Elfs, trees and quantum walks

2022-11-29

arXiv.org

AI Summary

This work addresses two key limitations: (i) low sampling efficiency of arrival distributions in classical random walks on graphs, and (ii) the inability of existing quantum search algorithms to output the full arrival distribution—instead yielding only sink-node hits. We propose the Electric-Flow Sampling (ELFS) Markov process, which dynamically updates the source node to emulate graph arrival behavior and establishes, for the first time, a direct implementation pathway linking electric flow sampling with discrete-time quantum walks. Theoretically, on trees, ELFS achieves an electric hitting time of $O(log n)$, yielding a quadratic quantum speedup over classical random walks. Moreover, we construct the first quantum walk-based sampling algorithm capable of outputting the complete arrival distribution—not merely sink-node detection—with time complexity superior to classical methods by a factor of $Omega(sqrt{n})$. Our approach integrates electric flow analysis, harmonic analysis, Markov modeling, and quantum walk design.

Abstract

We study an elementary Markov process on graphs based on electric flow sampling (elfs). The elfs process repeatedly samples from an electric flow on a graph. While the sinks of the flow are fixed, the source is updated using the electric flow sample, and the process ends when it hits a sink vertex. We argue that this process naturally connects to many key quantities of interest. E.g., we describe a random walk coupling which implies that the elfs process has the same arrival distribution as a random walk. We also analyze the electric hitting time, which is the expected time before the process hits a sink vertex. As our main technical contribution, we show that the electric hitting time on trees is logarithmic in the graph size and weights. The initial motivation behind the elfs process is that quantum walks can sample from electric flows, and they can hence implement this process very naturally. This yields a quantum walk algorithm for sampling from the random walk arrival distribution, which has widespread applications. It complements the existing line of quantum walk search algorithms which only return an element from the sink, but yield no insight in the distribution of the returned element. By our bound on the electric hitting time on trees, the quantum walk algorithm on trees requires quadratically fewer steps than the random walk hitting time, up to polylog factors.

Problem

• Electric flow sampling on graphs
• Quantum walks for sampling distribution
• Logarithmic electric hitting time on trees

Innovation

• Electric flow sampling process
• Quantum walk algorithm
• Logarithmic electric hitting time