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Quantum Information and Computation

A panoply of quantum algorithms

Bartholomew Furrow

Published Jan 1, 2010• 7 citations
Publisher

Abstract

This thesis’ aim is to explore improvements to, and applications of, a fundamental quantum algorithm invented by Grover. Grover’s algorithm is a basic tool that can be applied to a large number of problems in computer science, creating quantum algorithms that are polynomially faster than fastest known and fastest possible classical algorithms that solve the same problems. Our goal in this thesis is to make these techniques readily accessible to those without a strong background in quantum physics: we achieve this by providing a set of tools, each of which makes use of Grover’s algorithm or similar techniques, that can be used as subroutines in many quantum algorithms. The tools we provide are carefully constructed: they are easy to use, and they are asymptotically faster than the best tools previously available. The tools that we supersede include algorithms by Boyer, Brassard, Hoyer and Tapp, Buhrman, Cleve, de Witt and Zalka and Durr and Hoyer. After creating our tools, we create several new quantum algorithms, each of which is faster than the fastest known classical algorithm that accomplishes the same aim, and some of which are faster than the fastest possible classical algorithm. These algorithms come from graph theory, computational geometry and dynamic programming. We discuss a breadth-first search that is faster than (edges) (the classical limit) in a dense graph, maximum-points-on-a-line in (N3/2 lgN) (faster than the fastest classical algorithm known), as well as several other algorithms that are similarly illustrative of solutions in some class of problem. Through these new algorithms we illustrate the use of our tools, working to encourage their use and the study of quantum algorithms in general.

DOI
10.14288/1.0085242

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This thesis’ aim is to explore improvements to, and applications of, a fundamental quantum algorithm invented by Grover. Grover’s algorithm is a basic tool that can be applied to a large number of problems in computer science, creating quantum algorithms that are polynomially faster than fastest known and fastest possible classical algorithms that solve the same problems. Our goal in this thesis is to make these techniques readily accessible to those without a strong background in quantum physics: we achieve this by providing a set of tools, each of which makes use of Grover’s algorithm or similar techniques, that can be used as subroutines in many quantum algorithms. The tools we provide are carefully constructed: they are easy to use, and they are asymptotically faster than the best tools previously available. The tools that we supersede include algorithms by Boyer, Brassard, Hoyer and Tapp, Buhrman, Cleve, de Witt and Zalka and Durr and Hoyer. After creating our tools, we create several new quantum algorithms, each of which is faster than the fastest known classical algorithm that accomplishes the same aim, and some of which are faster than the fastest possible classical algorithm. These algorithms come from graph theory, computational geometry and dynamic programming. We discuss a breadth-first search that is faster than (edges) (the classical limit) in a dense graph, maximum-points-on-a-line in (N3/2 lgN) (faster than the fastest classical algorithm known), as well as several other algorithms that are similarly illustrative of solutions in some class of problem. Through these new algorithms we illustrate the use of our tools, working to encourage their use and the study of quantum algorithms in general.
This paper's aim is to explore improvements to, and applications of, a fundamental quantum algorithm invented by Grover\cite{grover}. Grover's algorithm is a basic tool that can be applied to a large number of problems in computer science, creating quantum algorithms that are polynomially faster than fastest known and fastest possible classical algorithms that solve the same problems. Our goal in this paper is to make these techniques readily accessible to those without a strong background in quantum physics: we achieve this by providing a set of tools, each of which makes use of Grover's algorithm or similar techniques, which can be used as subroutines in many quantum algorithms.}{The tools we provide are carefully constructed: they are easy to use, and in many cases they are asymptotically faster than the best tools previously available. The tools we build on include algorithms by Boyer, Brassard, Hoyer and Tapp, Buhrman, Cleve, de Witt and Zalka and Durr and Hoyer.}{After creating our tools, we create several new quantum algorithms, each of which is faster than the fastest known deterministic classical algorithm that accomplishes the same aim, and some of which are faster than the fastest possible deterministic classical algorithm. These algorithms solve problems from the fields of graph theory and computational geometry, and some employ dynamic programming techniques. We discuss a breadth-first search that is faster than Θ(edges)\Theta(\text{edges})Θ(edges) (the classical limit) in a dense graph, maximum-points-on-a-line in O(N3/2lg⁡N)O(N^{3/2}\lg N)O(N3/2lgN) (faster than the fastest classical algorithm known), as well as several other algorithms that are similarly illustrative of solutions in some class of problem. Through these new algorithms we illustrate the use of our tools, working to encourage their use and the study of quantum algorithms in general.
We create a variety of new quantum algorithms that use Grover's algorithm and similar techniques to give polynomial speedups over their classical counterparts. We begin by introducing a set of tools that carefully minimize the impact of errors on running time; those tools provide us with speedups to already-published quantum algorithms, such as improving Durr, Heiligman, Hoyer and Mhalla's algorithm for single-source shortest paths [quant-ph/0401091] by a factor of lg N. The algorithms we construct from scratch have a range of speedups, from O(E)->O(sqrt(VE lg V)) speedups in graph theory to an O(N^3)->O(N^2) speedup in dynamic programming.
Quantum algorithms have the potential to demonstrate that for some problems quantum computation is more efficient than classical computation. A goal of quantum computing is to determine for which problems quantum computers are faster than classical computers. In our survey we present recent quantum algorithms for basic problems from graph and algebra theory. The quantum algorithms for these problems use a combination of Grovers search algorithms, quantum amplitude amplification and quantum random walks. These quantum algorithms are faster than the best known classical algorithms for the corresponding problems. Keywords: Quantum computing, quantum algorithms, graph theory, algebra, quantum query complexity
This thesis studies strengths and weaknesses of quantum computers. In the first part we present three contributions to quantum algorithms. 1) consider a search space of N elements. One of these elements is "marked" and our goal is to find this. We describe a quantum algorithm to solve this problem using essentially sqrt{N} queries and other operations, improving over the gate count of Grover's algorithm. 2) We give a succinct characterization of quantum algorithms in terms of polynomials, and develop a new technique for showing upper and lower bounds on quantum query complexity based on this. 3) One generic technique used to compute the minimum of a given function is "gradient descent". We present a quantum gradient-calculation algorithm and a gradient-optimization algorithm that is quadratically faster than classical algorithms. In the second part of this thesis we look at quantum learning theory. 1) We survey quantum learning theory, describing the main results known for three models of learning, using classical as well as quantum data: exact learning from membership queries, the probably approximately correct (PAC) learning model, and the agnostic learning model. 2) We then consider if a quantum learner can PAC-learn a given concept class from fewer quantum examples. We give a negative answer by showing that quantum examples are not more powerful than classical random examples, both for PAC learning and for agnostic learning.