# Quantum Walk Sampling by Growing Seed Sets

## TL;DR

This paper introduces a quantum walk sampling algorithm that uses seed sets to efficiently generate superpositions over graph edges, improving quantum walk setup costs and enabling faster algorithms for graph connectivity and isomorphism problems.

## Contribution

It presents a novel seed set-based quantum walk algorithm that improves setup costs and offers new solutions for graph connectivity and isomorphism testing.

## Key findings

- Quantum walk steps scale as O(m^{1/3} \u03b4^{-1/3})
- New bounds on quantum walk search setup costs
- Superposition over graph isomorphisms in O(2^{n/3}) time

## Abstract

This work describes a new algorithm for creating a superposition over the edge set of a graph, encoding a quantum sample of the random walk stationary distribution. The algorithm requires a number of quantum walk steps scaling as $\widetilde{O}(m^{1/3} \delta^{-1/3})$, with $m$ the number of edges and $\delta$ the random walk spectral gap. This improves on existing strategies by initially growing a classical seed set in the graph, from which a quantum walk is then run. The algorithm leads to a number of improvements: (i) it provides a new bound on the setup cost of quantum walk search algorithms, (ii) it yields a new algorithm for $st$-connectivity, and (iii) it allows to create a superposition over the isomorphisms of an $n$-node graph in time $\widetilde{O}(2^{n/3})$, surpassing the $\Omega(2^{n/2})$ barrier set by index erasure.