In this thesis we present new quantum algorithms for graph and algebra problems. Our quantum algorithms for these problems use a combination of Grover search, amplitude amplification and quantum walk search. The quantum algorithms are faster than the best known classical algorithms for the corresponding problems.
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’ 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) (the classical limit) in a dense graph, maximum-points-on-a-line in 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.
[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
In this thesis we provide new upper and lower bounds on the quantum query complexity of a diverse set of problems. Specifically, we study quantum algorithms for Hamiltonian simulation, matrix multiplication, oracle identification, and graph-property recognition. \n \nFor the Hamiltonian simulation problem, we provide a quantum algorithm with query complexity sublogarithmic in the inverse error, an exponential improvement over previous methods. Our algorithm is based on a new quantum algorithm for implementing unitary matrices that can be written as linear combinations of efficiently implementable unitary gates. This algorithm uses a new form of ``oblivious amplitude amplification'' that can be applied even though the reflection about the input state is unavailable. \n \nIn the oracle identification problem, we are given oracle access to an unknown N-bit string x promised to belong to a known set of size M, and our task is to identify x. We present the first quantum algorithm for the problem that is optimal in its dependence on N and M. Our algorithm is based on ideas from classical learning theory and a new composition theorem for solutions of the filtered gamma_2-norm semidefinite program. \n \nWe then study the quantum query complexity of matrix multiplication and related problems over rings, semirings, and the Boolean semiring in particular. Our main result is an output-sensitive algorithm for Boolean matrix multiplication that multiplies two n x n Boolean matrices with query complexity O(n sqrt{l}), where l is the sparsity of the output matrix. The algorithm is based on a reduction to the graph collision problem and a new algorithm for graph collision. \n \nFinally, we study the quantum query complexity of minor-closed graph properties and show that most minor-closed properties---those that cannot be characterized by a finite set of forbidden subgraphs---have quantum query complexity Theta(n^{3/2}) and those that do have such a characterization can be solved strictly faster, with o(n^{3/2}) queries. Our lower bound is based on a detailed analysis of the structure of minor-closed properties with respect to forbidden topological minors and forbidden subgraphs. Our algorithms are a novel application of the quantum walk search framework and give improved upper bounds for several subgraph-finding problems.