The vast and complicated large-qubit state space forbids us to comprehensively capture the dynamics of modern quantum computers via classical simulations or quantum tomography. However, recent progress in quantum learning theory invokes a crucial question: given a quantum circuit containing d tunable RZ gates and G-d Clifford gates, can a learner perform purely classical inference to efficiently predict its linear properties using new classical inputs, after learning from data obtained by incoherently measuring states generated by the same circuit but with different classical inputs? In this work, we prove that the sample complexity scaling linearly in d is necessary and sufficient to achieve a small prediction error, while the corresponding computational complexity may scale exponentially in d. Building upon these derived complexity bounds, we further harness the concept of classical shadow and truncated trigonometric expansion to devise a kernel-based learning model capable of trading off prediction error and computational complexity, transitioning from exponential to polynomial scaling in many practical settings. Our results advance two crucial realms in quantum computation: the exploration of quantum algorithms with practical utilities and learning-based quantum system certification. We conduct numerical simulations to validate our proposals across diverse scenarios, encompassing quantum information processing protocols, Hamiltonian simulation, and variational quantum algorithms up to 60 qubits.

翻译：庞大而复杂的大比特态空间使我们无法通过经典模拟或量子层析成像全面捕捉现代量子计算机的动力学行为。然而，量子学习理论的最新进展引出了一个关键问题：给定一个包含d个可调RZ门和G-d个Clifford门的量子电路，学习者能否在从同一电路（但使用不同经典输入）生成并通过非相干测量获得的数据中学习后，执行纯经典推断来高效预测其线性性质（使用新的经典输入）？本工作中，我们证明了样本复杂度随d线性缩放是实现小预测误差的必要且充分条件，而相应的计算复杂度可能随d呈指数级增长。基于这些推导出的复杂度界限，我们进一步利用经典阴影和截断三角展开的概念，设计了一种基于核的学习模型，该模型能够在预测误差与计算复杂度之间进行权衡，在许多实际场景中实现从指数级到多项式级的过渡。我们的研究结果推动了量子计算中两个关键领域的发展：具有实际效用的量子算法探索以及基于学习的量子系统认证。我们通过数值模拟验证了所提方案在多种场景下的有效性，包括量子信息处理协议、哈密顿量模拟以及多达60个量子比特的变分量子算法。