Quantum query complexity of minor-closed graph properties

We study the quantum query complexity of minor-closed graph properties, which include such problems as determining whether an $n$-vertex graph is planar, is a forest, or does not contain a path of a given length. We 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}). To establish this, we prove an adversary lower bound using a detailed analysis of the structure of minor-closed properties with respect to forbidden topological minors and forbidden subgraphs. On the other hand, we show that minor-closed properties (and more generally, sparse graph properties) that can be characterized by finitely many forbidden subgraphs can be solved strictly faster, in o(n^{3/2}) queries. Our algorithms are a novel application of the quantum walk search framework and give improved upper bounds for several subgraph-finding problems.Comment: v1: 25 pages, 2 figures. v2: 26 page

Efficient algorithms in quantum query complexity

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. For 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. In 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. We 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. Finally, 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

Span programs and quantum algorithms for st-connectivity and claw detection

We introduce a span program that decides st-connectivity, and generalize the span program to develop quantum algorithms for several graph problems. First, we give an algorithm for st-connectivity that uses O(n d^{1/2}) quantum queries to the n x n adjacency matrix to decide if vertices s and t are connected, under the promise that they either are connected by a path of length at most d, or are disconnected. We also show that if T is a path, a star with two subdivided legs, or a subdivision of a claw, its presence as a subgraph in the input graph G can be detected with O(n) quantum queries to the adjacency matrix. Under the promise that G either contains T as a subgraph or does not contain T as a minor, we give O(n)-query quantum algorithms for detecting T either a triangle or a subdivision of a star. All these algorithms can be implemented time efficiently and, except for the triangle-detection algorithm, in logarithmic space. One of the main techniques is to modify the st-connectivity span program to drop along the way "breadcrumbs," which must be retrieved before the path from s is allowed to enter t.Comment: 18 pages, 4 figure

Quantum Query Complexity of Subgraph Containment with Constant-sized Certificates

We study the quantum query complexity of constant-sized subgraph containment. Such problems include determining whether an $n$-vertex graph contains a triangle, clique or star of some size. For a general subgraph $H$ with $k$ vertices, we show that $H$ containment can be solved with quantum query complexity $O(n^{2-\frac{2}{k}-g(H)})$, with $g(H)$ a strictly positive function of $H$. This is better than \tilde{O}\s{n^{2-2/k}} by Magniez et al. These results are obtained in the learning graph model of Belovs.Comment: 14 pages, 1 figure, published under title:"Quantum Query Complexity of Constant-sized Subgraph Containment

Applications of the Adversary Method in Quantum Query Algorithms

In the thesis, we use a recently developed tight characterisation of quantum query complexity, the adversary bound, to develop new quantum algorithms and lower bounds. Our results are as follows: * We develop a new technique for the construction of quantum algorithms: learning graphs. * We use learning graphs to improve quantum query complexity of the triangle detection and the $k$-distinctness problems. * We prove tight lower bounds for the $k$-sum and the triangle sum problems. * We construct quantum algorithms for some subgraph-finding problems that are optimal in terms of query, time and space complexities. * We develop a generalisation of quantum walks that connects electrical properties of a graph and its quantum hitting time. We use it to construct a time-efficient quantum algorithm for 3-distinctness.Comment: PhD Thesis, 169 page