Discrete-time quantum walk algorithm for ranking nodes on a network

Abstract

We present a quantum algorithm for ranking the nodes on a network in their order of importance. The algorithm is based on a directed discrete-time quantum walk, and works on all directed networks. This algorithm can theoretically be applied to the entire internet, and thus can function as a quantum PageRank algorithm. Our analysis shows that the hierarchy of quantum rank matches well with the hierarchy of classical rank for directed tree network and for non-trivial cyclic networks, the hierarchy of quantum ranks do not exactly match to the hierarchy of the classical rank. This highlights the role of quantum interference and fluctuations in networks and the importance of using quantum algorithms to rank nodes in quantum networks. Another application this algorithm can envision is to model the dynamics on networks mimicking the chemical complexes and rank active centers in order of reactivities. Since discrete-time quantum walks are implementable on current quantum processing systems, this algorithm will also be of practical relevance in analysis of quantum architecture.

Quantum walks in polycyclic aromatic hydrocarbons

“…( 7) the shift operation effects the traversal of the components of the probability amplitude in different directions. In order to obtain the order of reactivity of sites, the algorithm in [28] requires a variant of DTQW known as a directed DTQW (D-DTQW) [65]. In case of the D-DTQW, the shift operator only allows a single component of the probability amplitude to traverse the graph.…”

“…( 7), however in case of the procedure to arrange the nodes in order of reactivities, we use a specialized coin operation of the form defined in Ref. [28],…”

“…The dynamics of DTQW require a coin and a position Hilbert space in order to be defined, however, CTQW dynamics can be defined with only a position Hilbert space. A quantum walk spreads quadratically faster than a classical walk on its position space and its dynamics can be engineered using a set of evolution parameters, and therefore it has been used as a basis for design and implementation of quantum algorithms and quantum simulations [16][17][18][19][20][21][22][23][24][25][26][27][28][29], to study problems such as graph isomorphism [30], quantum percolation [31][32][33], and to develop schemes for implementation of universal quantum computation [34,35], among others. Quantum walks are thus very versatile tools indeed, and their practical significance has been demonstrated by way of implementation in many quantum systems, such as NMR [36], integrated photonics [37][38][39], ion traps [40,41], and cold atoms [42].…”

Logarithmic expansion of many-body wave packets in random potentials

“…They serve as a single particle version of a quantum cellular automaton [37,38]. DTQW became a useful tool to study various systems and phenomena such as relativistic particles, artificial gauge fields [39,40], various topological phases [41][42][43], percolation problems [44,45], localization phenonema [46], and implementation of quantum information tasks [47][48][49], among others. DTQW were implemented experimentally using NMR devices [50], optical devices [51,52], in the IBM quantum computer [53], and in a trapped ion quantum computer [54].…”

Quantum walks and Dirac cellular automata on a programmable trapped-ion quantum computer

“…uantum walks (QWs) are the quantum analog of classical random walks, in which the walker steps forward or backward along a line based on a coin flip. In a QW, the walker proceeds in a quantum superposition of paths, and the resulting interference forms the basis of a wide variety of quantum algorithms, such as quantum search [1][2][3][4][5] , graph isomorphism problems [6][7][8] , ranking nodes in a network [9][10][11][12] , and quantum simulations, which mimic different quantum systems at the low and high energy scale [13][14][15][16][17][18][19][20][21][22] . In the discrete-time QW (DQW) 23,24 , a quantum coin operation is introduced to prescribe the direction in which the particle moves in position space at each discrete step.…”