Elfs, trees and quantum walks

TL;DR

This work introduces the electric flow sampling (elfs) process, a Markov-like procedure that repeatedly samples from the unit electric flow between a fixed sink set and a source that moves to the sampled endpoint. It establishes a precise link between elfs and classical random walks via a coupling that preserves arrival distributions, and introduces the electric hitting time (EHT) as a key quantity tied to the classical hitting time (HT). The authors prove that on trees the elfs hitting time grows only logarithmically with graph size, and they connect EHT to the entropy of the arrival distribution, yielding practical bounds and algorithmic implications. Motivated by the quantum-network connection, they develop a quantum walk framework that can sample from the random-walk arrival distribution in time roughly the square root of the classical quantities, providing quadratic speedups especially on trees, and they give applications to quantum walk search and to estimating effective resistances. Overall, the work unifies electric-network theory, random walks, and quantum walks into a coherent framework with concrete bounds and quantum-speedup implications for sampling-based graph problems.

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.

Figures (13)

• Figure 1: Coupling between a random walk and elfs process through stopping rules $\nu_1<\dots<\nu_\rho$.
• Figure 2: $n$-vertex path graph with source $s$ and sink $M$. The electric hitxting time from $s$ to $M$ is $\mathrm{EHT}_s \in \Theta(\log n)$.
• Figure 3: Schur complement $G'$ of a tree graph $G$. We relate the number of steps of the elfs process on $G$ with source $s$ and sink $M$ (dotted circles) to the elfs process on $G'$.
• Figure 4: $n$-vertex complete graph with source $s$ and sink $M$. The electric hitting time from $s$ to $M$ is $\Theta(|M|(1-|M|/n))$.
• Figure 5: (identical to Figure \ref{['fig:intro-coupling']}) Coupling between a random walk $\{X_0 = s,X_1,\dots,X_\tau \in M\}$ and an elfs process $\{Y_0 = s,Y_1,\dots,Y_\rho \in M\}$ through a stopping rule $\nu$. From $s$ this ensures that $X_\nu = Y_1$.

Theorems & Definitions (41)

• Lemma 1
• proof
• Corollary 1
• proof
• proof
• Lemma 2: Coupling lemma
• proof : Proof of \ref{['lem:coupling']} (coupling lemma)
• Lemma 3: Edge coupling lemma
• Lemma 4: escape time identity
• proof