597 research outputs found
Parsimonious Segmentation of Time Series' by Potts Models
Typical problems in the analysis of data sets like time-series or images crucially rely on the extraction of primitive features based on segmentation. Variational approaches are a popular and convenient framework in which such problems can be studied. We focus on Potts models as simple nontrivial instances. The discussion proceeds along two data sets from brain mapping and functional genomics
Quantum walks on Cayley graphs
We address the problem of the construction of quantum walks on Cayley graphs.
Our main motivation is the relationship between quantum algorithms and quantum
walks. In particular, we discuss the choice of the dimension of the local
Hilbert space and consider various classes of graphs on which the structure of
quantum walks may differ. We completely characterise quantum walks on free
groups and present partial results on more general cases. Some examples are
given, including a family of quantum walks on the hypercube involving a
Clifford Algebra.Comment: J. Phys. A (accepted for publication
How to Spread a Rumor: Call Your Neighbors or Take a Walk?
We study the problem of randomized information dissemination in networks. We
compare the now standard PUSH-PULL protocol, with agent-based alternatives
where information is disseminated by a collection of agents performing
independent random walks. In the VISIT-EXCHANGE protocol, both nodes and agents
store information, and each time an agent visits a node, the two exchange all
the information they have. In the MEET-EXCHANGE protocol, only the agents store
information, and exchange their information with each agent they meet.
We consider the broadcast time of a single piece of information in an
-node graph for the above three protocols, assuming a linear number of
agents that start from the stationary distribution. We observe that there are
graphs on which the agent-based protocols are significantly faster than
PUSH-PULL, and graphs where the converse is true. We attribute the good
performance of agent-based algorithms to their inherently fair bandwidth
utilization, and conclude that, in certain settings, agent-based information
dissemination, separately or in combination with PUSH-PULL, can significantly
improve the broadcast time.
The graphs considered above are highly non-regular. Our main technical result
is that on any regular graph of at least logarithmic degree, PUSH-PULL and
VISIT-EXCHANGE have the same asymptotic broadcast time. The proof uses a novel
coupling argument which relates the random choices of vertices in PUSH-PULL
with the random walks in VISIT-EXCHANGE. Further, we show that the broadcast
time of MEET-EXCHANGE is asymptotically at least as large as the other two's on
all regular graphs, and strictly larger on some regular graphs.
As far as we know, this is the first systematic and thorough comparison of
the running times of these very natural information dissemination protocols.The authors would like to thank Thomas Sauerwald and Nicol\'{a}s Rivera for helpful discussions.
This research was undertaken, in part, thanks to funding from
the ANR Project PAMELA (ANR-16-CE23-0016-01),
the NSF Award Numbers CCF-1461559, CCF-0939370 and CCF-18107,
the Gates Cambridge Scholarship programme,
and the ERC grant DYNAMIC MARCH
The Complexity of Routing with Few Collisions
We study the computational complexity of routing multiple objects through a
network in such a way that only few collisions occur: Given a graph with
two distinct terminal vertices and two positive integers and , the
question is whether one can connect the terminals by at least routes (e.g.
paths) such that at most edges are time-wise shared among them. We study
three types of routes: traverse each vertex at most once (paths), each edge at
most once (trails), or no such restrictions (walks). We prove that for paths
and trails the problem is NP-complete on undirected and directed graphs even if
is constant or the maximum vertex degree in the input graph is constant.
For walks, however, it is solvable in polynomial time on undirected graphs for
arbitrary and on directed graphs if is constant. We additionally study
for all route types a variant of the problem where the maximum length of a
route is restricted by some given upper bound. We prove that this
length-restricted variant has the same complexity classification with respect
to paths and trails, but for walks it becomes NP-complete on undirected graphs
Three qubits can be entangled in two inequivalent ways
Invertible local transformations of a multipartite system are used to define
equivalence classes in the set of entangled states. This classification
concerns the entanglement properties of a single copy of the state.
Accordingly, we say that two states have the same kind of entanglement if both
of them can be obtained from the other by means of local operations and
classical communcication (LOCC) with nonzero probability. When applied to pure
states of a three-qubit system, this approach reveals the existence of two
inequivalent kinds of genuine tripartite entanglement, for which the GHZ state
and a W state appear as remarkable representatives. In particular, we show that
the W state retains maximally bipartite entanglement when any one of the three
qubits is traced out. We generalize our results both to the case of higher
dimensional subsystems and also to more than three subsystems, for all of which
we show that, typically, two randomly chosen pure states cannot be converted
into each other by means of LOCC, not even with a small probability of success.Comment: 12 pages, 1 figure; replaced with revised version; terminology
adapted to earlier work; reference added; results unchange
Continuous-time quantum walk on integer lattices and homogeneous trees
This paper is concerned with the continuous-time quantum walk on Z, Z^d, and
infinite homogeneous trees. By using the generating function method, we compute
the limit of the average probability distribution for the general isotropic
walk on Z, and for nearest-neighbor walks on Z^d and infinite homogeneous
trees. In addition, we compute the asymptotic approximation for the probability
of the return to zero at time t in all these cases.Comment: The journal version (save for formatting); 19 page
Asymptotic entanglement in a two-dimensional quantum walk
The evolution operator of a discrete-time quantum walk involves a conditional
shift in position space which entangles the coin and position degrees of
freedom of the walker. After several steps, the coin-position entanglement
(CPE) converges to a well defined value which depends on the initial state. In
this work we provide an analytical method which allows for the exact
calculation of the asymptotic reduced density operator and the corresponding
CPE for a discrete-time quantum walk on a two-dimensional lattice. We use the
von Neumann entropy of the reduced density operator as an entanglement measure.
The method is applied to the case of a Hadamard walk for which the dependence
of the resulting CPE on initial conditions is obtained. Initial states leading
to maximum or minimum CPE are identified and the relation between the coin or
position entanglement present in the initial state of the walker and the final
level of CPE is discussed. The CPE obtained from separable initial states
satisfies an additivity property in terms of CPE of the corresponding
one-dimensional cases. Non-local initial conditions are also considered and we
find that the extreme case of an initial uniform position distribution leads to
the largest CPE variation.Comment: Major revision. Improved structure. Theoretical results are now
separated from specific examples. Most figures have been replaced by new
versions. The paper is now significantly reduced in size: 11 pages, 7 figure
Nanoscale FasL Organization on DNA Origami to Decipher Apoptosis Signal Activation in Cells
Cell signaling is initiated by characteristic protein patterns in the plasma membrane, but tools to decipher their molecular organization and activation are hitherto lacking. Among the well-known signaling pattern is the death inducing signaling complex with a predicted hexagonal receptor architecture. To probe this architecture, DNA origami-based nanoagents with nanometer precise arrangements of the death receptor ligand FasL are introduced and presented to cells. Mimicking different receptor geometries, these nanoagents act as signaling platforms inducing fastest time-to-death kinetics for hexagonal FasL arrangements with 10 nm inter-molecular spacing. Compared to naturally occurring soluble FasL, this trigger is faster and 100x more efficient. Nanoagents with different spacing, lower FasL number or higher coupling flexibility impede signaling. The results present DNA origami as versatile signaling scaffolds exhibiting unprecedented control over molecular number and geometry. They define molecular benchmarks in apoptosis signal initiation and constitute a new strategy to drive particular cell responses
Almost uniform sampling via quantum walks
Many classical randomized algorithms (e.g., approximation algorithms for
#P-complete problems) utilize the following random walk algorithm for {\em
almost uniform sampling} from a state space of cardinality : run a
symmetric ergodic Markov chain on for long enough to obtain a random
state from within total variation distance of the uniform
distribution over . The running time of this algorithm, the so-called {\em
mixing time} of , is , where
is the spectral gap of .
We present a natural quantum version of this algorithm based on repeated
measurements of the {\em quantum walk} . We show that it
samples almost uniformly from with logarithmic dependence on
just as the classical walk does; previously, no such
quantum walk algorithm was known. We then outline a framework for analyzing its
running time and formulate two plausible conjectures which together would imply
that it runs in time when is
the standard transition matrix of a constant-degree graph. We prove each
conjecture for a subclass of Cayley graphs.Comment: 13 pages; v2 added NSF grant info; v3 incorporated feedbac
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