Overview: recent development and applications of reduction and lackadaisicalness techniques for spatial search quantum walk in the near term

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.