标题： 推理、推断和不变性：量子傅里叶变换如何帮助从数据中学习 显示英文标题

标题： Inference, interference and invariance: How the Quantum Fourier Transform can help to learn from data

摘要： 如何从典型的量子算法中获得灵感，以设计用于机器学习的启发式方法？ 一个常见的模板，从Deutsch-Josza算法到Shor算法都使用了该模板，即通过预言机将标记信息置于叠加态，在傅里叶空间中进行干涉，并进行测量。 在本文中，我们想了解这种干涉策略如何用于推理，即从有限的数据样本推广到真实情况。 我们的研究框架围绕隐藏子群问题（HSP）构建，我们通过用经典训练数据替换预言机，将HSP转化为一个学习任务。 解决HSP的标准量子算法使用量子傅里叶变换来揭示一个不变子空间，即希尔伯特空间中的一个子集，在其中隐藏的对称性是明显的。 基于这一见解，我们提出了一种“比较”数据与这个不变子空间的推理原则，并建议通过量子态的重叠来进行具体实现。 我们希望这能导致具有明确动机的量子启发式方法，这些方法可以利用对称性用于机器学习应用。 显示英文摘要

摘要： How can we take inspiration from a typical quantum algorithm to design heuristics for machine learning? A common blueprint, used from Deutsch-Josza to Shor's algorithm, is to place labeled information in superposition via an oracle, interfere in Fourier space, and measure. In this paper, we want to understand how this interference strategy can be used for inference, i.e. to generalize from finite data samples to a ground truth. Our investigative framework is built around the Hidden Subgroup Problem (HSP), which we transform into a learning task by replacing the oracle with classical training data. The standard quantum algorithm for solving the HSP uses the Quantum Fourier Transform to expose an invariant subspace, i.e., a subset of Hilbert space in which the hidden symmetry is manifest. Based on this insight, we propose an inference principle that "compares" the data to this invariant subspace, and suggest a concrete implementation via overlaps of quantum states. We hope that this leads to well-motivated quantum heuristics that can leverage symmetries for machine learning applications.