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.

中文翻译：

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应用Lackadaisical Quantum Walk算法在网格上搜索多解

量子计算有望将信息处理能力提高到经典计算无法达到的水平。量子行走正在引领量子算法的发展，用于比经典算法更有效地搜索图上的信息。在文献中脱颖而出的一种基于量子游走的算法是懒散的量子游走。懒散的量子行走是一种算法，用于搜索二维网格，其顶点具有权重 $l$ 的自循环。在本文中，我们解决了与应用乏味量子游走成功搜索网格上的多个解决方案相关的几个问题。首先，我们表明文献中发现的两个停止条件中只有一个适合模拟。我们还证明最终成功概率取决于解决方案的空间密度和解决方案之间的相对距离。此外，这项工作将懒惰的量子行走推广到在任意维度的网格上搜索多个解决方案。此外，我们针对这种任意维度的场景提出了自环权重 $l$ 的最佳调整。事实证明，文献中发现的其他 $l$ 拟合是特殊情况。最后，我们观察到惰性量子游走的步骤与 Grover 算法的步骤之间存在二对一的关系，这需要修改停止条件。总之，除了将该技术扩展到更广泛的搜索问题之外，这项工作还涉及在应用乏味量子行走时应考虑的实际问题。