神经网络为何能高效表示量子态？关键在测量关联。
When can classical neural networks represent quantum states?
- 用条件相关性分析量子态的测量分布，揭示神经表示的性能根源。
- 发现短程与长程条件相关性受纠缠、符号结构和测量基影响。
- 为理解神经量子态的表达能力提供理论框架，适合量子计算研究者。
n 个量子比特的量子态若用经典方式表示，需指定指数级多的计算基幅值。已有研究显示，经典神经网络可简洁表达许多物理相关的量子态，形成称为神经量子态的计算强大表示。其有效性背后的机制是什么？我们发现，量子态测量分布中的条件相关性决定了其神经表示的性能。这类条件相关性具有基依赖性，源于测量诱导的纠缠，揭示了传统多体关联分析中难以捕捉的特征。通过理论与数值结合，我们阐明了态的纠缠、符号结构及测量基选择如何导致短程或长程条件相关性的不同模式。这些发现为探索神经量子态的表达能力提供了严格框架。
原文摘要 · Abstract (English)
A naive classical representation of an n-qubit state requires specifying exponentially many amplitudes in the computational basis. Past works have demonstrated that classical neural networks can succinctly express these amplitudes for many physically relevant states, leading to computationally powerful representations known as neural quantum states. What underpins the efficacy of such representations? We show that conditional correlations present in the measurement distribution of quantum states control the performance of their neural representations. Such conditional correlations are basis dependent, arise due to measurement-induced entanglement, and reveal features not accessible through conventional few-body correlations often examined in studies of phases of matter. By combining theoretical and numerical analysis, we demonstrate how the state's entanglement and sign structure, along with the choice of measurement basis, give rise to distinct patterns of short- or long-range conditional correlations. Our findings provide a rigorous framework for exploring the expressive power of neural quantum states.
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