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arXiv:1305.1766

Google In A Photonic Lattice

Kallol Roy, Ji Liu, Le Luo, R. Srikanth, Tapan Mishra, Bhanu Pratap Das, Srinivas, T.

Published May 8, 2013• 2 citations
arXivHTMLPDF

Abstract

The quantum version of Google PageRank has recently been investigated by various groups and shown to be quadratically faster in time than the classical PageRank algorithm. In this paper we propose the implementation of Quantum PageRank by a stochastic quantum walk of correlated photons in a photonic waveguide lattice, where we allow the density matrix to evolve according to the Lindblad-Kossakowski master equation. This yields a stationary state, the diagonal of whose density matrix gives the page ranking.

DOI
10.48550/arxiv.1305.1766

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The quantum version of Google PageRank has recently been investigated by various groups and shown to be quadratically faster in time than the classical PageRank algorithm. In this paper we propose the implementation of Quantum PageRank by a stochastic quantum walk of correlated photons in a photonic waveguide lattice, where we allow the density matrix to evolve according to the Lindblad-Kossakowski master equation. This yields a stationary state, the diagonal of whose density matrix gives the page ranking.
The PageRank of a web page denotes popularity of a web page, higher the PageRank wider the popularity. The explosion of the number of web pages propel us to fasten the search time. The quantum version of Google PageRank has recently been investigated by various groups [1], [2] and shown to be quadratically faster in time than the classical PageRank algorithm. In this paper, we follow Quantum PageRank Class defined in [2]. We have modelled the web pages as quantum states and the search as a quantum stochastic walk. The evolution of the density matrix is computed using the Kossakowski-Lindblad master equation, which has reversible and irreversible evolution terms. The reversible term gives the coherent evolution, while the irreversible term adds noise to the system. The stochastic convergence of the diagonal elements of the density matrix gives the Quantum PageRank values, and to observe this it is essential to get consistent PageRank ordering, after the quantum fluctuations have been stabilized (insignificant). In our paper, we propose a method for faster convergence of the Quantum PageRanks by adding limited quantity of noise values during computation of the Kossakowski-Lindblad master equation, and the results obtained are an improvement over [2], where the quantum fluctuations do not stabilize and the ordering of PageRanks is not consistent. Addition of noise in limited amount during computation helps in limiting quantum fluctuations and in convergence of the Quantum PageRanks.
Integrated quantum photonics show monolithic waveguide chips to be a promising platform for realizing the next generation of quantum optical circuits. This work proposes the implementation of quantum page Rank algorithm on a photonic waveguide lattice. Our contributions are as follows: Continuous-time quantum stochastic walk(QSW)-an alternate paradigm of quantum computing, is a hybrid quantum walk that incorporates both unitary and non-unitary effects. We propose the use of QSW which necessitates the hopping of the quantum crawler on a directed graph, for the quantum page Rank problem. We propose the implementation of quantum page Rank on a photonic waveguide lattice, where we allow the density matrix to evolve according to the Lindblad-Kossakowski master equation, the diagonal of which gives the quantum page Rank. We have also shown the use of the metric of positional Kolmogorov Complexity as an efficient tool for determining whether or not the quantum channel has been compromised. We appositionally encode multi-photon decoy pulses within the stream of single photon pulses. This positional encoding is chosen in such a way as to have low Kolmogorov complexity. The PNS attack on the multi-photon decoy pulses causes a dip in the ratio of the transmittance of the decoy pulses to the signal pulses in the conventional analysis.
ABSTRACT In the modern day , we are witnessing two complementary trends, exponential growth in data and shrinking of chip size. The Data is approaching to 44 zettabytes by 2020 and the chips are now available with 10nm technology. The hyperconnectivity between machine-to-machine and humanto- machine creates multi-dimensional data which is more complex. Our thesis addresses the quantum meta layer abstraction which provides the interface to the Application layer to design quantum and classical algorithms. The first part of the thesis addresses the quantum algorithms and second part address classical algorithms running on top of quantum meta layer. In the first part of our thesis we explored quantum stochastic algorithm for ranking Quantum Webpages, analogous to the classical Google PageRank. The architecture is a six-waveguide photonic lattice that runs finely-tuned quantum stochastic walk. The evolution of density matrix solves the ranking of quantum webpages. We force the photon stochastic walk for quantum PageRank by matching the entries of Google matrix with parameters of the Kossakowski-Lindblad master equation. We have done extensive simulation to observe the density matrix evolution with different parameter settings. We have used noise in the Kossakowski-Lindblad master equation to break the symmetry (reciprocity) property of quantum system, which helps in distinguishable measurement of the quantum PageRank. We next propose a new quantum deep learning with photonic lattice waveguide as a feedforward neural network. The proposed deep photonic neural network uses the quantum properties for learning. The hidden layers of our deep photonic neural network can be designed to learn object representation and mentains the quantum quantum properties for longer time for optimal learning. The second part of the thesis discusses the data based learning. We have used data graph method which captures the system representation. The proposed data graph model captures and encodes the data efficiently and then the data graph is updated and trained with new data to provide efficient predictions. The model retains the previously learned knowledge by transfer learning and improves it with new training. The proposed method is highly adaptive and scalable for different real-time scenarios. Data graph models the system where every node (object) is associated with data and if two objects are related then they are linked with a data edge. The proposed algorithm is an incremental algorithm which learns hidden objects and hidden relationships through the data pattern over time and updates the model accordingly. We have used algebraic graph transformation methods to trigger the mutation of the Data Graph. This new updated Data Graph behaves differently for the data it observes. We explore more into machine learning algorithms and have proposed a complete framework to predict the state of the system based on the system parameters. We have proposed the discretization of the data points using the symbol algebra and used Bayesian machine…
PageRank is an algorithm used by Google Search to rank web pages in their search engine results. An important step for quantum networks is to quantize the classical protocol as quantum mechanics provides computational resources that can be used to outperform classical algorithms. In this paper, we experimentally realize continuous-time quantum walks for directed graphs with non-Hermitian adjacency matrices by using linear optical circuits and single photons. We find that the node classical centrality in a directed graph is correlated with the maximum node probability resulting from a continuous-time quantum walk and then demonstrate PageRank. Our work opens up an avenue of applications of quantum information in real-life tasks.
This article presents a new quantum PageRank algorithm on graphs using discrete-time open quantum walks. Google's PageRank is a widely used algorithm for ranking the web pages on the world wide web in classical computation. From a broader perspective, it is also a fundamental measure for quantifying the importance of vertices in a network. Similarly, the new quantum PageRank also serves to quantify the significance of a network's vertices. In this work, we extend the concept of discrete-time open quantum walk on arbitrary directed and undirected graphs by utilizing the Weyl operators as Kraus operators. This new model of quantum walk is useful for building up the quantum PageRank algorithm, discussed in this article. We compare the classical PageRank and the newly defined quantum PageRank for different types of complex networks, such as the scale-free network, Erdős-Rényi random network, Watts-Strogatz network, spatial network, Zachary Karate club network, GNC, GN, GNR networks, Barabási and Albert network, etc. In addition, we study the convergence of the quantum PageRank process and its dependency on the damping factor α. We observe that this quantum PageRank procedure is faster than many other proposals reported in the literature.
We present a quantum algorithm for obtaining a quantum state that encodes the PageRank vector of the Google matrix through multistep quantum resonant transition (mQRT). In the algorithm, the PageRank vector is encoded in the ground state of a problem Hamiltonian associated with the Google matrix. By dividing the web graph corresponding to the Google matrix into a series of subgraphs with nested structure, we construct a sequence of Hamiltonians based on the subgraphs to form a Hamiltonian evolution path from a simple initial Hamiltonian to the problem Hamiltonian. The ground state of the problem Hamiltonian is obtained by going through ground states of the intermediate Hamiltonians via QRT step by step. This algorithm requires only one ancillary qubit, and the runtime of the algorithm is proportional to the number of steps. It provides a new way for efficiently obtaining the quantum state of the PageRank vector of large-scale networks.