TY - JOUR
T1 - Spatial search by continuous-time quantum walks on crystal lattices
JF - Physical Review A
Y1 - 2014
A1 - Andrew M. Childs
A1 - Yimin Ge
AB - 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.
VL - 89
UR - http://arxiv.org/abs/1403.2676v2
CP - 5
J1 - Phys. Rev. A
U5 - 10.1103/PhysRevA.89.052337
ER -