On Applying the Lackadaisical Quantum Walk Algorithm to Search for Multiple Solutions on Grids

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.