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

Quantum Query Algorithms

Elektroniskā versija nesatur pielikumusLELDE LĀCE KVANTU VAICĀJOŠIE ALGORITMI ANOTĀCIJA Kvantu skaitļošana ir datorzinātņu apakšnozare, kurā tiek izmantotas kvantu mehānikas īpatnības, lai efektīvāk risinātu skaitļošanas uzdevumus. Šajā darbā tiek aplūkoti kvantu vaicājošie algoritmi Bula funkciju rēķināšanai. Darba sākumā tiek pierādīti kvantu algoritmu apakšējie novērtējumi dažādām funkcijām, kas apraksta grafu problēmas. Promocijas darba galvenais uzdevums ir izveidot efektīvus kvantu vaicājošos algoritmus. Ir nodefinēts kā veidot precīzus kvantu vaicājošos algoritmus ar sarežģītību n-1, 2n/3 un n/2. Darba turpinājumā tiek analizēti nedeterminētie kvantu algoritmi ar vienu jautājumu, to veidošanas iespējas un īpašības. Promocijas darbā tiek definēts jauns kvantu vaicājošo algoritmu veids - kvantu vaicājošie algoritmi ar pēcatlasi un tiek pierādīta šo algoritmu saistība ar nedeterminētajiem kvantu vaicājošajiem algoritmiem.LELDE LĀCE QUANTUM QUERY ALGORITHMS ANNOTATION Quantum computing is the subfield of computer science that aims to employ effects of quantum mechanics to efficiently perform computational tasks. The main research object of this work is quantum query model to compute Boolean functions. At first we prove higher lower bounds of quantum query algorithms for some of graph problems. Main purpose of the research is to find quantum query algorithms with complexity lower than deterministic one. The work presents a set of new exact quantum algorithms with quantum query complexity n-1, 2n/3 and n/2. We construct some nondeterministic quantum query algorithms with complexity 1 for Boolean functions with 2, 4 and 2n variables and study some properties of these functions. We propose definition of postselection quantum query algorithm and we propose one method how to make postselection quantum query algorithms

Improving Quantum Query Complexity of Boolean Matrix Multiplication Using Graph Collision

The quantum query complexity of Boolean matrix multiplication is typically studied as a function of the matrix dimension, n, as well as the number of 1s in the output, \ell. We prove an upper bound of O (n\sqrt{\ell}) for all values of \ell. This is an improvement over previous algorithms for all values of \ell. On the other hand, we show that for any \eps < 1 and any \ell <= \eps n^2, there is an \Omega(n\sqrt{\ell}) lower bound for this problem, showing that our algorithm is essentially tight. We first reduce Boolean matrix multiplication to several instances of graph collision. We then provide an algorithm that takes advantage of the fact that the underlying graph in all of our instances is very dense to find all graph collisions efficiently

Improved Quantum Algorithm for Triangle Finding via Combinatorial Arguments

In this paper we present a quantum algorithm solving the triangle finding problem in unweighted graphs with query complexity $\tilde O(n^{5/4})$, where $n$ denotes the number of vertices in the graph. This improves the previous upper bound $O(n^{9/7})=O(n^{1.285...})$ recently obtained by Lee, Magniez and Santha. Our result shows, for the first time, that in the quantum query complexity setting unweighted triangle finding is easier than its edge-weighted version, since for finding an edge-weighted triangle Belovs and Rosmanis proved that any quantum algorithm requires $\Omega(n^{9/7}/\sqrt{\log n})$ queries. Our result also illustrates some limitations of the non-adaptive learning graph approach used to obtain the previous $O(n^{9/7})$ upper bound since, even over unweighted graphs, any quantum algorithm for triangle finding obtained using this approach requires $\Omega(n^{9/7}/\sqrt{\log n})$ queries as well. To bypass the obstacles characterized by these lower bounds, our quantum algorithm uses combinatorial ideas exploiting the graph-theoretic properties of triangle finding, which cannot be used when considering edge-weighted graphs or the non-adaptive learning graph approach.Comment: 17 pages, to appear in FOCS'14; v2: minor correction

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

Quantum Algorithms for the Triangle Problem

We present two new quantum algorithms that either find a triangle (a copy of $K_{3}$) in an undirected graph $G$ on $n$ nodes, or reject if $G$ is triangle free. The first algorithm uses combinatorial ideas with Grover Search and makes $\tilde{O}(n^{10/7})$ queries. The second algorithm uses $\tilde{O}(n^{13/10})$ queries, and it is based on a design concept of Ambainis~\cite{amb04} that incorporates the benefits of quantum walks into Grover search~\cite{gro96}. The first algorithm uses only $O(\log n)$ qubits in its quantum subroutines, whereas the second one uses O(n) qubits. The Triangle Problem was first treated in~\cite{bdhhmsw01}, where an algorithm with $O(n+\sqrt{nm})$ query complexity was presented, where $m$ is the number of edges of $G$.Comment: Several typos are fixed, and full proofs are included. Full version of the paper accepted to SODA'0