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

On the relationship between continuous- and discrete-time quantum walk

Quantum walk is one of the main tools for quantum algorithms. Defined by analogy to classical random walk, a quantum walk is a time-homogeneous quantum process on a graph. Both random and quantum walks can be defined either in continuous or discrete time. But whereas a continuous-time random walk can be obtained as the limit of a sequence of discrete-time random walks, the two types of quantum walk appear fundamentally different, owing to the need for extra degrees of freedom in the discrete-time case. In this article, I describe a precise correspondence between continuous- and discrete-time quantum walks on arbitrary graphs. Using this correspondence, I show that continuous-time quantum walk can be obtained as an appropriate limit of discrete-time quantum walks. The correspondence also leads to a new technique for simulating Hamiltonian dynamics, giving efficient simulations even in cases where the Hamiltonian is not sparse. The complexity of the simulation is linear in the total evolution time, an improvement over simulations based on high-order approximations of the Lie product formula. As applications, I describe a continuous-time quantum walk algorithm for element distinctness and show how to optimally simulate continuous-time query algorithms of a certain form in the conventional quantum query model. Finally, I discuss limitations of the method for simulating Hamiltonians with negative matrix elements, and present two problems that motivate attempting to circumvent these limitations.Comment: 22 pages. v2: improved presentation, new section on Hamiltonian oracles; v3: published version, with improved analysis of phase estimatio

Quantum Algorithms for Finding Constant-sized Sub-hypergraphs

We develop a general framework to construct quantum algorithms that detect if a $3$-uniform hypergraph given as input contains a sub-hypergraph isomorphic to a prespecified constant-sized hypergraph. This framework is based on the concept of nested quantum walks recently proposed by Jeffery, Kothari and Magniez [SODA'13], and extends the methodology designed by Lee, Magniez and Santha [SODA'13] for similar problems over graphs. As applications, we obtain a quantum algorithm for finding a $4$-clique in a $3$-uniform hypergraph on $n$ vertices with query complexity $O(n^{1.883})$, and a quantum algorithm for determining if a ternary operator over a set of size $n$ is associative with query complexity $O(n^{2.113})$.Comment: 18 pages; v2: changed title, added more backgrounds to the introduction, added another applicatio

Search via Quantum Walk

We propose a new method for designing quantum search algorithms for finding a "marked" element in the state space of a classical Markov chain. The algorithm is based on a quantum walk \'a la Szegedy (2004) that is defined in terms of the Markov chain. The main new idea is to apply quantum phase estimation to the quantum walk in order to implement an approximate reflection operator. This operator is then used in an amplitude amplification scheme. As a result we considerably expand the scope of the previous approaches of Ambainis (2004) and Szegedy (2004). Our algorithm combines the benefits of these approaches in terms of being able to find marked elements, incurring the smaller cost of the two, and being applicable to a larger class of Markov chains. In addition, it is conceptually simple and avoids some technical difficulties in the previous analyses of several algorithms based on quantum walk.Comment: 21 pages. Various modifications and improvements, especially in Section

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