Quantum walks can find a marked element on any graph

We solve an open problem by constructing quantum walks that not only detect but also find marked vertices in a graph. In the case when the marked set $M$ consists of a single vertex, the number of steps of the quantum walk is quadratically smaller than the classical hitting time $HT(P,M)$ of any reversible random walk $P$ on the graph. In the case of multiple marked elements, the number of steps is given in terms of a related quantity $HT^+(\mathit{P,M})$ which we call extended hitting time. Our approach is new, simpler and more general than previous ones. We introduce a notion of interpolation between the random walk $P$ and the absorbing walk $P'$, whose marked states are absorbing. Then our quantum walk is simply the quantum analogue of this interpolation. Contrary to previous approaches, our results remain valid when the random walk $P$ is not state-transitive. We also provide algorithms in the cases when only approximations or bounds on parameters $p_M$ (the probability of picking a marked vertex from the stationary distribution) and $HT^+(\mathit{P,M})$ are known.Comment: 50 page

On the coalescence time of reversible random walks

Consider a system of coalescing random walks where each individual performs random walk over a finite graph G, or (more generally) evolves according to some reversible Markov chain generator Q. Let C be the first time at which all walkers have coalesced into a single cluster. C is closely related to the consensus time of the voter model for this G or Q. We prove that the expected value of C is at most a constant multiple of the largest hitting time of an element in the state space. This solves a problem posed by Aldous and Fill and gives sharp bounds in many examples, including all vertex-transitive graphs. We also obtain results on the expected time until only k>1 clusters remain. Our proof tools include a new exponential inequality for the meeting time of a reversible Markov chain and a deterministic trajectory, which we believe to be of independent interest.Comment: 29 pages in 11pt font with 3/2 line spacing. v2 has an extra reference and corrects a minor error in the proof of the last claim. To appear in Transactions of the AM

Robotic Surveillance and Deployment Strategies

Autonomous mobile systems are becoming more common place, and have the opportunity to revolutionize many modern application areas. They include, but are not limited to, tasks such as search and rescue operations, ad-hoc mobile wireless networks and warehouse management; each application having its own complexities and challenging problems that need addressing. In this thesis, we explore and characterize two application areas in particular. First, we explore the problem of autonomous stochastic surveillance. In particular, we study random walks on a finite graph that are described by a Markov chain. We present strategies that minimize the first hitting time of the Markov chain, and look at both the single agent and multi-agent cases. In the single agent case, we provide a formulation and convex optimization scheme for the hitting time on graphs with travel distances. In addition, we provide detailed simulation results showing the effectiveness of our strategy versus other well-known Markov chain design strategies. In the multi-agent case, we provide the first characterization of the hitting time for multiple random walkers, which we denote the "group hitting time". We also provide a closed form solution for calculating the hitting time between specified nodes for both the single and multiple random walker cases. Our results allow for the multiple random walks to be different and, moreover, for the random walks to operate on different subgraphs. Finally, we use sequential quadratic programming to find the transition matrices that generate minimal "group hitting time".Second, we consider the problem of optimal coverage with a group of mobile agents. For a planar environment with an associated density function, this problem is equivalent to dividing the environment into optimal subregions such that each agent is responsible for the coverage of its own region. We study this problem for the discrete time and space case and the continuous time and space case. For the discrete time and space case, we present algorithms that provide optimal coverage control in a non-convex environment when each robot has only asynchronous and sporadic communication with a base station. We introduce the notion of coverings, a generalization of partitions, to do this. For the continuous time and space case, we present a continuous-time distributed policy which allows a team of agents to achieve a convex area-constrained partition in a convex workspace. This work is related to the classic Lloyd algorithm, and makes use of generalized Voronoi diagrams. For both cases we provide detailed simulation results and discuss practical implementation issues

Decoherence in quantum walks - a review

The development of quantum walks in the context of quantum computation, as generalisations of random walk techniques, led rapidly to several new quantum algorithms. These all follow unitary quantum evolution, apart from the final measurement. Since logical qubits in a quantum computer must be protected from decoherence by error correction, there is no need to consider decoherence at the level of algorithms. Nonetheless, enlarging the range of quantum dynamics to include non-unitary evolution provides a wider range of possibilities for tuning the properties of quantum walks. For example, small amounts of decoherence in a quantum walk on the line can produce more uniform spreading (a top-hat distribution), without losing the quantum speed up. This paper reviews the work on decoherence, and more generally on non-unitary evolution, in quantum walks and suggests what future questions might prove interesting to pursue in this area.Comment: 52 pages, invited review, v2 & v3 updates to include significant work since first posted and corrections from comments received; some non-trivial typos fixed. Comments now limited to changes that can be applied at proof stag

Hitting time for quantum walks on the hypercube

Hitting times for discrete quantum walks on graphs give an average time before the walk reaches an ending condition. To be analogous to the hitting time for a classical walk, the quantum hitting time must involve repeated measurements as well as unitary evolution. We derive an expression for hitting time using superoperators, and numerically evaluate it for the discrete walk on the hypercube. The values found are compared to other analogues of hitting time suggested in earlier work. The dependence of hitting times on the type of unitary ``coin'' is examined, and we give an example of an initial state and coin which gives an infinite hitting time for a quantum walk. Such infinite hitting times require destructive interference, and are not observed classically. Finally, we look at distortions of the hypercube, and observe that a loss of symmetry in the hypercube increases the hitting time. Symmetry seems to play an important role in both dramatic speed-ups and slow-downs of quantum walks.Comment: 8 pages in RevTeX format, four figures in EPS forma