纳瓦斯奎斯 - 皮罗尼奥 - 阿辛层次结构中的矩优化

中文摘要 AI 辅助

纳瓦斯奎斯 - 皮罗尼奥 - 阿辛（NPA）层次结构为非对易多项式优化提供了收敛的半定规划（SDP）松弛序列，在量子物理中普遍存在。但其实际应用受限于各级所需算子矩的组合增长。由于并非所有矩对边界紧性的贡献相同，在固定计算预算内选择矩是个相关问题。我们将矩选择重新构建为组合子集选择，并表明它受矩之间强大的高阶协同相互作用支配，通过从复杂系统理论改编的边际协同诊断来量化。我们开发并比较了三种优化方法：并行回火（PT）、基于受限玻尔兹曼机（RBM）的强化学习策略和贝叶斯优化（BO）。在\(I_{3322}\)贝尔不等式基准测试中表现出色，应用该框架到\((4,4,2,2)\)场景中的174个贝尔不等式及一维海森堡自旋链，结果表明在非对易多项式优化中建立了可扩展的矩选择框架，在量子物理和量子信息中有广泛应用。

英文摘要

The Navascués-Pironio-Acín (NPA) hierarchy provides a convergent sequence of semidefinite programming (SDP) relaxations for noncommutative polynomial optimisation, ubiquitous in quantum physics. However, its practical applicability is limited by the combinatorial growth in operator moments required at each level. Since not all moments contribute equally to bound tightness, selecting moments within a fixed computational budget is a relevant problem. We reframe moment selection as combinatorial subset selection and show it is governed by strong higher-order synergistic interactions among moments, quantified through a marginal synergy diagnostic adapted from complex systems theory. We develop and compare three optimisation methods: Parallel Tempering (PT), an RBM-based reinforcement learning policy, and Bayesian Optimisation (BO). On the $I_{3322}$ Bell inequality benchmark, all three substantially outperform greedy approaches at costs around two orders of magnitude below brute force, with the RBM achieving the closest approach to optimal throughout the hard transition regime. We apply the framework to the 174 Bell inequalities in the $(4,4,2,2)$ scenario, finding heterogeneous convergence behaviour across inequalities, and to the one-dimensional Heisenberg spin chain, demonstrating that physically motivated monomial bases are internally compressible and are not globally optimal in general. A budget-aware search over a broader pool improves certified bounds on long-range correlations by nearly two orders of magnitude. These results establish a scalable framework for moment selection in noncommutative polynomial optimisation, with broad applications across quantum physics and quantum information.

发表机构

• ICFO - Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology（ICFO光子科学研究所，巴塞罗那科学技术研究院）
• Eurecat, Centre Tecnològic de Catalunya（Eurecat加泰罗尼亚技术中心）
• École Polytechnique, Institut Polytechnique de Paris（巴黎综合理工学院，巴黎理工学院）
• Institute for Theoretical Physics, ETH Zurich（苏黎世联邦理工学院理论物理研究所）
• ICREA - Institució Catalana de Recerca i Estudis Avançats（加泰罗尼亚研究与高级研究学院）

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