拟设，在规模为[n]^m的单热积空间内操作，其中m为区块数量，每个区块初始化为n量子比特的W_n态。我们给出一种无需辅助量子比特、深度最优的编码器，每个区块使用n-1个双量子比特旋转制备W_n态，并采用限制在同一n量子比特区块内的双局域XY混合器，该混合器具有恒定谱隙。算法层面，我们将恒定深度采样与确定性经典验证器结合，构建多项式时间混合量子-经典求解器（PHQC），该求解器在O(S n^2)时间内返回观测到的最佳可行解，其中S为采样次数。我们获得两项优势：首先，当CE-QAOA固定r≥1个不同于起始城市的位置时，即使相较于从可行集均匀采样的经典采样器，我们实现了采样复杂度Theta(n^r)的降低；其次，相较于仅限于原始比特串采样的经典基线，我们在极小极大意义上展示了exp(Theta(n^2))的分离优势。在基于QOPTLib基准库中4至10个位置的TSP实例的无噪声电路模拟中，我们使用多项式采样预算和由问题规模定义的粗粒度参数网格，在深度p=1时恢复了全局最优解。