Electric flow sampling (elfs) is a new tool in the quantum walk toolbox and a useful primitive for solving search, sampling and optimization problems on graphs. We refine this tool by showing that there exists a zero-error transducer for implementing elfs. More broadly, we establish a zero-error transducer for reflecting about the intersection of two subspaces, yielding an errorfree transducer version of the effective gap lemma. Building on this result, we obtain improved quantum walk algorithms for estimating effective resistances and span program witness sizes with an optimal error scaling, and for sampling from the random walk arrival distribution, via the composition of many elfs. Using this last algorithm, we obtain an up-to-quadratic quantum speedup for semi-supervised learning on expander graphs.

翻译：电通量采样（elfs）是量子游走工具箱中的新工具，也是解决图上搜索、采样与优化问题的基础性基本操作。我们通过证明存在零误差换能器可实现elfs，从而精炼了这一工具。更广泛地，我们建立了关于两个子空间交集反射的零误差换能器，由此导出有效间隙引理的无误换能器版本。基于此结果，我们改进了量子游走算法：在最优误差标度下估算有效电阻与跨越程序见证者规模，并通过组合多个elfs实现从随机游走到达分布中采样。利用最后这个算法，我们在扩展图上的半监督学习中获得至多二次的量子加速。