We introduce the graph composition framework, a generalization of the $st$-connectivity framework for constructing quantum algorithms. Our framework constructs algorithms that solve a connectivity problem on an undirected graph, where the availability of each edge is computed by a span program. The key novelty of our framework is that the construction allows for amortization of the span programs' costs, while at the same time avoiding build-up of errors due to composition. We give generic time-efficient implementations of algorithms generated through the graph composition framework in the quantum read-only memory model, which is a weaker assumption than the more common quantum random-access model. Along the way, we also simplify the span program algorithm by converting it to a transducer, and remove the dependence of its analysis on the effective spectral gap lemma. We use graph composition to unify existing quantum algorithmic frameworks. Surprisingly, we show that any randomized algorithm can be converted into an instance of the $st$-connectivity framework. Furthermore, we show that the $st$-connectivity framework subsumes the learning graph framework, and the weighted-decision-tree framework. We show that the graph composition framework subsumes part of the quantum divide-and-conquer framework, and that it is itself subsumed by the multidimensional quantum walk framework. Moreover, we show polynomial relations and separations between the optimal query complexities that can be achieved with several of these frameworks. Finally, we apply our techniques to give improved algorithms for various string-search problems.

翻译：本文提出图组合框架，作为构建量子算法的$st$连通性框架的推广。该框架构建的算法用于解决无向图上的连通性问题，其中每条边的可用性通过跨度程序计算。本框架的核心创新在于：其构造方式允许对跨度程序的成本进行摊销，同时避免因组合而产生的误差累积。我们在量子只读存储器模型中给出了通过图组合框架生成算法的通用时间高效实现，该模型是比常见的量子随机存取模型更弱的假设。在此过程中，我们通过将跨度程序算法转换为转换器来简化其结构，并消除了其分析对有效谱间隙引理的依赖。我们利用图组合框架统一了现有的量子算法框架。令人惊讶的是，我们证明任何随机化算法均可转换为$st$连通性框架的实例。此外，我们证明$st$连通性框架包含了学习图框架和加权决策树框架。我们证明图组合框架包含了量子分治框架的部分内容，而其本身又被多维量子游走框架所包含。进一步，我们展示了这些框架间可实现的最优查询复杂度之间的多项式关系与分离性。最后，我们将所提技术应用于多种字符串搜索问题，给出了改进算法。