纠缠数据集的非自由午餐定理的改进
Reformulation of the No-Free-Lunch Theorem for Entangled Data Sets
摘要：
没有免费的午餐（NFL）定理是学习理论中的一个著名结果，它限制了人们使用训练数据集学习函数的能力。随着最近量子机器学习的兴起，很自然地会问是否存在NFL定理的量子模拟，这将限制量子计算机使用量子训练数据学习酉过程（函数的量子模拟）的能力。然而，在量子环境中，训练数据可以具有纠缠，一种没有经典模拟的强相关性。在这项工作中，我们表明，纠缠的数据集导致一个明显的违反（经典）NFL定理。这激发了重新表述，以说明训练集中的纠缠程度。作为我们的主要结果，我们证明了一个量子NFL定理，其中一个单一的可学习性的基本限制减少纠缠。我们采用Rigetti的量子计算机来测试经典和量子NFL定理。我们的工作表明，纠缠是量子机器学习中的一种商品。
英文摘要：
The no-free-lunch (NFL) theorem is a celebrated result in learning theory that limits one's ability to learn a function with a training data set. With the recent rise of quantum machine learning, it is natural to ask whether there is a quantum analog of the NFL theorem, which would restrict a quantum computer's ability to learn a unitary process (the quantum analog of a function) with quantum training data. However, in the quantum setting, the training data can possess entanglement, a strong correlation with no classical analog. In this work, we show that entangled data sets lead to an apparent violation of the (classical) NFL theorem. This motivates a reformulation that accounts for the degree of entanglement in the training set. As our main result, we prove a quantum NFL theorem whereby the fundamental limit on the learnability of a unitary is reduced by entanglement. We employ Rigetti's quantum computer to test both the classical and quantum NFL theorems. Our work establishes that entanglement is a commodity in quantum machine learning.
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