资源可扩展的完全量子Metropolis-Hastings算法用于整数线性规划
Resource-Scalable Fully Quantum Metropolis-Hastings for Integer Linear Programming
AI总结：
本文提出一种完全量子Metropolis-Hastings算法，用于高效解决整数线性规划问题，通过相干随机游走和可逆量子电路实现线性资源扩展。
摘要：
整数线性规划（ILP）由于其NP难的性质仍然在计算上具有挑战性，尽管它在调度、物流和设计优化中起着核心作用。我们介绍了一种完全量子Metropolis-Hastings算法用于ILP，该算法仅使用可逆量子电路在离散可行区域上实现相干随机游走，而不依赖于量子-RAM假设或经典预处理。每个游走步骤是一个单位ary更新，准备相干候选移动，可逆地评估目标函数和约束（包括一个约束满足计数器以确保可行性），并通过低开销线性化规则编码Metropolis接受振幅。在逻辑层面，该构造使用O(n log₂N)个量子比特来表示n个整数变量，范围在[-N,N-1]之间，每个Metropolis步骤的Toffoli等效成本与总逻辑量子比特数成线性增长。使用显式的进位链加法构造，我们支持线性目标函数和混合等式/不等式约束。对广泛随机生成实例的数值电路级模拟验证了预测的线性资源扩展，并在退火调度下表现出逐步热化向低成本可行解的倾向。总体而言，该方法为完全量子约束编程提供了一个相干、资源特征化的基准，并为组合优化中引入额外量子加速奠定了基础。
英文摘要：
Integer linear programming (ILP) remains computationally challenging due to its NP-complete nature despite its central role in scheduling, logistics, and design optimization. We introduce a fully quantum Metropolis-Hastings algorithm for ILP that implements a coherent random walk over the discrete feasible region using only reversible quantum circuits, without quantum-RAM assumptions or classical pre/post-processing. Each walk step is a unitary update that prepares coherent candidate moves, evaluates the objective and constraints reversibly -- including a constraint-satisfaction counter to enforce feasibility -- and encodes Metropolis acceptance amplitudes via a low-overhead linearized rule. At the logical level, the construction uses $\mathcal{O}(n\log_2 N)$ qubits to represent $n$ integer variables over the interval $[-N,\,N-1]$, and the Toffoli-equivalent cost per Metropolis step grows linearly with the total logical qubit count. Using explicit ripple-carry adder constructions, we support linear objectives and mixed equality/inequality constraints. Numerical circuit-level simulations on a broad ensemble of randomly generated instances validate the predicted linear resource scaling and exhibit progressive thermalization toward low-cost feasible solutions under the annealing schedule. Overall, the method provides a coherent, resource-characterized baseline for fully quantum constraint programming and a foundation for incorporating additional quantum speedups in combinatorial optimization.
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