Grover Search: We introduce the basics of discrete quantum walks, describing some of the underlying physics. One of the most important algorithms in quantum computing is Grover’s search algorithm, we show how one can implement this algorithm using a discrete walk on the arcs of a graph.
Discrete quantum walks are quantum analogues of classical random walks. They are an important tool in quantum computing and a number of algorithms can be viewed as discrete quantum walks, in particular Grover's search algorithm. These walks are constructed on an underlying graph, and so there is a relation between properties of walks and properties of the graph. This book studies the mathematical problems that arise from this connection, and the different classes of walks that arise. Written at a level suitable for graduate students in mathematics, the only prerequisites are linear algebra and basic graph theory; no prior knowledge of physics is required. The text serves as an introduction to this important and rapidly developing area for mathematicians and as a detailed reference for computer scientists and physicists working on quantum information theory.
In the sub-field of quantum algorithms, physicists and computer scientist take classical computing algorithms and principles and see if there is a more efficient or faster approach implementable on a quantum computer, i.e. a ”quantum advantage”. We take random walks, a widely applicable group of classical algorithms, and move them into the quantum computing paradigm. Additionally, an introduction to a popular quantum search algorithm called Grover’s search is included to guide the reader to the development of a quantum search algorithm using quantum random walks. To close the gap between algorithm and hardware, we will look at using neutral-atom (also known as cold-atom) quantum computing to physically implement two types of quantum random walks and Grover’s algorithm.
Quantum computing is a system of computation that exploits the quantum-mechanical nature of reality at nano-scales in order to perform certain operations in a way that can scale better than is possible with classical computing. While the hardware sideof quantum computing is still in its infancy, many quantum algorithms - sets of instructions to run on a quantum computer in order to complete particular tasks - have been designed and analyzed. One of the most famous of these is Grover’s algorithm, an algorithm for searching a database. Two types of generalizations of Grover’s algorithm have been studied: spatial search algorithms (algorithms for searching discrete spaces) and “analog” search algorithms (algorithms performed by the evolution of physical systems). We develop and analyze algorithms that are generalizations of both those generalizations – an analog search algorithm for searching continuous, rather than discrete, spaces.
Quantum walk search on graphs is primarily limited to real evolution operators, although there is no restriction for using the complex unitary matrix as the evolution operator. In the context of quantum search, several studies on the Grover’s search utilize the complex unitary matrix as an evolution operator, which allows for efficient searching of marked elements. However, the generalization of the quantum walk search to complex space has largely been overlooked. In this paper, we extend existing discrete-time coined quantum walk search algorithms on one- and two-dimensional periodic lattices to complex space by introducing two phase parameters, [Formula: see text], and [Formula: see text], into the coin and oracle operators, respectively, for four different discrete-time coined quantum walk search algorithms. One of the advantages of this generalization is that it enables the Grover and SKW coins with specific values of the phase parameters [Formula: see text], and [Formula: see text] to search for the exceptional configurations on a two-dimensional periodic lattice with high success probability, which is otherwise impossible to search using the real Grover and SKW coins operators.
Abstract In this paper, we study Grover’s search algorithm focusing on continuous-time quantum walk on graphs. We propose an alternative optimization approach to Grover’s algorithm on graphs that can be summarized as follows: Instead of finding specific graph topologies convenient for the related quantum walk, we fix the graph topology and vary the underlying graph Laplacians. As a result, we search for the most appropriate analytical structure on graphs endowed with fixed topologies yielding better search outcomes. We discuss strategies to investigate the optimality of Grover’s algorithm and provide an example with an easy tunable graph Laplacian to investigate our ideas.