Faster quantum-walk algorithm for the two-dimensional spatial search
Avatar Tulsi · Physical Review A · 2008
We consider the problem of finding a desired item out of $N$ items arranged on the sites of a two-dimensional lattice of size $\sqrt{N}\ifmmode\times\else\texttimes\fi{}\sqrt{N}$. The previous quantum-walk based algorithms take $O(\sqrt{N}\text{ }\text{ln}\text{ }N)$ steps to solve this problem, and it is an open question whether the performance can be improved. We present an algorithm which solves the problem in $O(\sqrt{N\text{ }\text{ln}\text{ }N})$ steps, thus giving an $O(\sqrt{\text{ln}\text{ }N})$ improvement over the known algorithms. The improvement is achieved by controlling the quantum walk on the lattice using an ancilla qubit.