可学习但不可模拟：学习模型的量子资源理论
Learnable yet not simulable: a quantum resource theory of learning models
浏览论文内容
中文总结 AI 辅助
该研究提出动力学稳定子熵（DSE）作为新资源度量，构建对比经典模拟器与量子数据辅助代理模型的计算相图，证明存在可被代理模型高效学习但无法仅由电路描述高效模拟的电路族，为量子系统模拟与学习的边界提供定量框架。
中文摘要 AI 辅助
量子资源理论深化了我们对量子系统固有复杂性的理解，尤其是其经典可模拟性。然而，目前尚不清楚在超出高效经典模拟的范畴时，是哪种量子资源决定了量子电路的经典可学习性。本文通过研究可调量子电路族的期望值函数来填补这一知识空白，这类函数在数字量子模拟、量子计量学和量子系统表征中具有诸多应用。具体而言，我们引入了一种新的资源度量——动力学稳定子熵（\text{DSE}），它量化了期望值函数在其频率模式中的分布广度。通过将\textbf{DSE}与算子稳定子熵关联，我们构建了一个计算相图，用于对比经典模拟器与量子数据辅助的经典代理模型。我们首先通过推导经典代理模型的样本复杂度和运行时间边界，并开发一种\textbf{DSE}引导的代理模型，确定了该图中依赖于\textbf{DSE}的可学习性边界；随后，在标准复杂性理论假设下，我们证明了存在这样的电路族，它们可以被该代理模型高效学习，但无法仅通过其电路描述被高效模拟，从而完善了该相图。对多达80个量子比特的随机和结构化电路进行的数值实验，验证了所预测的依赖于\textbf{DSE}的计算格局。这些结果建立了一个定量的资源理论框架，用于界定经典模拟与学习之间的边界，推动了将量子资源与可学习性关联起来的资源度量研究，并为无法通过直接经典模拟实现的可扩展量子系统设计基于学习的算法提供了指导。
英文摘要
Quantum resource theory has sharpened our understanding of the intrinsic complexity of quantum systems, particularly their classical simulability. However, it remains unclear which quantum resource governs the classical learnability of quantum circuits, especially beyond the regime of efficient classical simulation. Here we close this knowledge gap by studying the expectation-value functions of families of tunable quantum circuits, with many applications in digital quantum simulation, quantum metrology, and quantum-system characterization. Specifically, we introduce a new resource measure, the dynamical stabilizer entropy (\DSE), which quantifies how broadly an expectation-value function is distributed across its frequency modes. By relating \DSE to operator stabilizer entropy, we establish a computational phase diagram that compares classical simulators with quantum-data-assisted classical surrogates. We first determine the \DSE-dependent learnability boundary of this diagram by deriving bounds on the sample complexity and runtime of classical surrogates, and by developing a \DSE-guided surrogate. We then complete the diagram by proving, under standard complexity-theoretic assumptions, the existence of circuit families that can be efficiently learned by this surrogate but cannot be efficiently emulated from their circuit descriptions alone. Numerical experiments on random and structured circuits with up to 80 qubits support the predicted \DSE-dependent computational landscape. These results establish a quantitative resource-theoretic framework for delineating the boundary between classical simulation and learning, motivate resource measures linking quantum resources to learnability, and guide the design of learning-based algorithms for scalable quantum systems beyond the reach of direct classical simulation.
