Clustering-induced localization of quantum walks on networks

TL;DR

It is shown that local clustering, which is a key structural feature of networks, can induce localization of quantum walks, and it is shown that localization also occurs in Kleinberg navigable small-world networks and Holme--Kim power-law cluster networks.

Abstract

Quantum walks on networks are a paradigmatic model in quantum information theory. Quantum-walk algorithms have been developed for various applications, including spatial-search problems, element-distinctness problems, and node centrality analysis. Unlike their classical counterparts, the evolution of quantum walks is unitary, so they do not converge to a stationary distribution. However, for many applications, it is important to understand the long-time behavior of quantum walks and the impact of network structure on their evolution. In the present paper, we study the localization of quantum walks on networks. We demonstrate how localization emerges in highly clustered networks that we construct by recursively attaching triangles, and we derive an analytical expression for the long-time inverse participation ratio that depends on products of eigenvectors of the quantum-walk Hamiltonian. Building on the insights from this example, we then show that localization also occurs in Kleinberg navigable small-world networks and Holme--Kim power-law cluster networks. Our results illustrate that local clustering, which is a key structural feature of networks, can induce localization of quantum walks.

Figures (5)

• Figure 1: Localization in recursive triangle networks. (a) Recursive triangle networks of depths 1, 2, and 3. The long-time mean inverse participation ratios (IPRs) of the nodes with red rings are substantially larger than those of other nodes. (b) The long-time mean transition probabilities $\bar{\pi}_{ij}$ [see Eq. \ref{['eq:pi_bar_ij']}] for the networks in (a). (c) The long-time mean IPR $\hbox{$\mathrm{IPR}$}_j$ [see Eq. \ref{['eq:IPR_bar_j']}] as a function of the initially excited node $j$. The hollow red circles indicate the maximum value of $\hbox{$\mathrm{IPR}$}_j$. (d) The probability density $|\psi_i(t)|^2 = |\braket{i|\psi(t)}|^2$ as a function of the node $i$ and time $t$ for a CTQW that starts at node 61 in a recursive triangle network of depth 6. In this example, the quantum walker predominantly alternates between two nodes.
• Figure 2: The absolute and relative gaps between the maximum and minimum long-time mean IPRs for recursive triangle networks. (a) The absolute gap $\Delta \hbox{$\mathrm{IPR}$}$ between the maximum and minimum long-time mean IPRs [see Eq. \ref{['eq:Delta_IPR']}] as a function of the network depth $d$. (b) The relative gap $\delta \hbox{$\mathrm{IPR}$}$ between the maximum and minimum long-time mean IPRs [see Eq. \ref{['eq:delta_IPR']}] as a function of the network depth $d$.
• Figure 3: Localization on networks that we construct using the network geometry with flavor (NGF) model. (a) Evolution of the IPR for both NGF networks (with $s = -1$) and a recursive triangle network with depth $d = 5$. For the NGF model, the curve is the mean of the results for 1000 NGF networks, which each have 100 nodes. The CTQW starts at node 51 in the NGF networks. The recursive triangle network has 96 nodes, and the CTQW starts at node 36, where localization is strongest. (b) The probability density $|\psi_i(t)|^2 = |\braket{i|\psi(t)}|^2$ as a function of the node $i$ and time $t$ for a CTQW that starts at node 51 in one 100-node NGF network (with $s = -1$).
• Figure 4: Evolution of the IPR for several types of networks. (a) A Newman--Watts--Strogatz (NWS) network in which each node is adjacent to its two nearest neighbors in a ring. For each edge, the probabilities of adding a new edge are $p \in \{0, 0.01, 0.02, 0.05, 1\}$. (b, c) A Kleinberg navigable small-world network with clustering exponent $\alpha$ and $q \in \{0,1,2,3,4\}$ additional connections for each node. In (b), $\alpha = 2$; in (c), $\alpha = 3$. (d) A Holme--Kim (HK) power-law cluster network with probability $p' \in \{0,0.25,0.5,0.75,1\}$ of adding a triangle after adding a random edge. We construct this network by starting with an empty dyad (i.e., two isolated nodes) and iteratively adding new nodes until the network has 100 nodes. Each new node connects to two existing nodes using linear preferential attachment. All networks have $N = 100$ nodes, and all curves are means of results for 1000 networks. The CTQWs start at node 51.
• Figure 5: The probability density $|\psi_i(t)|^2 = |\braket{i|\psi(t)}|^2$ as a function of the node $i$ and time $t$ for a CTQW that starts at node 51 in (a) a 100-node Kleinberg network (with $\alpha = 2$ and $q = 1$) and (b) a 100-node Holme--Kim network (with $p' = 1$).