2,945 research outputs found
Quantum Algorithms for Finding Constant-sized Sub-hypergraphs
We develop a general framework to construct quantum algorithms that detect if
a -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 -clique in a -uniform hypergraph on
vertices with query complexity , and a quantum algorithm for
determining if a ternary operator over a set of size is associative with
query complexity .Comment: 18 pages; v2: changed title, added more backgrounds to the
introduction, added another applicatio
Data Structures in Classical and Quantum Computing
This survey summarizes several results about quantum computing related to
(mostly static) data structures. First, we describe classical data structures
for the set membership and the predecessor search problems: Perfect Hash tables
for set membership by Fredman, Koml\'{o}s and Szemer\'{e}di and a data
structure by Beame and Fich for predecessor search. We also prove results about
their space complexity (how many bits are required) and time complexity (how
many bits have to be read to answer a query). After that, we turn our attention
to classical data structures with quantum access. In the quantum access model,
data is stored in classical bits, but they can be accessed in a quantum way: We
may read several bits in superposition for unit cost. We give proofs for lower
bounds in this setting that show that the classical data structures from the
first section are, in some sense, asymptotically optimal - even in the quantum
model. In fact, these proofs are simpler and give stronger results than
previous proofs for the classical model of computation. The lower bound for set
membership was proved by Radhakrishnan, Sen and Venkatesh and the result for
the predecessor problem by Sen and Venkatesh. Finally, we examine fully quantum
data structures. Instead of encoding the data in classical bits, we now encode
it in qubits. We allow any unitary operation or measurement in order to answer
queries. We describe one data structure by de Wolf for the set membership
problem and also a general framework using fully quantum data structures in
quantum walks by Jeffery, Kothari and Magniez
Quantum Algorithm for Triangle Finding in Sparse Graphs
This paper presents a quantum algorithm for triangle finding over sparse
graphs that improves over the previous best quantum algorithm for this task by
Buhrman et al. [SIAM Journal on Computing, 2005]. Our algorithm is based on the
recent -query algorithm given by Le Gall [FOCS 2014] for
triangle finding over dense graphs (here denotes the number of vertices in
the graph). We show in particular that triangle finding can be solved with
queries for some constant whenever the graph
has at most edges for some constant .Comment: 13 page
Graph Convolutional Neural Networks based on Quantum Vertex Saliency
This paper proposes a new Quantum Spatial Graph Convolutional Neural Network
(QSGCNN) model that can directly learn a classification function for graphs of
arbitrary sizes. Unlike state-of-the-art Graph Convolutional Neural Network
(GCNN) models, the proposed QSGCNN model incorporates the process of
identifying transitive aligned vertices between graphs, and transforms
arbitrary sized graphs into fixed-sized aligned vertex grid structures. In
order to learn representative graph characteristics, a new quantum spatial
graph convolution is proposed and employed to extract multi-scale vertex
features, in terms of quantum information propagation between grid vertices of
each graph. Since the quantum spatial convolution preserves the grid structures
of the input vertices (i.e., the convolution layer does not change the original
spatial sequence of vertices), the proposed QSGCNN model allows to directly
employ the traditional convolutional neural network architecture to further
learn from the global graph topology, providing an end-to-end deep learning
architecture that integrates the graph representation and learning in the
quantum spatial graph convolution layer and the traditional convolutional layer
for graph classifications. We demonstrate the effectiveness of the proposed
QSGCNN model in relation to existing state-of-the-art methods. The proposed
QSGCNN model addresses the shortcomings of information loss and imprecise
information representation arising in existing GCN models associated with the
use of SortPooling or SumPooling layers. Experiments on benchmark graph
classification datasets demonstrate the effectiveness of the proposed QSGCNN
model
Nesting statistics in the loop model on random planar maps
In the loop model on random planar maps, we study the depth -- in
terms of the number of levels of nesting -- of the loop configuration, by means
of analytic combinatorics. We focus on the `refined' generating series of
pointed disks or cylinders, which keep track of the number of loops separating
the marked point from the boundary (for disks), or the two boundaries (for
cylinders). For the general loop model, we show that these generating
series satisfy functional relations obtained by a modification of those
satisfied by the unrefined generating series. In a more specific model
where loops cross only triangles and have a bending energy, we explicitly
compute the refined generating series. We analyze their non generic critical
behavior in the dense and dilute phases, and obtain the large deviations
function of the nesting distribution, which is expected to be universal. Using
the framework of Liouville quantum gravity (LQG), we show that a rigorous
functional KPZ relation can be applied to the multifractal spectrum of extreme
nesting in the conformal loop ensemble () in the Euclidean
unit disk, as obtained by Miller, Watson and Wilson, or to its natural
generalization to the Riemann sphere. It allows us to recover the large
deviations results obtained for the critical random planar map models.
This offers, at the refined level of large deviations theory, a rigorous check
of the fundamental fact that the universal scaling limits of random planar map
models as weighted by partition functions of critical statistical models are
given by LQG random surfaces decorated by independent CLEs.Comment: 71 pages, 11 figures. v2: minor text and abstract edits, references
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