With the rapid development of quantum computers, quantum algorithms have been studied extensively. However, quantum algorithms tackling statistical problems are still lacking. In this paper, we propose a novel non-oracular quantum adaptive search (QAS) method for the best subset selection problems. QAS performs almost identically to the naive best subset selection method but reduces its computational complexity from $O(D)$ to $O(\sqrt{D}\log_2D)$, where $D=2^p$ is the total number of subsets over $p$ covariates. Unlike existing quantum search algorithms, QAS does not require the oracle information of the true solution state and hence is applicable to various statistical learning problems with random observations. Theoretically, we prove QAS attains any arbitrary success probability $q \in (0.5, 1)$ within $O(\log_2D)$ iterations. When the underlying regression model is linear, we propose a quantum linear prediction method that is faster than its classical counterpart. We further introduce a hybrid quantum-classical strategy to avoid the capacity bottleneck of existing quantum computing systems and boost the success probability of QAS by majority voting. The effectiveness of this strategy is justified by both theoretical analysis and extensive empirical experiments on quantum and classical computers.
翻译:随着量子计算机的迅速发展,量子算法已经得到了广泛的研究,然而,解决统计问题的量子算法仍然缺乏。在本文中,我们建议对最佳子选择问题采用一种新的非整体量子适应性搜索(QAS)方法。QAS几乎与天真的最佳子集选择方法完全相同,但将其计算复杂性从O(D)美元降至$O(O)美元(log_log_2D)美元($D=2P美元),因为美元是超过美元共变数子数的总数。与现有的量子搜索算法不同,QAS并不要求真正解决方案状态的灵巧信息,因此适用于随机观测的各种统计学习问题。理论上,我们证明QAS取得了任意的成功概率$(0.5,1美元),但在O(log_2D)美元的范围内,将其计算复杂性降低到$O($)(0.5,1美元)。当基本回归模型为直线性预测方法比其古典对应方法要快。我们进一步引入混合量子级算法战略,以避免现有量子计算系统的能力瓶颈,从而通过模拟实验多数和大规模分析,提高典型计算机的成功概率。