New Developments in Quantum Algorithms

In this survey, we describe two recent developments in quantum algorithms. The first new development is a quantum algorithm for evaluating a Boolean formula consisting of AND and OR gates of size N in time O(\sqrt{N}). This provides quantum speedups for any problem that can be expressed via Boolean formulas. This result can be also extended to span problems, a generalization of Boolean formulas. This provides an optimal quantum algorithm for any Boolean function in the black-box query model. The second new development is a quantum algorithm for solving systems of linear equations. In contrast with traditional algorithms that run in time O(N^{2.37...}) where N is the size of the system, the quantum algorithm runs in time O(\log^c N). It outputs a quantum state describing the solution of the system.Comment: 11 pages, 1 figure, to appear as an invited survey talk at MFCS'201

Central limit theorem for reducible and irreducible open quantum walks

In this work we aim at proving central limit theorems for open quantum walks on $\mathbb{Z}^d$. We study the case when there are various classes of vertices in the network. Furthermore, we investigate two ways of distributing the vertex classes in the network. First we assign the classes in a regular pattern. Secondly, we assign each vertex a random class with a uniform distribution. For each way of distributing vertex classes, we obtain an appropriate central limit theorem, illustrated by numerical examples. These theorems may have application in the study of complex systems in quantum biology and dissipative quantum computation.Comment: 20 pages, 4 figure

Quantum walks: a comprehensive review

Quantum walks, the quantum mechanical counterpart of classical random walks, is an advanced tool for building quantum algorithms that has been recently shown to constitute a universal model of quantum computation. Quantum walks is now a solid field of research of quantum computation full of exciting open problems for physicists, computer scientists, mathematicians and engineers. In this paper we review theoretical advances on the foundations of both discrete- and continuous-time quantum walks, together with the role that randomness plays in quantum walks, the connections between the mathematical models of coined discrete quantum walks and continuous quantum walks, the quantumness of quantum walks, a summary of papers published on discrete quantum walks and entanglement as well as a succinct review of experimental proposals and realizations of discrete-time quantum walks. Furthermore, we have reviewed several algorithms based on both discrete- and continuous-time quantum walks as well as a most important result: the computational universality of both continuous- and discrete- time quantum walks.Comment: Paper accepted for publication in Quantum Information Processing Journa

Application of quantum walks on graph structures to quantum computing

Quantum computation is a new computational paradigm which can provide fundamentally faster computation than in the classical regime. This is dependent on finding efficient quantum algorithms for problems of practical interest. One of the most successful tools in developing new quantum algorithms is the quantum walk. In this thesis, we explore two applications of the discrete time quantum walk. In addition, we introduce an experimental scheme for generating cluster states, a universal resource for quantum computation. We give an explicit construction which provides a link between the circuit model of quantum computation, and a graph structure on which the discrete time quantum walk traverses, performing the same computation. We implement a universal gate set, proving the discrete time quantum walk is universal for quantum computation, thus confirming any quantum algorithm can be recast as a quantum walk algorithm. In addition, we study factors affecting the efficiency of the quantum walk search algorithm. Although there is a strong dependence on the spatial dimension of the structure being searched, we find secondary dependencies on other factors including the connectivity and disorder (symmetry). Fairly intuitively, as the connectivity increases, the efficiency of the algorithm increases, as the walker can coalesce on the marked state with higher probability in a quicker time. In addition, we find as disorder in the system increases, the algorithm can maintain the quantum speed up for a certain level of disorder before gradually reverting to the classical run time. Finally, we give an abstract scheme for generating cluster states. We see a linear scaling, better than many schemes, as doubling the size of the generating grid in our scheme produces a cluster state which is double the depth. Our scheme is able to create other interesting topologies of entangled states, including the unit cell for topological error correcting schemes