The adjacency matrices of graphs provide the foundation for constructing the Hamiltonians of Continuous-Time Quantum Walks (CTQWs). Various classes of graphs have been identified to be highly reducible and the reduced Hamiltonian preserves the dynamics of the original system. This makes the CTQW implementation feasible in the near term for search problems of large size. Highly reducible Hamiltonians are desirable because existing quantum devices are of limited size in terms of the number of qubits. In this work, we review the recent developments of dimensionality reduction and coupling factor value finding techniques. The CTQWs based on a reduced Hamiltonian can search optimally when the correctly calculated coupling factor is used. We list identified highly reducible graphs and include their optimality proofs when correct coupling factors are used. In addition, we discuss the recent developments on Lackadaisical Quantum Walkers (LQW) (a type of coin-based discrete-time quantum walk) for one- and two-dimensional spatial search. The optimal lower upper bound remains open in one- and two-dimensional Discrete-Time Quantum Walk.

图的邻接矩阵为构造连续时间量子游动（CTQW）的哈密顿量提供了基础。各种类型的图已被确定为高度可归约的，并且简化的哈密顿量保留了原始系统的动态。这使得CTQW实施在短期内可解决大型搜索问题。高度可还原的哈密顿量是合乎需要的，因为就量子位的数量而言，现有的量子装置的大小有限。在这项工作中，我们回顾了降维和耦合因子值查找技术的最新发展。当使用正确计算的耦合因子时，基于简化哈密顿量的CTQW可以最佳搜索。我们列出了已识别的高度可约化图，并在使用正确的耦合因子时包括了它们的最优性证明。此外，我们讨论了用于一维和二维空间搜索的Lackadaisical Quantum Walkers（LQW）（一种基于硬币的离散时间量子行走）的最新发展。最佳下限在一维和二维离散时间量子遍历中保持打开状态。