Coins make quantum walks faster

Abstract

We show how to search N items arranged on a $\\sqrt{N}\\times\\sqrt{N}$ grid in time $O(\\sqrt N \\log N)$, using a discrete time quantum walk. This result for the first time exhibits a significant difference between discrete time and continuous time walks without coin degrees of freedom, since it has been shown recently that such a continuous time walk needs time $\\Omega(N)$ to perform the same task. Our result furthermore improves on a previous bound for quantum local search by Aaronson and Ambainis. We generalize our result to 3 and more dimensions where the walk yields the optimal performance of $O(\\sqrt{N})$ and give several extensions of quantum walk search algorithms for general graphs. The coin-flip operation needs to be chosen judiciously: we show that another ``natural'' choice of coin gives a walk that takes $\\Omega(N)$ steps. We also show that in 2 dimensions it is sufficient to have a two-dimensional coin-space to achieve the time $O(\\sqrt{N} \\log N)$.