We introduce the characterization of a class of quantum PageRank algorithms in a scenario in which some kind of quantum network is realizable out of the current classical internet web, but no quantum computer is yet available. This class represents a quantization of the PageRank protocol currently employed to list web pages according to their importance. We have found an instance of this class of quantum protocols that outperforms its classical counterpart and may break the classical hierarchy of web pages depending on the topology of the web.

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Google in a Quantum Network

The quantum walk formalism is a widely used and highly successful framework for modeling quantum systems, such as simulations of the Dirac equation, different dynamics in both the low and high energy regime, and for developing a wide range of quantum algorithms. Here we present the circuit-based implementation of a discrete-time quantum walk in position space on a five-qubit trapped-ion quantum processor. We encode the space of walker positions in particular multi-qubit states and program the system to operate with different quantum walk parameters, experimentally realizing a Dirac cellular automaton with tunable mass parameter. The quantum walk circuits and position state mapping scale favorably to a larger model and physical systems, allowing the implementation of any algorithm based on discrete-time quantum walks algorithm and the dynamics associated with the discretized version of the Dirac equation.

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

Quantum walk and its application domains: A systematic review

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

Quantum walk has been regarded as a primitive to universal quantum computation. By using the operations required to describe the single particle discrete-time quantum walk on a position space we demonstrate the realization of the universal set of quantum gates on two- and three-qubit systems. The idea is to utilize the effective Hilbert space of the single qubit and the position space on which it evolves in order to realize multi-qubit states and universal set of quantum gates on them. Realization of many non-trivial gates and engineering arbitrary states is simpler in the proposed quantum walk model when compared to the circuit based model of computation. We will also discuss the scalability of the model and some propositions for using lesser number of qubits in realizing larger qubit systems.

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Universal quantum computing using single-particle discrete-time quantum walk

We present a quantum algorithm for ranking the nodes of 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 ranks matches well with the hierarchy of classical ranks for directed trees and other acyclic networks. For cyclic networks, however, the hierarchy of quantum ranks does not exactly match the hierarchy of the classical ranks. 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 chemical complexes and rank active centres in the order of reactivities. Since discrete-time quantum walks are implementable on current quantum processing systems, this algorithm will also be of practical relevance in the analysis of quantum architecture.

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

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.

Roopesh Mangal

Papers

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

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