Elfs, transducers and quantum walks
This work provides a foundational tool (zero-error transducer for elfs) that enhances quantum walk algorithms, offering practical improvements for search, sampling, and optimization problems on graphs.
The authors refine electric flow sampling (elfs) by introducing a zero-error transducer, leading to improved quantum walk algorithms for estimating effective resistances and span program witness sizes with optimal error scaling, and achieving an up-to-quadratic quantum speedup for semi-supervised learning on expander graphs.
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.