量子学习布尔线性函数产物分布
Quantum Learning Boolean Linear Functions w.r.t. Product Distributions
摘要：
从量子例子中学习布尔线性函数的问题。可以使用Bernstein-Vazirani算法在量子计算机上求解均匀分布。类似的策略可以应用于有噪声的量子训练数据的情况，如arXiv：1702.08255v2 [quant-ph]中所观察到的。然而，这些学习算法的扩展超出了均匀分布还没有被研究。我们采用arXiv：1802.05690v2 [quant-ph]中引入的偏置量子傅里叶变换来开发有效的量子算法，用于从量子示例w.r.t.有偏见的产品分布。我们的第一个程序适用于任何（除了完全）的偏见，并需要$\mathcal{O}(\ln (n))$量子的例子。我们的第二个算法所使用的量子例子的数量是独立的$n$，但该策略只适用于小偏差。此外，我们表明，第二个程序是稳定的w.r.t.噪声训练数据和w.r.t.有缺陷的量子门。这也使我们能够解决一个版本的学习问题，其中潜在的分布是事先不知道的。最后，我们证明了学习问题的经典和量子样本复杂性的下界。而经典的，$Ω(n)$的例子是必要的独立的偏见，我们能够建立一个量子样本的复杂性下限的$Ω(\ln (n))$只有在假设大的偏见。然而，这允许我们建议的学习算法w.r.t.的性能进行讨论。样本复杂度通过我们的分析，我们有助于更定量地了解量子训练数据在学习经典函数方面的能力和局限性。
英文摘要：
The problem of learning Boolean linear functions from quantum examples w.r.t. the uniform distribution can be solved on a quantum computer using the Bernstein-Vazirani algorithm. A similar strategy can be applied in the case of noisy quantum training data, as was observed in arXiv:1702.08255v2 [quant-ph]. However, extensions of these learning algorithms beyond the uniform distribution have not yet been studied. We employ the biased quantum Fourier transform introduced in arXiv:1802.05690v2 [quant-ph] to develop efficient quantum algorithms for learning Boolean linear functions on $n$ bits from quantum examples w.r.t. a biased product distribution. Our first procedure is applicable to any (except full) bias and requires $\mathcal{O}(\ln (n))$ quantum examples. The number of quantum examples used by our second algorithm is independent of $n$, but the strategy is applicable only for small bias. Moreover, we show that the second procedure is stable w.r.t. noisy training data and w.r.t. faulty quantum gates. This also enables us to solve a version of the learning problem in which the underlying distribution is not known in advance. Finally, we prove lower bounds on the classical and quantum sample complexities of the learning problem. Whereas classically, $Ω(n)$ examples are necessary independently of the bias, we are able to establish a quantum sample complexity lower bound of $Ω(\ln (n))$ only under an assumption of large bias. Nevertheless, this allows for a discussion of the performance of our suggested learning algorithms w.r.t. sample complexity. With our analysis we contribute to a more quantitative understanding of the power and limitations of quantum training data for learning classical functions.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
