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.
Journal reference
Journal of Computational and Applied Mathematics 491 (2027) 118059
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.
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.
Abstract We present a quantum algorithm for finding the minimum of a function based on multistep quantum computation, and apply the algorithm for solving optimization problems with continuous variables. We construct the state space of the problem by discretizing the variables of the problem, and divide the state space according to the function values of the vectors of the state space. By comparing the function values of the vectors with a series of threshold values in decreasing order, we construct a sequence of Hamiltonians where the search space of a Hamiltonian is nested in that of the previous one. By applying a multistep quantum computation process for finding the ground state of the last Hamiltonian, the optimal vector of the state space of the problem is located in a small search space and can be determined efficiently. One of the most difficult problems in optimization algorithms is that a trial vector is trapped in a deep local minimum while the global minimum is missed, this problem can be alleviated in our algorithm and the run time is proportional to the number of the steps of the algorithm, provided that the reduction rate of the search spaces is polynomial large. We discuss the implementation of the algorithm and test the algorithm for some test functions.
In classical computation, a problem can be solved in multiple steps where the calculated results of each step can be copied and used repeatedly.However, in quantum computation, it is difficult to realize a similar multistep computation process because the no-cloning theorem forbids making copies of an unknown quantum state perfectly.We find a method based on a quantum resonant transition to protect and reuse an unknown quantum state that encodes the calculated results of an intermediate step without copying it, and present a quantum algorithm that solves a problem via multistep quantum computation.We demonstrate that this algorithm can solve a type of structured search problems efficiently.
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.
The quantum SearchRank algorithm is a promising tool for a future quantum search engine based on PageRank quantization. However, this algorithm loses its functionality when the N / M ratio between the network size N and the number of marked nodes M is sufficiently large. We propose a modification of the algorithm, replacing the underlying Szegedy quantum walk with a semiclassical walk. To maintain the same time complexity as the quantum SearchRank algorithm we propose a simplification of the algorithm. This new algorithm is called randomized SearchRank, since it corresponds to a quantum walk over a randomized mixed state. The performance of the SearchRank algorithms is first analyzed on an example network and then statistically on a set of different networks of increasing size and different number of marked nodes. On the one hand, to test the search ability of the algorithms, it is computed how the probability of measuring the marked nodes decreases with N / M for the quantum SearchRank, but remarkably it remains at a high value around 0.9 for our semiclassical algorithms, solving the quantum SearchRank problem. The time complexity of the algorithms is also analyzed, obtaining a quadratic speedup with respect to the classical ones. On the other hand, the ranking functionality of the algorithms has been investigated, obtaining a good agreement with the classical PageRank distribution. Finally, the dependence of these algorithms on the intrinsic PageRank damping parameter has been clarified. Our results suggest that this parameter should be below a threshold so that the execution time does not increase drastically. Published by the American Physical Society 2024
Abstract The internet is one of the most valuable technologies invented to date. Among them, Google is the most widely used search engine. The PageRank algorithm is the backbone of Google search, ranking web pages according to relevance and recency. We employ quantum stochastic walks (QSWs) with the hope of bettering the classical PageRank (CPR) algorithm, which is based on classical continuous time random walks. We implement QSW via two schemes: only incoherence and dephasing with incoherence. PageRank using QSW with only incoherence or QSW with dephasing and incoherence best resolves degeneracies that are unresolvable via CPR and with a convergence time comparable to that for CPR, which is generally the minimum. For some networks, the two QSW schemes obtain a convergence time lower than CPR and an almost degeneracy-free ranking compared to CPR.