Lower Bounds on the Localisation Length of Balanced Random Quantum Walks

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)