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Quantum and classical algorithms for graph classification and search problems

B. L. Douglas

Published Jan 1, 2011• 0 citations
OpenAlex

Abstract

[Truncated abstract] This thesis is split into three parts, each addressing the overarching theme of applying graph theoretic techniques to classi cation and search problems. Part I explores this theme in the context of quantum algorithms, in particular involving quantum walks, coined by Aharonov et al. in 1993 [1]. As a quantum analogue to classical random walks, quantum walks represent a promising candidate for quantum algorithms providing speed-ups over known classical methods. I will discuss two categories of quantum walk algorithms; quantum search and graph isomorphism. In particular, the e ciency of quantum walk based search algorithms on various families of graphs will be investigated, linking both e ciency of quantum circuit implementation and the complexity of the search problem to the symmetry of the underlying graphs. Several families of graphs are presented that are amenable to efficient quantum walk based searching, together with explicit quantum circuit implementations. In the case of the n-cube, I show that O(p2n) steps of a quanum walk are suffcient to nd a marked sub-d-cube (where d

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[Truncated abstract] This thesis is split into three parts, each addressing the overarching theme of applying graph theoretic techniques to classi cation and search problems. Part I explores this theme in the context of quantum algorithms, in particular involving quantum walks, coined by Aharonov et al. in 1993 [1]. As a quantum analogue to classical random walks, quantum walks represent a promising candidate for quantum algorithms providing speed-ups over known classical methods. I will discuss two categories of quantum walk algorithms; quantum search and graph isomorphism. In particular, the e ciency of quantum walk based search algorithms on various families of graphs will be investigated, linking both e ciency of quantum circuit implementation and the complexity of the search problem to the symmetry of the underlying graphs. Several families of graphs are presented that are amenable to efficient quantum walk based searching, together with explicit quantum circuit implementations. In the case of the n-cube, I show that O(p2n) steps of a quanum walk are suffcient to nd a marked sub-d-cube (where d
Given the extensive application of classical random walks to classical algorithms in a variety of fields, their quantum analogue in quantum walks is expected to provide a fruitful source of quantum algorithms. So far, however, such algorithms have been scarce. In this work, we enumerate some important differences between quantum and classical walks, leading to their markedly different properties. We show that for many practical purposes, the implementation of quantum walks can be efficiently achieved using a classical computer. We then develop both classical and quantum graph isomorphism algorithms based on discrete-time quantum walks. We show that they are effective in identifying isomorphism classes of large databases of graphs, in particular groups of strongly regular graphs. We consider this approach to represent a promising candidate for an efficient solution to the graph isomorphism problem, and believe that similar methods employing quantum walks, or derivatives of these walks, may prove beneficial in constructing other algorithms for a variety of purposes.
This thesis is on quantum algorithms. It has three main themes: \n(1) quantum walk based search algorithms, \n(2) quantum rejection sampling, and \n(3) the Boolean function hidden shift problem. \nThe first two parts deal with generic techniques for constructing quantum algorithms, and the last part is on quantum algorithms for a specific algebraic problem. \n \nIn the first part of this thesis we show how certain types of random walk search algorithms can be transformed into quantum algorithms that search quadratically faster. More formally, given a random walk on a graph with an unknown set of marked vertices, we construct a quantum walk that finds a marked vertex in a number of steps that is quadratically smaller than the hitting time of the random walk. The main idea of our approach is to interpolate the random walk from one that does not stop when a marked vertex is found to one that stops. The quantum equivalent of this procedure drives the initial superposition over all vertices to a superposition over marked vertices. We present an adiabatic as well as a circuit version of our algorithm, and apply it to the spatial search problem on the 2D grid. \n \nIn the second part we study a quantum version of the problem of resampling one probability distribution to another. More formally, given query access to a black box that produces a coherent superposition of unknown quantum states with given amplitudes, the problem is to prepare a coherent superposition of the same states with different specified amplitudes. Our main result is a tight characterization of the number of queries needed for this transformation. By utilizing the symmetries of the problem, we prove a lower bound using a hybrid argument and semidefinite programming. For the matching upper bound we construct a quantum algorithm that generalizes the rejection sampling method first formalized by von~Neumann in~1951. We describe quantum algorithms for the linear equations problem and quantum Metropolis sampling as applications of quantum rejection sampling. \n \nIn the third part we consider a hidden shift problem for Boolean functions: given oracle access to f(x+s), where f(x) is a known Boolean function, determine the hidden shift s. We construct quantum algorithms for this problem using the "pretty good measurement" and quantum rejection sampling. Both algorithms use the Fourier transform and their complexity can be expressed in terms of the Fourier spectrum of f (in particular, in the second case it relates to "water-filling" of the spectrum). We also construct algorithms for variations of this problem where the task is to verify a given shift or extract only a single bit of information about it.
This paper explores the application of quantum walk algorithms for graph search problems. Classical graph search algorithms, such as Dijkstra's algorithm, frequently encounter limitations when dealing with large graphs, often exhibiting performance bottlenecks due to exhaustive exploration of the graph's nodes. We propose a novel approach utilizing quantum walks to address these challenges. Quantum walks exploit quantum superposition and interference to facilitate a more efficient exploration of the graph space compared to their classical counterparts. The theoretical framework presented investigates the potential for exponential speedups in graph search operations. We formally define the quantum walk model and its implementation for graph traversal, outlining the key steps involved in constructing a quantum algorithm. This work focuses on the theoretical aspects of the algorithm, providing a foundation for future experimental investigations and optimization.
We present several families of graphs that allow both efficient quantum walk implementations and efficient quantum walk based search algorithms. For these graphs, we construct quantum circuits that explicitly implement the full quantum walk search algorithm, without reference to a `black box' oracle. These circuits provide a practically implementable method to explore quantum walk based search algorithms with the aim of eventual real-world applications. We also provide a numerical analysis of a quantum walk based search along a twisted toroid family of graphs, which requires O(n\sqrt{n}n​ log(nnn)) elementary 2-qubit quantum gate operations to find a marked node.
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.
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.