522 research outputs found
Total tessellation cover and quantum walk
We propose the total staggered quantum walk model and the total tessellation
cover of a graph. This model uses the concept of total tessellation cover to
describe the motion of the walker who is allowed to hop both to vertices and
edges of the graph, in contrast with previous models in which the walker hops
either to vertices or edges. We establish bounds on , which is the
smallest number of tessellations required in a total tessellation cover of .
We highlight two of these lower bounds and , where is the size of a maximum clique and is the
number of edges of a maximum induced star subgraph. Using these bounds, we
define the good total tessellable graphs with either or
. The -total tessellability problem aims to decide whether a
given graph has . We show that -total tessellability is
in for good total tessellable graphs. We establish the
-completeness of the following problems when restricted to the
following classes: ()-total tessellability for graphs with ; -total tessellability for graphs with ;
-total tessellability for graphs with far
from ; and -total tessellability for graphs with . As a consequence, we establish hardness results for bipartite graphs,
line graphs of triangle-free graphs, universal graphs, planar graphs, and
-chordal graphs
The tessellation problem of quantum walks
Quantum walks have received a great deal of attention recently because they
can be used to develop new quantum algorithms and to simulate interesting
quantum systems. In this work, we focus on a model called staggered quantum
walk, which employs advanced ideas of graph theory and has the advantage of
including the most important instances of other discrete-time models. The
evolution operator of the staggered model is obtained from a tessellation
cover, which is defined in terms of a set of partitions of the graph into
cliques. It is important to establish the minimum number of tessellations
required in a tessellation cover, and what classes of graphs admit a small
number of tessellations. We describe two main results: (1) infinite classes of
graphs where we relate the chromatic number of the clique graph to the minimum
number of tessellations required in a tessellation cover, and (2) the problem
of deciding whether a graph is -tessellable for is NP-complete.Comment: 10 pages, 7 fig
Element Distinctness Revisited
The element distinctness problem is the problem of determining whether the
elements of a list are distinct, that is, if is a list with
elements, we ask whether the elements of are distinct or not. The
solution in a classical computer requires queries because it uses sorting
to check whether there are equal elements. In the quantum case, it is possible
to solve the problem in queries. There is an extension which asks
whether there are colliding elements, known as element -distinctness
problem.
This work obtains optimal values of two critical parameters of Ambainis'
seminal quantum algorithm [SIAM J.~Comput., 37, 210-239, 2007]. The first
critical parameter is the number of repetitions of the algorithm's main block,
which inverts the phase of the marked elements and calls a subroutine. The
second parameter is the number of quantum walk steps interlaced by oracle
queries. We show that, when the optimal values of the parameters are used, the
algorithm's success probability is , quickly approaching 1.
The specification of the exact running time and success probability is
important in practical applications of this algorithm.Comment: 14 page
The role of tessellation intersection in staggered quantum walks
The staggered quantum walk (SQW) model is defined by partitioning the graph
into cliques, which are called polygons. We analyze the role that the size of
the polygon intersection plays on the dynamics of SQWs on graphs. We introduce
two processes (intersection reduction and intersection expansion), that change
the number of vertices in some intersection of polygons, and we compare the
behavior of the SQW on the reduced or expanded graph in relation to the SQW on
the original graph. We describe how the eigenvectors and eigenvalues of the
evolution operators relate to each other. This processes can help to establish
the equivalence between SQWs on different graphs and to simplify the analysis
of SQWs. We also show an example of a SQW on a graph that is not included in
Szegedy's model, but which is equivalent to an instance of Szegedy's model
after applying the intersection reduction.Comment: 14 page
Staggered Quantum Walks on Graphs
The staggered quantum walk model allows to establish an unprecedented
connection between discrete-time quantum walks and graph theory. We call
attention to the fact that a large subclass of the coined model is included in
Szegedy's model, which in its turn is entirely included in the staggered model.
In order to compare those three quantum walk models, we put them in the
staggered formalism and we show that the Szegedy and coined models are defined
on a special subclass of graphs. This inclusion scheme is also true when the
searching framework is added. We use graph theory to characterize which
staggered quantum walks can be reduced to the Szegedy or coined quantum walk
model. We analyze a staggered-based search that cannot be included in Szegedy's
model and we show numerically that this search is more efficient than a
random-walk-based search.Comment: 14 pages, 9 fig
Establishing the equivalence between Szegedy's and coined quantum walks using the staggered model
Coined Quantum Walks (QWs) are being used in many contexts with the goal of
understanding quantum systems and building quantum algorithms for quantum
computers. Alternative models such as Szegedy's and continuous-time QWs were
proposed taking advantage of the fact that quantum theory seems to allow
different quantized versions based on the same classical model, in this case,
the classical random walk. In this work, we show the conditions upon which
coined QWs are equivalent to Szegedy's QWs. Those QW models have in common a
large class of instances, in the sense that the evolution operators are equal
when we convert the graph on which the coined QW takes place into a bipartite
graph on which Szegedy's QW takes place, and vice versa. We also show that the
abstract search algorithm using the coined QW model can be cast into Szegedy's
searching framework using bipartite graphs with sinks.Comment: 23 pages, 7 figures, Quantum Information Processing, 201
Eigenbasis of the Evolution Operator of 2-Tessellable Quantum Walks
Staggered quantum walks on graphs are based on the concept of graph
tessellation and generalize some well-known discrete-time quantum walk models.
In this work, we address the class of 2-tessellable quantum walks with the goal
of obtaining an eigenbasis of the evolution operator. By interpreting the
evolution operator as a quantum Markov chain on an underlying multigraph, we
define the concept of quantum detailed balance, which helps to obtain the
eigenbasis. A subset of the eigenvectors is obtained from the eigenvectors of
the double discriminant matrix of the quantum Markov chain. To obtain the
remaining eigenvectors, we have to use the quantum detailed balance conditions.
If the quantum Markov chain has a quantum detailed balance, there is an
eigenvector for each fundamental cycle of the underlying multigraph. If the
quantum Markov chain does not have a quantum detailed balance, we have to use
two fundamental cycles linked by a path in order to find the remaining
eigenvectors. We exemplify the process of obtaining the eigenbasis of the
evolution operator using the kagome lattice (the line graph of the hexagonal
lattice), which has symmetry properties that help in the calculation process.Comment: 21 pages, 3 figure
Experimental Implementation of Quantum Walks on IBM Quantum Computers
The development of universal quantum computers has achieved remarkable
success in recent years, culminating with the quantum supremacy reported by
Google. Now is possible to implement short-depth quantum circuits with dozens
of qubits and to obtain results with significant fidelity. Quantum walks are
good candidates to be implemented on the available quantum computers. In this
work, we implement discrete-time quantum walks with one and two interacting
walkers on cycles, two-dimensional lattices, and complete graphs on IBM quantum
computers. We are able to obtain meaningful results using the cycle, the
two-dimensional lattice, and the complete graph with 16 nodes each, which
require 4-qubit quantum circuits up to depth 100
Quantum Walks via Quantum Cellular Automata
Very much as its classical counterpart, quantum cellular automata are
expected to be a great tool for simulating complex quantum systems. Here we
introduce a partitioned model of quantum cellular automata and show how it can
simulate, with the same amount of resources (in terms of effective Hilbert
space dimension), various models of quantum walks. All the algorithms developed
within quantum walk models are thus directly inherited by the quantum cellular
automata. The latter, however, has its structure based on local interactions
between qubits, and as such it can be more suitable for present (and future)
experimental implementations.Comment: 10 pages, 3 figures. Comments are welcom
The Tessellation Cover Number of Good Tessellable Graphs
A tessellation of a graph is a partition of its vertices into vertex disjoint
cliques. A tessellation cover of a graph is a set of tessellations that covers
all of its edges, and the tessellation cover number, denoted by , is the
size of a smallest tessellation cover. The \textsc{-tessellability} problem
aims to decide whether a graph has and is
-complete for . Since the number of edges of a maximum
induced star of , denoted by , is a lower bound on , we define
good tessellable graphs as the graphs~ such that . The
\textsc{good tessellable recognition (gtr)} problem aims to decide whether
is a good tessellable graph. We show that \textsc{gtr} is
-complete not only if is known or is fixed, but
also when the gap between and is large. As a byproduct, we
obtain graph classes that obey the corresponding computational complexity
behaviors.Comment: 14 pages, 3 figure
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