多光子量子神经网络的表达能力与局限性
Expressive Power and Limitations of Multi-photon Quantum Neural Networks
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中文总结 AI 辅助
该研究定量探究多光子量子神经网络的表达能力，明确其在固定与可训练可观测量下的光子数影响规律，经数值模拟验证，为量子机器学习利用多光子优势提供指导。
中文摘要 AI 辅助
量子神经网络（QNNs）在利用量子计算完成机器学习任务方面展现出潜力。多光子量子神经网络（MPQNNs）以多个全同光子作为输入，有望通过增加光子数提升表达能力，但光子数增加对MPQNNs表达能力的具体影响、能否通过无限增加光子数实现表达能力的无限提升，目前尚未得到探究。本研究通过推导两种情形下的近似误差上界，定量评估该模型的表达能力：在固定可观测量的情形中，存在一个与模式数呈线性缩放的阈值，低于该阈值时，MPQNNs的表达能力可通过增加光子数实现多项式级提升，高于该阈值时，增加光子数不会对表达能力产生影响；在可训练可观测量的情形中，表达能力始终可通过增加光子数实现多项式级提升。上述发现通过数值模拟得到验证。本研究阐明了量子神经网络中多光子量子特征的性能提升机制及其局限性，为在量子机器学习中利用多光子优势提供了指导。
英文摘要
Quantum neural networks (QNNs) have shown promise in leveraging quantum computation for machine learning tasks. Utilizing multiple identical photons as input, multi-photon quantum neural networks (MPQNNs) have the potential to enhance the expressivity through increasing the photon number. However, how precisely the expressivity of an MPQNN is affected by an increase in photon number, and whether it can be infinitely enhanced by increasing the photon number, remains unexplored. In this work, we quantitatively estimate the expressivity of this model by deriving upper bounds on approximation error in two cases. In the case of a fixed observable, there exists a threshold that scales linearly with the mode number. Below the threshold, the expressivity of an MPQNN can be enhanced polynomially by increasing the photon number. Above the threshold, however, increasing the photon number does not affect the expressivity. In the case of a trainable observable, the expressivity can always be enhanced polynomially by increasing the photon number. These findings are then validated by numerical simulations. Our work elucidates the performance enhancement of multi-photon quantum feature in QNNs, as well as its limitations, offering guidance for leveraging multi-photon advantages in quantum machine learning.
