Today I read a paper titled “Coins Make Quantum Walks Faster”
The abstract is:
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 $O(\sqrt{N} \log N)$.