量子学习算法的改进界
Improved Bounds on Quantum Learning Algorithms
摘要：
在这篇文章中，我们给出了几个新的结果，从量子查询和量子示例中学习布尔函数的算法的复杂性。Hunziker等人证明，对于任何布尔函数类C，从C中精确识别未知函数所需的量子黑盒查询的数量是$O(\frac{\log |C|}{\sqrt{{\hatγ}^{C}}})$，其中$\hatγ^{C}$是类C的组合参数。我们基本上解决了这个猜想的肯定，给出了一个量子算法，对于任何类C，确定任何未知的功能，从C使用$O(\frac{\log |C| \log \log |C|}{\sqrt{{\hatγ}^{C}}})$量子黑箱查询。我们考虑一系列介于精确学习问题（学习者必须获得关于黑盒函数的所有信息）和计算谓词的通常问题（学习者必须仅获得关于黑盒函数的一位信息）之间的自然问题。我们给出了积极和消极的结果时，这些中间问题的量子和经典的查询复杂性是多项式相关的。最后，我们改进了已知的量子例子（相对于量子黑盒查询）的数量的下界$(ε,δ)$ -PAC学习任何概念类的Vapnik-Chervonenkis维D域$\{0,1\}^n$从$Ω(\frac{d}{n})$到$Ω(\frac{1}ε\log \frac{1}δ+d+\frac{\sqrt{d}}ε)$。这个新的下限更接近于经典PAC学习的已知上限。
英文摘要：
In this article we give several new results on the complexity of algorithms that learn Boolean functions from quantum queries and quantum examples. Hunziker et al. conjectured that for any class C of Boolean functions, the number of quantum black-box queries which are required to exactly identify an unknown function from C is $O(\frac{\log |C|}{\sqrt{{\hatγ}^{C}}})$, where $\hatγ^{C}$ is a combinatorial parameter of the class C. We essentially resolve this conjecture in the affirmative by giving a quantum algorithm that, for any class C, identifies any unknown function from C using $O(\frac{\log |C| \log \log |C|}{\sqrt{{\hatγ}^{C}}})$ quantum black-box queries. We consider a range of natural problems intermediate between the exact learning problem (in which the learner must obtain all bits of information about the black-box function) and the usual problem of computing a predicate (in which the learner must obtain only one bit of information about the black-box function). We give positive and negative results on when the quantum and classical query complexities of these intermediate problems are polynomially related to each other. Finally, we improve the known lower bounds on the number of quantum examples (as opposed to quantum black-box queries) required for $(ε,δ)$-PAC learning any concept class of Vapnik-Chervonenkis dimension d over the domain $\{0,1\}^n$ from $Ω(\frac{d}{n})$ to $Ω(\frac{1}ε\log \frac{1}δ+d+\frac{\sqrt{d}}ε)$. This new lower bound comes closer to matching known upper bounds for classical PAC learning.
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