发表机构
Technion – Israel Institute of Technology; MIGAL – Galilee Research Institute / Tel-Hai Academic College（以色列理工学院; 米加尔-加利利研究所 / 泰尔海学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对固定车队车辆路径问题的仅度弧QUBO模型存在的连通性缺陷，提出带上限单商品流的QUBO修复项，通过经典基准和量子设备实验验证了方法的有效性，解决了基态可能与 depot 断开的问题。
AI 中文摘要
本文重新研究了用于固定车队、同构、无容量车辆路径问题的仅度弧哈密顿量。由于其局部惩罚仅定义了一个覆盖圈，基态可能包含与 depot 断开的客户圈。我们构造了一个多项式规模的无约束二次二元优化（QUBO）修复项，使用带上限的单商品流，并证明在显式惩罚假设下，每个基态路径都是连通且成本最优的。对于 N-1 个客户和 K 条非空路径，未简化编码恰好使用 |E|(1+⌈log₂(N-K+1)⌉) 个逻辑问题量子比特。可逆计算-相位-反计算实现，在具有 O(log N) 可重用工作空间且无乘积寄存器的完全图上，以 O(N²log N + N log²N) 个逻辑门评估流惩罚。在完全无环图上，当流词长度增加时，depot 分隔的单序列位置编码使用更少的问题量子比特和更少的写入项。相反，流模型在常见可逆核算模型下实现了更小的结构化逻辑门上界。对哈密顿量和电路实现的精确审计，结合 1200 矩阵的经典基准，验证了该公式并量化了连通性差距。最后，在 IQM Emerald 上进行的 32000 次 shot 的 Amazon Braket 任务，在诊断 N=4、K=1 的反例实例上表征了深度 1 的逐项 Ising 电路。在仅度电路中，78.05% 的选定 p=1 shot 实现了无效的断开基态；简化后的 14 量子比特流增强电路未产生完全可行的样本。这些设备结果表征了映射的哈密顿量和编译，而非渐近路径解决方案的优势。
英文摘要
We revisit a degree-only arc Hamiltonian for fixed-fleet, homogeneous, uncapacitated vehicle routing. Its local penalties define only a cycle cover, so ground states may contain customer cycles disconnected from the depot. We construct a polynomial-size quadratic unconstrained binary optimization (QUBO) repair using capped single-commodity flow and prove connected, cost-optimal ground-state routings under explicit penalty assumptions. For $N-1$ customers and $K$ nonempty routes, the unreduced encoding uses exactly $|E|(1+\lceil\log_2(N-K+1)\rceil)$ logical problem qubits. A reversible compute-phase-uncompute implementation evaluates the flow penalties in $O(N^2\log N+N\log^2N)$ logical gates on a complete graph, using $O(\log N)$ reusable workspace and no product register. On complete loopless graphs, a position encoding that arranges all routes in one sequence separated by depot occurrences uses fewer problem qubits and written terms when the number of bits per flow register grows. Under the same reversible accounting model, the flow formulation admits a smaller logical-gate upper bound. Exact Hamiltonian and circuit audits and a benchmark of 1,200 cost matrices verify the formulation and quantify the connectivity gap. A 32,000-shot Amazon Braket task on IQM Emerald characterizes depth-one termwise Ising circuits for a diagnostic $N=4$, $K=1$ counterexample. In the degree-only circuit, 78.05% of selected $p=1$ shots realize the invalid disconnected ground state; the reduced 14-qubit flow-augmented circuit yields no fully feasible sample. These device results characterize mapped Hamiltonians and compilation, not an asymptotic advantage in solving routing problems.
Comments29 pages, 3 figures, 5 tables. Reproducibility package: https://doi.org/10.5281/zenodo.21595142
