Search by lackadaisical quantum walks with nonhomogeneous weights

Abstract

The lackadaisical quantum walk, which is a quantum walk with a weighted self-loop at each vertex, has been shown to speed up dispersion on the line and improve spatial search on the complete graph and periodic square lattice. In these investigations, each self-loop had the same weight, owing to each graph's vertex-transitivity. In this paper, we propose lackadaisical quantum walks where the self-loops have different weights. We investigate spatial search on the complete bipartite graph, which can be irregular with N1 and N2 vertices in each partite set, and this naturally leads to self-loops in each partite set having different weights l1 and l2, respectively. We analytically prove that for large N1 and N2, if the k marked vertices are confined to, say, the first partite set, then with the typical initial uniform state over the vertices, the success probability is improved from its non-lackadaisical value when l1 = kN2/2N1 and N2 > (3 − 2 √ 2)N1, regardless of l2. When the initial state is stationary under the quantum walk, however, then the success probability is improved when l1 = kN2/2N1, now without a constraint on the ratio of N1 and N2, and again independent of l2. Next, when marked vertices lie in both partite sets, then for either initial state, there are many configurations for which the self-loops yield no improvement in quantum search, no matter what weights they take.

Search by Lackadaisical Quantum Walk with Symmetry Breaking

“…We do this by finding the eigenvectors and eigenvalues of U = SCQ for large N using degenerate pertur- bation theory, expressing the initial state in this eigenbasis, and then evolving the state by multiplying each eigenvector by its eigenvalue. This proves that some of the symmetry of the graph can be broken while preserving the speedup, as for the regular complete bipartite graph [21]. Proving the general case with every self-loop taking a different value, as in Fig.…”

“…Lackadaisical quantum walks with nonhomogeneous weights were introduced for searching complete bipartite graphs [21], where the self-loops in one partite set had one weight, and the self-loops in the other partite set had another weight. Regular complete bipartite graphs are vertex transitive, and although the nonhomogeneous weights broke this symmetry, not all the symmetry was broken, as vertices within a partite set still evolved identically.…”

“…The solid black curve is without self-loops [23]. With each self-loop weight equal to the optimal value of 1/2 [21], we get the dashed red curve. Randomly choosing the self-loops at the unmarked vertices to have weights between 0 and 10, which breaks the symmetries of the graph, we get the dotted green curve, and it closely matches the dashed red curve, indicating that only the weight at the marked vertex matters, asymptotically.…”

“…Prior work on the irregular complete bipartite graph answers this in the negative. From [21], the weights of the self-loops at unmarked vertices can affect the success probability. This work was supported by startup funds from Creighton University.…”

Coherence evolution of three-state quantum walk on lattice

“…The three-state quantum walk has been demonstrated to increase the success probability of quantum search tasks from 50% to 100%. Wong has shown that lazy three-state quantum walk on a two-dimensional grid are faster than their acyclic counterparts when searching for marked vertices [21]. Additionally, Rhodes and Wong have developed a numerical expression for self-loop appropriate weights in undirected quantum walk spatial search [22].…”