Improving feasibility in the quantum approximate optimization algorithm for vehicle routing via constraint-aware initialization and hybrid XY–X mixing

The Quantum Approximate Optimization Algorithm (QAOA) is a leading framework for quantum combinatorial optimization. The vehicle routing problem (VRP), a core problem in logistics and transportation, is a natural application target but poses a major feasibility challenge for standard QAOA: feasible solutions typically occupy only a tiny fraction of the full binary search space, and the conventional Pauli- X mixer can disrupt partial solution structures that already satisfy key local constraints. To address this issue, we propose a constraint-aware QAOA framework with two complementary components. First, we design an initialization strategy that encodes selected constraints into the initial state. This reduces the size of the initial superposition without explicitly constructing the complete feasible set and concentrates probability mass on states that satisfy the selected constraints. Second, we introduce a hybrid XY – X mixer whose preservation property is intentionally partial. The XY -exchange terms preserve only the selected structures chosen for protection, while the X -mixer terms retain exploration on qubits outside those structures, and the remaining VRP constraints are addressed through QUBO penalty terms and variational optimization. We evaluate the proposed framework on one three-node benchmark instance and two four-node benchmark instances under three regimes: exact-statevector objective optimization with noiseless finite-shot final sampling, ideal finite-shot optimization and sampling, and noisy finite-shot optimization and sampling. The experiments compare the proposed method with standard QAOA across all cases and with a block-wise XY-QAOA baseline for the four-node cases. Across the tested instances and regimes, the proposed method increases optimal-solution probability and reduces expected energy gap relative to standard QAOA. The four-node comparisons further show that block-wise XY-QAOA is a stronger feasibility-aware baseline than standard QAOA, but the proposed overlap-aware initialization and hybrid mixer generally provide a better balance between selected-constraint preservation and search flexibility. The advantage of the proposed ansatz is attenuated under hardware-inspired noise, in part because the structured mixers introduce additional two-qubit operations. These results highlight that the practical benefit of the proposed method depends on gate fidelity, readout quality, state-preparation cost, and circuit execution reliability.