Spatial search by continuous-time quantum walks on crystal lattices

We consider the problem of searching a general $d$-dimensional lattice of $N$
vertices for a single marked item using a continuous-time quantum walk. We
demand locality, but allow the walk to vary periodically on a small scale. By
constructing lattice Hamiltonians exhibiting Dirac points in their dispersion
relations and exploiting the linear behaviour near a Dirac point, we develop
algorithms that solve the problem in a time of $O(\sqrt N)$ for $d>2$ and
$O(\sqrt N \log N)$ in $d=2$. In particular, we show that such algorithms exist
even for hypercubic lattices in any dimension. Unlike previous continuous-time
quantum walk algorithms on hypercubic lattices in low dimensions, our approach
does not use external memory.