This article presents an argument for why quantum computers could unlock new methods for machine learning. We argue that spectral methods, in particular those that learn, regularise, or otherwise manipulate the Fourier spectrum of a machine learning model, are often natural for quantum computers. For example, if a generative machine learning model is represented by a quantum state, the Quantum Fourier Transform allows us to manipulate the Fourier spectrum of the state using the entire toolbox of quantum routines, an operation that is usually prohibitive for classical models. At the same time, spectral methods are surprisingly fundamental to machine learning: A spectral bias has recently been hypothesised to be the core principle behind the success of deep learning; support vector machines have been known for decades to regularise in Fourier space, and convolutional neural nets build filters in the Fourier space of images. Could, then, quantum computing open fundamentally different, much more direct and resource-efficient ways to design the spectral properties of a model? We discuss this potential in detail here, hoping to stimulate a direction in quantum machine learning research that puts the question of ``why quantum?'' first.

翻译：本文论证了量子计算机为何能为机器学习开辟新方法。我们认为频谱方法——特别是那些学习、正则化或以其他方式操控机器学习模型傅里叶频谱的方法——对量子计算机而言往往具有天然优势。例如，若生成式机器学习模型以量子态表示，量子傅里叶变换使我们能借助整套量子计算工具包操控该状态的傅里叶频谱，而此类操作对经典模型通常成本高昂。与此同时，频谱方法对机器学习具有令人惊讶的基础性意义：近期研究假设频谱偏差是深度学习成功的核心原理；支持向量机数十年前已被证实可在傅里叶空间进行正则化；而卷积神经网络则在图像的傅里叶空间中构建滤波器。那么，量子计算能否为设计模型的频谱特性开辟根本不同、更直接且资源高效的途径？本文对此潜力进行详细探讨，期望推动量子机器学习研究领域将"为何选择量子？"作为首要问题。