Elfs, transducers and quantum walks

2026-05-28

AI Summary

This work addresses the challenge of efficiently implementing errorless electrical flow sampling (ELFS) on graphs to enhance the performance of quantum walk algorithms in search, sampling, and optimization tasks. We introduce, for the first time, an errorless transformer that establishes a framework supporting subspace intersection reflections, combined with an effective gap lemma to achieve optimal error scaling. This approach yields optimal estimates of effective resistance and Span program witness size, and demonstrates up to a quadratic quantum speedup in semi-supervised learning tasks on graphs.

Abstract

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.

Problem

• Electric Flow Sampling
• Quantum Walks
• Zero-error Transducer
• Effective Resistance
• Semi-supervised Learning

Innovation

• electric flow sampling
• zero-error transducer
• quantum walks
• effective resistance estimation
• semi-supervised learning