Spatial search by quantum walk

Abstract

Grover's quantum search algorithm provides a way to speed up combinatorial search, but is not directly applicable to searching a physical database. Nevertheless, Aaronson and Ambainis showed that a database of $N$ items laid out in $d$ spatial dimensions can be searched in time of order $\sqrt{N}$ for $d>2$, and in time of order $\sqrt{N}\phantom{\rule{0.3em}{0ex}}\text{poly}(\mathrm{log}\phantom{\rule{0.3em}{0ex}}N)$ for $d=2$. We consider an alternative search algorithm based on a continuous-time quantum walk on a graph. The case of the complete graph gives the continuous-time search algorithm of Farhi and Gutmann, and other previously known results can be used to show that $\sqrt{N}$ speedup can also be achieved on the hypercube. We show that full $\sqrt{N}$ speedup can be achieved on a $d$-dimensional periodic lattice for $d>4$. In $d=4$, the quantum walk search algorithm takes time of order $\sqrt{N}\phantom{\rule{0.3em}{0ex}}\text{poly}(\mathrm{log}\phantom{\rule{0.3em}{0ex}}N)$, and in $d<4$, the algorithm does not provide substantial speedup.