Quantum computing holds the unparalleled potentials to enhance, speed up or innovate machine learning. However, an unambiguous demonstration of quantum learning advantage has not been achieved so far. Here, we rigorously establish a noise-robust, unconditional quantum learning advantage in terms of expressivity, inference speed, and training efficiency, compared to commonly-used classical machine learning models. Our proof is information-theoretic and pinpoints the origin of this advantage: quantum entanglement can be used to reduce the communication required by non-local machine learning tasks. In particular, we design a fully classical task that can be solved with unit accuracy by a quantum model with a constant number of variational parameters using entanglement resources, whereas commonly-used classical models must scale at least linearly with the size of the task to achieve a larger-than-exponentially-small accuracy. We further show that the quantum model can be trained with constant time and a number of samples inversely proportional to the problem size. We prove that this advantage is robust against constant depolarization noise. We show through numerical simulations that even though the classical models can have improved performance as their sizes are increased, they would suffer from overfitting. The constant-versus-linear separation, bolstered by the overfitting problem, makes it possible to demonstrate the quantum advantage with relatively small system sizes. We demonstrate, through both numerical simulations and trapped-ion experiments on IonQ Aria, the desired quantum-classical learning separation. Our results provide a valuable guide for demonstrating quantum learning advantages in practical applications with current noisy intermediate-scale quantum devices.

翻译：量子计算在增强、加速或革新机器学习方面具有无与伦比的潜力。然而，迄今为止尚未实现量子学习优势的明确证明。在此，我们严格建立了一种在表达能力、推理速度和训练效率方面，相较于常用经典机器学习模型而言，具有噪声稳健性且无条件的量子学习优势。我们的证明基于信息论，并精确指出了该优势的起源：量子纠缠可用于减少非局部机器学习任务所需的通信。具体而言，我们设计了一个完全经典的任务，该任务可以通过一个使用纠缠资源、具有恒定数量变分参数的量子模型以单位精度求解，而常用的经典模型必须至少随任务规模线性扩展才能实现大于指数级小的精度。我们进一步证明，该量子模型可以在恒定时间内，使用与问题规模成反比数量的样本进行训练。我们证明了该优势对恒定去极化噪声具有稳健性。我们通过数值模拟表明，尽管经典模型随着其规模增大可以提升性能，但它们会遭受过拟合问题。这种恒定与线性的分离，加之过拟合问题，使得在相对较小的系统规模下演示量子优势成为可能。我们通过数值模拟以及在IonQ Aria上的囚禁离子实验，演示了所期望的量子-经典学习分离。我们的结果为在当前含噪声中等规模量子设备上，于实际应用中演示量子学习优势提供了有价值的指导。