2,256 research outputs found
Improved Quantum Algorithm for Triangle Finding via Combinatorial Arguments
In this paper we present a quantum algorithm solving the triangle finding
problem in unweighted graphs with query complexity , where
denotes the number of vertices in the graph. This improves the previous
upper bound recently obtained by Lee, Magniez and
Santha. Our result shows, for the first time, that in the quantum query
complexity setting unweighted triangle finding is easier than its edge-weighted
version, since for finding an edge-weighted triangle Belovs and Rosmanis proved
that any quantum algorithm requires queries.
Our result also illustrates some limitations of the non-adaptive learning graph
approach used to obtain the previous upper bound since, even over
unweighted graphs, any quantum algorithm for triangle finding obtained using
this approach requires queries as well. To
bypass the obstacles characterized by these lower bounds, our quantum algorithm
uses combinatorial ideas exploiting the graph-theoretic properties of triangle
finding, which cannot be used when considering edge-weighted graphs or the
non-adaptive learning graph approach.Comment: 17 pages, to appear in FOCS'14; v2: minor correction
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
Lower Bounds on Quantum Query Complexity
Shor's and Grover's famous quantum algorithms for factoring and searching
show that quantum computers can solve certain computational problems
significantly faster than any classical computer. We discuss here what quantum
computers_cannot_ do, and specifically how to prove limits on their
computational power. We cover the main known techniques for proving lower
bounds, and exemplify and compare the methods.Comment: survey, 23 page
Quantum query complexity of minor-closed graph properties
We study the quantum query complexity of minor-closed graph properties, which
include such problems as determining whether an -vertex graph is planar, is
a forest, or does not contain a path of a given length. We show that most
minor-closed properties---those that cannot be characterized by a finite set of
forbidden subgraphs---have quantum query complexity \Theta(n^{3/2}). To
establish this, we prove an adversary lower bound using a detailed analysis of
the structure of minor-closed properties with respect to forbidden topological
minors and forbidden subgraphs. On the other hand, we show that minor-closed
properties (and more generally, sparse graph properties) that can be
characterized by finitely many forbidden subgraphs can be solved strictly
faster, in o(n^{3/2}) queries. Our algorithms are a novel application of the
quantum walk search framework and give improved upper bounds for several
subgraph-finding problems.Comment: v1: 25 pages, 2 figures. v2: 26 page
Towards a fully automated computation of RG-functions for the 3- O(N) vector model: Parametrizing amplitudes
Within the framework of field-theoretical description of second-order phase
transitions via the 3-dimensional O(N) vector model, accurate predictions for
critical exponents can be obtained from (resummation of) the perturbative
series of Renormalization-Group functions, which are in turn derived
--following Parisi's approach-- from the expansions of appropriate field
correlators evaluated at zero external momenta.
Such a technique was fully exploited 30 years ago in two seminal works of
Baker, Nickel, Green and Meiron, which lead to the knowledge of the
-function up to the 6-loop level; they succeeded in obtaining a precise
numerical evaluation of all needed Feynman amplitudes in momentum space by
lowering the dimensionalities of each integration with a cleverly arranged set
of computational simplifications. In fact, extending this computation is not
straightforward, due both to the factorial proliferation of relevant diagrams
and the increasing dimensionality of their associated integrals; in any case,
this task can be reasonably carried on only in the framework of an automated
environment.
On the road towards the creation of such an environment, we here show how a
strategy closely inspired by that of Nickel and coworkers can be stated in
algorithmic form, and successfully implemented on the computer. As an
application, we plot the minimized distributions of residual integrations for
the sets of diagrams needed to obtain RG-functions to the full 7-loop level;
they represent a good evaluation of the computational effort which will be
required to improve the currently available estimates of critical exponents.Comment: 54 pages, 17 figures and 4 table
Lorentzian and Euclidean Quantum Gravity - Analytical and Numerical Results
We review some recent attempts to extract information about the nature of
quantum gravity, with and without matter, by quantum field theoretical methods.
More specifically, we work within a covariant lattice approach where the
individual space-time geometries are constructed from fundamental simplicial
building blocks, and the path integral over geometries is approximated by
summing over a class of piece-wise linear geometries. This method of
``dynamical triangulations'' is very powerful in 2d, where the regularized
theory can be solved explicitly, and gives us more insights into the quantum
nature of 2d space-time than continuum methods are presently able to provide.
It also allows us to establish an explicit relation between the Lorentzian- and
Euclidean-signature quantum theories. Analogous regularized gravitational
models can be set up in higher dimensions. Some analytic tools exist to study
their state sums, but, unlike in 2d, no complete analytic solutions have yet
been constructed. However, a great advantage of our approach is the fact that
it is well-suited for numerical simulations. In the second part of this review
we describe the relevant Monte Carlo techniques, as well as some of the
physical results that have been obtained from the simulations of Euclidean
gravity. We also explain why the Lorentzian version of dynamical triangulations
is a promising candidate for a non-perturbative theory of quantum gravity.Comment: 69 pages, 16 figures, references adde
Using 1-Factorization from Graph Theory for Quantum Speedups on Clique Problems
The clique problems, including -CLIQUE and Triangle Finding, form an
important class of computational problems; the former is an NP-complete
problem, while the latter directly gives lower bounds for Matrix
Multiplication. A number of previous efforts have approached these problems
with Quantum Computing methods, such as Amplitude Amplification. In this paper,
we provide new Quantum oracle designs based on the 1-factorization of complete
graphs, all of which have depth instead of the presented in
previous studies. Also, we discuss the usage of one of these oracles in
bringing the Triangle Finding time complexity down to , compared to the classical record. Finally, we benchmark the
number of required Amplitude Amplification iterations for another presented
oracle, for solving -CLIQUE.Comment: 14 pages, 8 figure
- …