在受限量子优化中分离几何与干涉

中文摘要 AI 辅助

我们研究量子优化算法中几何效应与量子干涉的分离。诸如路由、分配和调度等受限优化问题常被编码为局部变量的乘积空间以及全局可行性惩罚。我们解决的核心算法问题是，在存在局部和全局约束的情况下，一个保持约束的混合算子如何在指数搜索空间中传输量子振幅。我们开发了一个框架，分离了通常相互混合的三种效应：振幅传输、传输振幅之间的相干干涉以及与问题相关的经典后处理。我们表明仅混合算子本身没有目标寻求能力。具体而言，其振幅传输诱导的归一化分布趋向于均匀随机配置的距离分布。因此，只有当到达目标配置的许多计算路径的相位充分对齐以使它们的振幅增强时，量子采样优势才可能出现。我们还展示了，当设计代价相位使这些路径相干相加时，数量仅随问题规模对数增长的电路交替足以将它们绝对贡献的总和转换为目标振幅的下限，产生与环境希尔伯特空间维度、搜索空间大小或可行集基数无关的认证成功概率，并开发了针对特定问题的转译诊断、可扩展硬件探测、量子生成样本的约束诱导经典映射、混合量子 - 经典工作流程中量子分布与经典后处理之间的解质量归因以及与经典编码理论中距离划分乘积空间的联系等应用。

英文摘要

We study the separation of geometric effects from quantum interference in quantum optimization algorithms. Constrained optimization problems such as routing, assignment, and scheduling are often encoded as product spaces of local variables, together with global feasibility penalties. The central algorithmic question we address is how a constraint-preserving mixing operator transports quantum amplitude across an exponential search space in the presence of local and global constraints. We develop a framework that separates three effects that are usually intermixed: amplitude transport, coherent interference among transported amplitudes, and problem-dependent classical postprocessing. We show that the mixing operator alone does not have a target-seeking ability. Concretely, the normalized distribution induced by its amplitude transport moves toward the distance profile of a uniformly random configuration. Thus, quantum sampling advantage may only arise when the phases of the many computational paths reaching a target configuration are sufficiently aligned for their amplitudes to reinforce. We show that, when the cost phases are engineered so that these paths add coherently, a number of circuit alternations growing only logarithmically with problem size suffices to convert the sum of their absolute contributions into a lower bound on the target amplitude, yielding a certified success probability independent of the ambient Hilbert-space dimension, the search-space size, or the feasible-set cardinality. We develop applications to problem-specific transpilation diagnostics, scalable hardware probes, constraint-induced classical maps of quantum-generated samples, the attribution of solution quality between the quantum distribution and classical post-processing in hybrid quantum-classical workflows and connections to distance-partitioned product spaces from classical coding theory.

发表机构

• Volkswagen AG（大众汽车股份公司）
• RWTH Aachen University（亚琛工业大学）
• USRA Research Institute for Advanced Computer Science (RIACS)（USRA高级计算机科学研究所（RIACS））
• Forschungszentrum Jülich（于利希研究中心）
• Universität zu Köln（科隆大学）

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