There exist several types of configurations of marked vertices, referred to
as the exceptional configurations, on one- and two-dimensional periodic
lattices with additional long-range edges of the HN4 network, which are
challenging to find using discrete-time quantum walk algorithms. In this
article, we conduct a comparative analysis of the discrete-time quantum walk
algorithm utilizing various coin operators to search for these exceptional
configurations. First, we study the emergence of several new exceptional
configurations/vertices due to the additional long-range edges of the HN4
network on both one- and two-dimensional lattices. Second, our study shows that
the diagonal configuration on a two-dimensional lattice, which is exceptional
in the case without long-range edges, no longer remains an exceptional
configuration. Third, it is also shown that a recently proposed modified coin
can search all these configurations, including any other configurations in one-
and two-dimensional lattices with very high success probability. Additionally,
we construct stationary states for the exceptional configurations caused by the
additional long-range edges, which explains why the standard and lackadaisical
quantum walks with the Grover coin cannot search these configurations.

Cet article explore les excursions dans le temps et leurs implications.

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