We illustrate how quantum algorithms leveraging linear algebra can be used to solve important problems more efficiently than classical computers and then introduce a new family of quantum algorithms based on graph theory that naturally scale to massive data. We first show how this class of quantum graph search algorithms allows us to scale classical graph-search algorithms such as Breadth-First Search, Depth-First Search, and Minimum Spanning Tree Algorithm to massive graphs. We then present a new algorithm for finding a Steiner Tree using quantum annealing. Experimentally, we demonstrate quantum speedup using a 49-qubit device, and also classically search for Steiner Trees up to 3000 vertices on ten 1-TB nodes by exploiting the I/O scaling properties of our algorithm. We also discuss the application of these results to understanding better classical algorithms based on two-dimensional matrices and graphs. Finally, we propose several directions for future research in this area.
The applications of linear algebra have grown from a mere handful to literally hundreds of thousands, including number theory, mathematical economics, computer science, applied physics, and even the social sciences. Quantum information theory is one of very few disciplines within which the understanding has been pushed in new directions rather than deepening what we already know. This book is an advanced introduction to quantum information theory based on an original approach that uses linear algebra as its basic tool.
introduction to quantum algorithms via linear algebra pdf
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