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Coins Make Quantum Walks Faster

Andris Ambainis·Julia Kempe·Alexander Rivosh
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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 $Ω(N)$ 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 $Ω(N)$ steps. We also show that in 2 dimensions it is sufficient to have a two-dimensional coin-space to achieve the time $O(\sqrt{N} \log N)$.

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Andris Ambainis, Julia Kempe, Alexander Rivosh. 2004-02-16. Coins Make Quantum Walks Faster. https://arxiv.org/abs/quant-ph/0402107

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