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arXiv · 2201.11735

Strong dispersion property for the quantum walk on the hypercube

Martins Kokainis·Krišjānis Prūsis·Jevgēnijs Vihrovs·Vyacheslavs Kashcheyevs·Andris Ambainis
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Abstract

We show that the discrete time quantum walk on the Boolean hypercube of dimension $n$ has a strong dispersion property: if the walk is started in one vertex, then the probability of the walker being at any particular vertex after $O(n)$ steps is of an order $O(1.4818^{-n})$. This improves over the known mixing results for this quantum walk which show that the probability distribution after $O(n)$ steps is close to uniform but do not show that the probability is small for every vertex. A rigorous proof of this result involves an intricate argument about analytic properties of Bessel functions.

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Martins Kokainis, Krišjānis Prūsis, Jevgēnijs Vihrovs, Vyacheslavs Kashcheyevs, Andris Ambainis. 2022-01-27. Strong dispersion property for the quantum walk on the hypercube. https://doi.org/10.1088/1751-8121%2Faca6b9

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