# Lower Bounds on the Localisation Length of Balanced Random Quantum Walks

## TL;DR

This paper investigates the localization length of balanced random quantum walks on lattices and trees, deriving lower bounds and showing divergence on trees as the degree increases, with implications for quantum transport.

## Contribution

It introduces a combinatorial expression for localization length and establishes lower bounds, including divergence on trees and bounds on cubic lattices, advancing understanding of quantum walk localization.

## Key findings

- Localization length diverges on the tree as degree squared (d^2).
- Lower bound of 1/ln(2) for the cubic lattice localization length.
- Localization length is bounded below by the correlation length of self-avoiding walks.

## Abstract

We consider the dynamical properties of Quantum Walks defined on the d-dimensional cubic lattice, or the homogeneous tree of coordination number 2d, with site dependent random phases, further characterised by transition probabilities between neighbouring sites equal to 1/(2d). We show that the localisation length for these Balanced Random Quantum Walks can be expressed as a combinatorial expression involving sums over weighted paths on the considered graph. This expression provides lower bounds on the localisation length by restriction to paths with weight 1, which allows us to prove the localisation length diverges on the tree as d^2. On the cubic lattice, the method yields the lower bound 1/ln(2) for all d, and allows us to bound the localisation length from below by the correlation length of self-avoiding walks computed at 1/(2d)