Quantum computing holds the promise of improving the information processing
power to levels unreachable by classical computation. Quantum walks are heading
the development of quantum algorithms for searching information on graphs more
efficiently than their classical counterparts. A quantum-walk-based algorithm
that is standing out in the literature is the lackadaisical quantum walk. The
lackadaisical quantum walk is an algorithm developed to search two-dimensional
grids whose vertices have a self-loop of weight l. In this paper, we address
several issues related to the application of the lackadaisical quantum walk to
successfully search for multiple solutions on grids. Firstly, we show that only
one of the two stopping conditions found in the literature is suitable for
simulations. We also demonstrate that the final success probability depends on
the space density of solutions and the relative distance between solutions.
Furthermore, this work generalizes the lackadaisical quantum walk to search for
multiple solutions on grids of arbitrary dimensions. In addition, we propose an
optimal adjustment of the self-loop weight l for such scenarios of arbitrary
dimensions. It turns out the other fits of l found in the literature are
particular cases. Finally, we observe a two-to-one relation between the steps
of the lackadaisical quantum walk and the ones of Grover's algorithm, which
requires modifications in the stopping condition. In conclusion, this work
deals with practical issues one should consider when applying the lackadaisical
quantum walk, besides expanding the technique to a wider range of search
problems.Comment: Extended version of the conference paper available at
https://doi.org/10.1007/978-3-030-61377-8_9 . 21 pages, 6 figure