In this work, we unify several quantum algorithmic frameworks for boolean functions that are based on the quantum adversary bound. First, we show that the $st$-connectivity framework subsumes the (adaptive/extended) learning graph framework, and the weighted-decision-tree framework. Additionally, we show that every randomized algorithm can be turned into an $st$-connectivity problem as well, with the same complexity up to constants. This situates the $st$-connectivity framework in between randomized and quantum algorithms, indicating that it's an intermediate computational model between classical and quantum. We also introduce a generalization of the $st$-connectivity framework, the \textit{graph composition framework}, and show that it subsumes part of the quantum divide and conquer framework, and it is itself subsumed by the multidimensional quantum walk framework. Second, we investigate these frameworks' power by investigating the most efficient algorithms they can produce, in terms of the number of queries to the input. We show that the weighted-decision-tree framework's power is polynomially related to deterministic query complexity, showing that the quantum speed-ups that can be obtained with this framework are at most quadratic. Finally, we turn our attention to time-efficient implementations of the algorithms constructed through the $st$-connectivity and graph composition frameworks. To that end, we convert instances to the two-subspace phase estimation framework, and we show how we can implement these as transducers. This has the added benefit of removing the effective spectral gap lemma from the construction, significantly simplifying the analysis. We showcase the techniques developed in this work to give improved algorithms for various string search problems.

翻译：在本工作中，我们统一了若干基于量子敌手界的布尔函数量子算法框架。首先，我们证明$st$-连通性框架包含了（自适应/扩展型）学习图框架与加权决策树框架。此外，我们证明任意随机算法均可转化为$st$-连通性问题，且其复杂度在常数因子内保持不变。这确立了$st$-连通性框架介于随机算法与量子算法之间的地位，表明其是经典计算与量子计算之间的中间计算模型。我们还引入了$st$-连通性框架的推广形式——\textit{图组合框架}，并证明该框架包含了量子分治框架的部分内容，同时其本身又被多维量子游走框架所包含。其次，我们通过考察各框架在输入查询次数维度上所能产生的最优算法，来探究这些框架的计算能力。我们证明加权决策树框架的能力与确定性查询复杂度呈多项式关联，表明通过该框架所能获得的量子加速最多仅为二次加速。最后，我们将研究重点转向通过$st$-连通性与图组合框架构建算法的时间高效实现。为此，我们将问题实例转化为双子空间相位估计框架，并展示如何将其实现为转换器。这种方法还具有消除构造过程中有效谱隙引理的优势，显著简化了分析过程。我们运用本工作发展的技术，为多种字符串搜索问题提供了改进算法。