发表机构
Shanghai Institute for Mathematics and Interdisciplinary Sciences (SIMIS); Research Institute of Intelligent Complex Systems, Fudan University; Jmuse Technologies; Center for Quantum Information, IIIS, Tsinghua University; Inria, Université Grenoble Alpes; CNRS, Grenoble INP, LIG, Université Grenoble Alpes; Center for Mathematics and Interdisciplinary Sciences, Fudan University（上海数学与交叉学科研究院; 复旦大学智能复杂系统研究院; 聚摩科技; 清华大学交叉信息研究院量子信息中心; 法国国家信息与自动化研究所，格勒诺布尔阿尔卑斯大学; 法国国家科学研究中心，格勒诺布尔理工学院，信息实验室，格勒诺布尔阿尔卑斯大学; 复旦大学数学与交叉学科学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对大规模贝尔不等式优化难题，提出基于批量梯度下降的量子值优化器，利用GPU并行加速，显著快于跷跷板方法，可处理上千输入输出，并支持经典值高效计算。
AI 中文摘要
探索贝尔不等式世界需要能够随参与方、输入和输出数量有效扩展的数值方法。此类方法可用于研究超出解析方法范围的大规模贝尔不等式，设计违反更复杂贝尔不等式的实验，并推动贝尔不等式违反的应用，例如随机数生成和受限通信下的多智能体协调。然而，当前通用优化器（如跷跷板方法）难以处理仅含十几个输入和输出的贝尔不等式。我们引入一种基于批量梯度下降（BGD）的贝尔不等式量子值优化器。更精确地说，我们的BGD优化器是对可行量子策略的可微搜索，其中状态和投影测量由无约束变量生成。贝尔表达式通过直接张量收缩求值，无需构造规模过大的贝尔算子。该公式使并行随机重启成本低廉，因此自然适用于GPU实现。我们将我们的优化器与跷跷板方法进行比较，发现对多种贝尔不等式族有显著加速。在GPU上实现时，我们的BGD优化器能在几分钟内优化我们数值实验中具有超过一千个输入或输出的贝尔不等式。对于这些大规模贝尔不等式，我们还通过使用GPU枚举可能策略或将优化表达为混合整数线性规划来高效计算经典值。我们的优化器可作为贝尔实验和多智能体协调问题的数值评估工具，这些问题由具有大量输入和输出的贝尔不等式建模，这是高频交易和分布式系统等现实场景中的常见特征。
英文摘要
Exploring the world of Bell inequalities requires numerical methods that scale effectively with the number of parties, inputs, and outputs. Such methods can be used to study large-scale Bell inequalities beyond the reach of analytic methods, to design experiments for violating more complicated Bell inequalities, and to enable applications of Bell inequality violation. Such applications include randomness generation and multi-agent coordination with restricted communication. However, current general purpose optimizers such as the see-saw method struggle to handle Bell inequalities with only a dozen inputs and outputs. We introduce an optimizer for the quantum value of a Bell inequality based on batched gradient descent (BGD). More precisely, our BGD optimizer is a differentiable search over feasible quantum strategies in which states and projective measurements are generated from unconstrained variables. The Bell expression is evaluated by a direct tensor contraction without forming the prohibitively large Bell operator. This formulation makes parallel random restarts inexpensive and is thus naturally implemented by a GPU. We evaluate our optimizer against the see-saw method and find a significant speedup for a wide variety of Bell inequality families. Implemented on a GPU, our BGD optimizer can optimize Bell inequalities in our numerical experiments with more than a thousand inputs or outputs within minutes. For these large-scale Bell inequalities, we also efficiently compute the classical value by enumerating possible strategies using a GPU or expressing the optimization as a mixed-integer linear program. Our optimizers can serve as numerical evaluation tools for Bell experiments and multi-agent coordination problems modeled by Bell inequalities with many inputs and outputs, a common feature of real-world scenarios such as high frequency trading and distributed systems.
Comments54 pages, 8 figures
