Related papers: Power law scaling for the adiabatic algorithm for …
Quantum Algorithms have long captured the imagination of computer scientists and physicists primarily because of the speed up achieved by them over their classical counterparts using principles of quantum mechanics. Entanglement is believed…
Large-scale eigenvalue problems pose a significant challenge to classical computers. While there are efficient quantum algorithms for unitary or Hermitian matrices, eigenvalue problems for non-normal matrices remain open in quantum…
An efficient algorithm for computing eigenvectors of a matrix of integers by exact computation is proposed. The components of calculated eigenvectors are expressed as polynomials in the eigenvalue to which the eigenvector is associated, as…
We discuss a definition of robust dominant eigenvector of a family of stochastic matrices. Our focus is on application to ranking problems, where the proposed approach can be seen as a robust alternative to the standard PageRank technique.…
The PageRank is a popularity measure designed by Google to rank Web pages. Experiments confirm that the PageRank obeys a `power law' with the same exponent as the In-Degree. This paper presents a novel mathematical model that explains this…
We present a stochastic quantum computing algorithm that can prepare any eigenvector of a quantum Hamiltonian within a selected energy interval $[E-\epsilon, E+\epsilon]$. In order to reduce the spectral weight of all other eigenvectors by…
Eigenfactor.org, a journal evaluation tool which uses an iterative algorithm to weight citations (similar to the PageRank algorithm used for Google) has been proposed as a more valid method for calculating the impact of journals. The…
Kernel methods augmented with random features give scalable algorithms for learning from big data. But it has been computationally hard to sample random features according to a probability distribution that is optimized for the data, so as…
We build up a directed network tracing links from a given integer to its divisors and analyze the properties of the Google matrix of this network. The PageRank vector of this matrix is computed numerically and it is shown that its…
We consider the problem of search of an unstructured list for a marked element, when one is given advice as to where this element might be located, in the form of a probability distribution. The goal is to minimise the expected number of…
Eigenvectors of large matrices (and graphs) play an essential role in combinatorics and theoretical computer science. The goal of this survey is to provide an up-to-date account on properties of eigenvectors when the matrix (or graph) is…
This paper introduces an efficient algorithm for finding the dominant generalized eigenvectors of a pair of symmetric matrices. Combining tools from approximation theory and convex optimization, we develop a simple scalable algorithm with…
Complex networks have emerged as a simple yet powerful framework to represent and analyze a wide range of complex systems. The problem of ranking the nodes and the edges in complex networks is critical for a broad range of real-world…
Search-base algorithms have widespread applications in different scenarios. Grover's quantum search algorithms and its generalization, amplitude amplification, provide a quadratic speedup over classical search algorithms for unstructured…
We study the typical behavior of a generalized version of Google's PageRank algorithm on a large family of inhomogeneous random digraphs. This family includes as special cases directed versions of classical models such as the…
Quantum walks have been useful for designing quantum algorithms that outperform their classical versions for a variety of search problems. Most of the papers, however, consider a search space containing a single marked element only. We show…
Over the last decade, PageRank has gained importance in a wide range of applications and domains, ever since it first proved to be effective in determining node importance in large graphs (and was a pioneering idea behind Google's search…
We demonstrate the possibility of (sub)exponential quantum speedup via a quantum algorithm that follows an adiabatic path of a gapped Hamiltonian with no sign problem. This strengthens the superpolynomial separation recently proved by…
We introduce a new framework for web page ranking -- reinforcement ranking -- that improves the stability and accuracy of Page Rank while eliminating the need for computing the stationary distribution of random walks. Instead of relying on…
It has previously been established that adiabatic quantum computation, operating based on a continuous Zeno effect due to dynamical phases between eigenstates, is able to realise an optimal Grover-like quantum speedup. In other words, is…