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arXiv:1710.05084 · Physical Review A

Finding paths in tree graphs with a quantum walk

Daniel Koch, Mark S. Hillery

Published Jan 12, 2018• 22 citations
arXivHTMLPDF

Abstract

We analyze the potential for different types of searches using the formalism of scattering random walks on quantum computers. Given a particular type of graph consisting of nodes and connections, a ``tree maze,'' we would like to find a selected final node as quickly as possible, faster than any classical search algorithm. We show that this can be done using a quantum random walk, both through numerical calculations as well as by using the eigenvectors and eigenvalues of the quantum system.

DOI
10.1103/physreva.97.012308

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We analyze the potential for different types of searches using the formalism of scattering random walks on quantum computers. Given a particular type of graph consisting of nodes and connections, a ``tree maze,'' we would like to find a selected final node as quickly as possible, faster than any classical search algorithm. We show that this can be done using a quantum random walk, both through numerical calculations as well as by using the eigenvectors and eigenvalues of the quantum system.
This paper investigates the application of quantum walk algorithms to large-scale graph search problems. Classical algorithms often struggle with the computational complexity of exploring vast graphs, particularly when searching for specific patterns or nodes. Quantum walks, inspired by the probabilistic nature of random walks, provide a promising alternative by utilizing quantum superposition and interference to accelerate the search process. We explore the theoretical framework and potential advantages of employing quantum walks for graph traversal, focusing on scenarios where exponential speedups are theoretically possible. This work aims to demonstrate the potential of quantum computing to tackle computationally demanding graph search tasks that are currently intractable for classical methods. The core concept leverages the ability to simultaneously explore multiple paths within a graph, drastically reducing the search time compared to sequential exploration. We present a foundational understanding of quantum walk implementation and analyze its suitability for various graph search algorithms.
This paper explores the potential of quantum-inspired algorithms for graph search problems. Traditional random walk approaches often suffer from inefficiencies, particularly in complex graphs. We propose a novel framework that leverages concepts from quantum walks – specifically interference and tunneling – within a classical computational environment. Our algorithms dynamically adjust the probability distribution of node visits based on graph connectivity and the history of previous steps, mimicking the probabilistic nature of quantum mechanics. We demonstrate that this approach can, in certain scenarios, outperform standard random walks, offering a more adaptable and potentially faster search strategy. The core of the method lies in approximating the quantum effects through carefully designed classical simulations, without requiring actual quantum hardware. We present a mathematical formulation of the core algorithm and discuss its key parameters, focusing on the balance between classical simulation fidelity and computational cost. The results suggest a promising direction for graph search optimization, drawing inspiration from the principles of quantum mechanics.
The scattering-quantum-random-walk scheme has found success as a basis for search algorithms on highly symmetric graph structures. In this paper we examine its effectiveness at locating a specially marked vertex on square grid graphs, consisting of N2{N}^{2}N2 nodes. We simulate these quantum systems using classical computational methods and find that the probability distributions that arise are very favorable for a hybrid quantum-classical algorithm. We then examine how this hybrid algorithm handles varying types of randomness in both location of the special vertex and later random obstacles placed throughout the geometry, showing that the algorithm is resilient to both cases.
Using numerical simulation, we measured the performance of several potential quantum algorithms, based on quantum random walks, to solve Boolean satisfiability (SAT) problems. We develop the fundamentals of quantum computing and the theory of classical algorithms to indicate how these algorithms could be implemented. We also discuss the development of quantum random walks and the principles that go into creating them, presenting existing work on search algorithms using coined discrete-time quantum random walks. Although our quantum algorithms do not solve SAT problems faster than may be done classically, as a step toward that objective, we present the first example of a quantum walk on a directed graph that has a significant increase in speed over the analogous classical walk.
Quantum walk, the quantum counterpart of random walk, is an important model and widely studied to develop new quantum algorithms. This paper studies the relationship between the continuous-time quantum walk and the symmetry of a graph, especially that of a tree. Firstly, we prove in mathematics that the symmetry of a graph is highly related to quantum walk. Secondly, we propose an algorithm based on the continuous-time quantum walk to compute the symmetry of a tree. Our algorithm has better time complexity O( N 3 ) than the current best algorithm. Finally, through testing three types of 10024 trees, we find that the symmetry of a tree can be found with an extremely high efficiency with the help of the continuous-time quantum walk.
Quantum walks, being the quantum analogue of classical random walks, are expected to provide a fruitful source of quantum algorithms. A few such algorithms have already been developed, including the `glued trees' algorithm, which provides an exponential speedup over classical methods, relative to a particular quantum oracle. Here, we discuss the possibility of a quantum walk algorithm yielding such an exponential speedup over possible classical algorithms, without the use of an oracle. We provide examples of some highly symmetric graphs on which efficient quantum circuits implementing quantum walks can be constructed, and discuss potential applications to quantum search for marked vertices along these graphs.