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Time and Query Optimal Quantum Algorithms Based on Decision Trees
arXiv - CS - Computational Complexity Pub Date : 2021-05-18 , DOI: arxiv-2105.08309 Salman Beigi, Leila Taghavi, Artin Tajdini
arXiv - CS - Computational Complexity Pub Date : 2021-05-18 , DOI: arxiv-2105.08309 Salman Beigi, Leila Taghavi, Artin Tajdini
It has recently been shown that starting with a classical query algorithm
(decision tree) and a guessing algorithm that tries to predict the query
answers, we can design a quantum algorithm with query complexity $O(\sqrt{GT})$
where $T$ is the query complexity of the classical algorithm (depth of the
decision tree) and $G$ is the maximum number of wrong answers by the guessing
algorithm [arXiv:1410.0932, arXiv:1905.13095]. In this paper we show that,
given some constraints on the classical algorithms, this quantum algorithm can
be implemented in time $\tilde O(\sqrt{GT})$. Our algorithm is based on
non-binary span programs and their efficient implementation. We conclude that
various graph theoretic problems including bipartiteness, cycle detection and
topological sort can be solved in time $O(n^{3/2}\log n)$ and with $O(n^{3/2})$
quantum queries. Moreover, finding a maximal matching can be solved with
$O(n^{3/2})$ quantum queries in time $O(n^{3/2}\log n)$, and maximum bipartite
matching can be solved in time $O(n^2\log n)$.
中文翻译:
基于决策树的时间和查询最优量子算法
最近显示,从经典查询算法(决策树)和试图预测查询答案的猜测算法开始,我们可以设计一种查询复杂度为$ O(\ sqrt {GT})$的量子算法,其中$ T $是经典算法的查询复杂度(决策树的深度),而$ G $是猜测算法的最大错误答案数[arXiv:1410.0932,arXiv:1905.13095]。在本文中,我们证明,在经典算法受某些约束的情况下,该量子算法可以在$ \ tilde O(\ sqrt {GT})$的时间内实现。我们的算法基于非二进制跨度程序及其有效实现。我们得出结论,可以在$ O(n ^ {3/2} \ log n)$的时间和$ O(n ^ {3/2})$量子的情况下解决各种图形理论问题,包括二部性,周期检测和拓扑排序查询。
更新日期:2021-05-19
中文翻译:
基于决策树的时间和查询最优量子算法
最近显示,从经典查询算法(决策树)和试图预测查询答案的猜测算法开始,我们可以设计一种查询复杂度为$ O(\ sqrt {GT})$的量子算法,其中$ T $是经典算法的查询复杂度(决策树的深度),而$ G $是猜测算法的最大错误答案数[arXiv:1410.0932,arXiv:1905.13095]。在本文中,我们证明,在经典算法受某些约束的情况下,该量子算法可以在$ \ tilde O(\ sqrt {GT})$的时间内实现。我们的算法基于非二进制跨度程序及其有效实现。我们得出结论,可以在$ O(n ^ {3/2} \ log n)$的时间和$ O(n ^ {3/2})$量子的情况下解决各种图形理论问题,包括二部性,周期检测和拓扑排序查询。