Exponential speedups for quantum walks in random hierarchical graphs

TL;DR

This paper generalizes exponential quantum walk speedups from welded-tree graphs to a broad class of hierarchical graphs with random interconnections, revealing new quantum advantages in traversal times.

Contribution

It introduces a framework for exponential quantum speedups on hierarchical graphs with random connections, extending prior specific cases to a wider class of structures.

Findings

Quantum walks on these graphs exhibit superpolynomial to exponential speedups.

Hitting times relate to localization in disordered Hamiltonians.

Concrete hierarchical graph models with efficient quantum traversal are provided.

Abstract

There are few known exponential speedups for quantum algorithms and these tend to fall into even fewer families. One speedup that has mostly resisted generalization is the use of quantum walks to traverse the welded-tree graph, due to Childs, Cleve, Deotto, Farhi, Gutmann, and Spielman. We show how to generalize this to a large class of hierarchical graphs in which the vertices are grouped into "supervertices" which are arranged according to a $d$-dimensional lattice. Supervertices can have different sizes, and edges between supervertices correspond to random connections between their constituent vertices. The hitting times of quantum walks on these graphs are related to the localization properties of zero modes in certain disordered tight binding Hamiltonians. The speedups range from superpolynomial to exponential, depending on the underlying dimension and the random graph model. We also provide concrete realizations of these hierarchical graphs, and introduce a general method for constructing graphs with efficient quantum traversal times using graph sparsification.