638 research outputs found
On the probability of finding marked connected components using quantum walks
Finding a marked vertex in a graph can be a complicated task when using
quantum walks. Recent results show that for two or more adjacent marked
vertices search by quantum walk with Grover's coin may have no speed-up over
classical exhaustive search. In this paper, we analyze the probability of
finding a marked vertex for a set of connected components of marked vertices.
We prove two upper bounds on the probability of finding a marked vertex and
sketch further research directions.Comment: 13 pages. To appear at Lobachevskii Journal of Mathematic
Quantum walks can unitarily match random walks on finite graphs
Quantum and random walks were proven to be equivalent on finite graphs by
demonstrating how to construct a time-dependent random walk sharing the exact
same evolution of vertex probability of any given discrete-time coined quantum
walk. Such equivalence stipulated a deep connection between the processes that
is far stronger than simply considering quantum walks as quantum analogues of
random walks. This article expands on the connection between quantum and random
walks by demonstrating a procedure that constructs a time-dependent quantum
walk matching the evolution of vertex probability of any given random walk in a
unitary way. It is a trivial fact that a quantum walk measured at all time
steps of its evolution degrades to a random walk. More interestingly, the
method presented describes a quantum walk that matches a random walk without
measurement operations, such that the unitary evolution of the quantum walk
captures the probability evolution of the random walk. The construction
procedure is general, covering both homogeneous and non-homogeneous random
walks. For the homogeneous random walk case, the properties of unitary
evolution imply that the quantum walk described is time-dependent since
homogeneous quantum walks do not converge for arbitrary initial conditionsComment: 9 pages, 1 figur
Green function approach for scattering quantum walks
In this work a Green function approach for scattering quantum walks is
developed. The exact formula has the form of a sum over paths and always can be
cast into a closed analytic expression for arbitrary topologies and position
dependent quantum amplitudes. By introducing the step and path operators, it is
shown how to extract any information about the system from the Green function.
The method relevant features are demonstrated by discussing in details an
example, a general diamond-shaped graph.Comment: 13 pages, 6 figures, this article was selected by APS for Virtual
Journal of Quantum Information, Vol 11, Iss 11 (2011
Entanglement in coined quantum walks on regular graphs
Published versio
Projective simulation with generalization
The ability to generalize is an important feature of any intelligent agent.
Not only because it may allow the agent to cope with large amounts of data, but
also because in some environments, an agent with no generalization capabilities
cannot learn. In this work we outline several criteria for generalization, and
present a dynamic and autonomous machinery that enables projective simulation
agents to meaningfully generalize. Projective simulation, a novel, physical
approach to artificial intelligence, was recently shown to perform well in
standard reinforcement learning problems, with applications in advanced
robotics as well as quantum experiments. Both the basic projective simulation
model and the presented generalization machinery are based on very simple
principles. This allows us to provide a full analytical analysis of the agent's
performance and to illustrate the benefit the agent gains by generalizing.
Specifically, we show that already in basic (but extreme) environments,
learning without generalization may be impossible, and demonstrate how the
presented generalization machinery enables the projective simulation agent to
learn.Comment: 14 pages, 9 figure
Quantum kernels for unattributed graphs using discrete-time quantum walks
In this paper, we develop a new family of graph kernels where the graph structure is probed by means of a discrete-time quantum walk. Given a pair of graphs, we let a quantum walk evolve on each graph and compute a density matrix with each walk. With the density matrices for the pair of graphs to hand, the kernel between the graphs is defined as the negative exponential of the quantum Jensen–Shannon divergence between their density matrices. In order to cope with large graph structures, we propose to construct a sparser version of the original graphs using the simplification method introduced in Qiu and Hancock (2007). To this end, we compute the minimum spanning tree over the commute time matrix of a graph. This spanning tree representation minimizes the number of edges of the original graph while preserving most of its structural information. The kernel between two graphs is then computed on their respective minimum spanning trees. We evaluate the performance of the proposed kernels on several standard graph datasets and we demonstrate their effectiveness and efficiency
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