The analysis of the computational power of single-query quantum algorithms is important because they must extract maximal information from one oracle call, revealing fundamental limits of quantum advantage and enabling optimal, resource-efficient quantum computation. This paper proposes a formulation of single-query quantum decision trees as weighted graphs. This formulation has the advantage that it facilitates the analysis of the $L_1$ spectral norm of the algorithm output. This advantage is based on the fact that a high $L_1$ spectral norm of the output of a quantum decision tree is a necessary condition to outperform its classical counterpart. We propose heuristics for maximizing the $L_{1}$ spectral norm, show how to combine weighted graphs to generate sequences with strictly increasing norm, and present functions exhibiting exponential quantum advantage. Finally, we establish a necessary condition linking single-query quantum advantage to the asymptotic growth of measurement projector dimensions.

翻译：单查询量子算法的计算能力分析具有重要意义，因为它们必须从一次预言机调用中提取最大信息，这揭示了量子优势的基本限制，并支持实现资源最优的高效量子计算。本文提出将单查询量子决策树表述为加权图。该表述的优势在于便于分析算法输出的$L_1$谱范数。这一优势基于以下事实：量子决策树输出具有较高的$L_1$谱范数是其超越经典对应算法的必要条件。我们提出了最大化$L_{1}$谱范数的启发式方法，展示了如何组合加权图以生成谱范数严格递增的序列，并给出了呈现指数级量子优势的函数实例。最后，我们建立了一个将单查询量子优势与测量投影子维度的渐近增长联系起来的必要条件。