发表机构
National University of Science and Technology POLITEHNICA Bucharest（布加勒斯特理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种自适应贝叶斯量子搜索方法，统一经典、Grover与广义定点搜索，在未知解数量下最大化期望成功概率，并证明其渐近最优性及混合策略优势。
AI 中文摘要
非结构化数据库搜索是一个基础性问题，要求在一组候选项中找到一个或多个满足特定属性的元素。当解的数量已知时，Grover算法为寻找标记元素提供了二次加速，但当解的数量未知时，寻找所有解则带来额外的挑战。在本文中，我们通过考虑经典搜索、Grover搜索和广义定点量子搜索实例的所有不同组合机制，深入分析了这一问题。我们将这些统一为一种自适应贝叶斯方法，该方法在每次搜索实例中最大化单位执行时间的期望成功概率。对于已知解数量远小于数据库规模的问题，我们证明了在所考虑的搜索模型内，我们提出的方法是渐近最优的。此外，我们证明：对于任何表征可能解数量的概率分布，在经典验证时间至多为量子振幅放大时间25%的机制下，经典与Grover混合方法优于广义定点方法。在此机制之外，我们推导了定点搜索相对于最佳基于Grover的得分所能实现的最大改进的上界。另外，我们利用解数量概率分布的积分近似，确定了经典采样与量子搜索之间的最优过渡点。分析结果也得到了所提算法的数值模拟以及广义定点与基于Grover的搜索策略之间比较的支持。
英文摘要
Unstructured database search is a fundamental problem that requires finding one or more elements satisfying a specific property within a set of candidates. While Grover's algorithm provides a quadratic speedup for finding a marked element when the number of solutions is known, finding all solutions when their number is unknown presents additional challenges. In this paper, we thoroughly analyze this problem by considering all the different regimes of combinations of classical, Grover and generalized fixed-point quantum search instances. We unify these into an adaptive Bayesian approach that maximizes the expected success probability per unit execution time at each searching instance. For a problem with a known number of solutions, much lower than the database size, we prove that our proposed approach is asymptotically optimal within the considered search model. Furthermore, we prove that for any probability distribution characterizing the likely number of solutions, in regimes where the classical verification time is at most 25% of the quantum amplitude amplification time, hybrid classical and Grover approaches are preferred to generalized fixed-point. Outside of this regime, we derive an upper bound on the maximal improvement achievable by fixed-point search compared to the best Grover-based score. Additionally, we determine the optimal transition point between classical sampling and quantum searching using an integral approximation of the probability distribution of the number of solutions. The analytical results are also supported by numerical simulations of the proposed algorithm and by comparisons between the generalized fixed-point and Grover-based search policies.
Comments35 pages, 8 figures
