Search of clustered marked states with lackadaisical quantum walks

Search on Vertex-Transitive Graphs by Lackadaisical Quantum Walk

“…We note that l = 4/N is the degree of the loopless graph divided by the total number of vertices, i.e., d/N , since d = 4. With multiple marked vertices [20][21][22], it is no longer true that d/N is the best value of l.…”

“…More recent results have disproved this speculation, however. Search on the torus with multiple marked vertices [20][21][22], and search on the complete bipartite graph with one or more marked vertices [23], indicate that l should not be the degree centrality at each vertex. In this paper, our observation is specifically restricted to vertex-transitive graphs with a single marked vertex, where the observation does seem to hold.…”

“…Search using the lackadaisical quantum walk has been explored on the complete graph [16], discrete torus with one marked vertex [19] and multiple marked vertices [20][21][22], cycle [22], and complete bipartite graph [23]. Depending on the weight of the self-loop l, the success probability can be improved.…”

Search by Lackadaisical Quantum Walk with Symmetry Breaking

“…(N − 1) + M + (N − M )From the denominator of the overall coefficient, for large N ,N (N − 1) + M + (N − M ) ≈ N 2 .Then, for large N ,(12) …”

“…Later, the unweighted self-loops at each vertex were replaced by a single self-loop of real-valued weight at each vertex, such that if is an integer, it is equivalent to the original definition of integer self-loops per vertex [10]. This generalization to real-valued weights led to speedups for spatial search on a variety of graphs, including the discrete torus with one marked vertex [11] and multiple marked vertices [12][13][14][15][16], periodic square lattices of arbitrary dimension [17,18], strongly regular graphs [18], Johnson graphs [18], the hypercube [18], regular locally arc-transitive graphs [19], the triangular lattice [20], and the honeycomb lattice [20]. All of these graphs are vertex transitive, meaning they have symmetries such that each vertex has the same structure.…”