380 research outputs found
Isotropic quantum walks on lattices and the Weyl equation
We present a thorough classification of the isotropic quantum walks on
lattices of dimension for cell dimension . For there exist
two isotropic walks, namely the Weyl quantum walks presented in Ref. [G. M.
D'Ariano and P. Perinotti, Phys. Rev. A 90, 062106 (2014)], resulting in the
derivation of the Weyl equation from informational principles. The present
analysis, via a crucial use of isotropy, is significantly shorter and avoids a
superfluous technical assumption, making the result completely general.Comment: 16 pages, 1 figur
Quantum cellular automata and free quantum field theory
In a series of recent papers it has been shown how free quantum field theory
can be derived without using mechanical primitives (including space-time,
special relativity, quantization rules, etc.), but only considering the easiest
quantum algorithm encompassing a countable set of quantum systems whose network
of interactions satisfies the simple principles of unitarity, homogeneity,
locality, and isotropy. This has opened the route to extending the axiomatic
information-theoretic derivation of the quantum theory of abstract systems to
include quantum field theory. The inherent discrete nature of the informational
axiomatization leads to an extension of quantum field theory to a quantum
cellular automata theory, where the usual field theory is recovered in a regime
where the discrete structure of the automata cannot be probed. A simple
heuristic argument sets the scale of discreteness to the Planck scale, and the
customary physical regime where discreteness is not visible is the relativistic
one of small wavevectors. In this paper we provide a thorough derivation from
principles that in the most general case the graph of the quantum cellular
automaton is the Cayley graph of a finitely presented group, and showing how
for the case corresponding to Euclidean emergent space (where the group resorts
to an Abelian one) the automata leads to Weyl, Dirac and Maxwell field dynamics
in the relativistic limit. We conclude with some perspectives towards the more
general scenario of non-linear automata for interacting quantum field theory.Comment: 10 pages, 2 figures, revtex style. arXiv admin note: substantial text
overlap with arXiv:1601.0483
Classes of Symmetric Cayley Graphs over Finite Abelian Groups of Degrees 4 and 6
The present work is devoted to characterize the family of symmetric
undirected Cayley graphs over finite Abelian groups for degrees 4 and 6.Comment: 12 pages. A previous version of some of the results in this paper
where first announced at 2010 International Workshop on Optimal
Interconnection Networks (IWONT 2010). It is accessible at
http://upcommons.upc.edu/revistes/handle/2099/1037
On the spectral gap of some Cayley graphs on the Weyl group
The Laplacian of a (weighted) Cayley graph on the Weyl group is a
matrix with equal to the order of the group. We show
that for a class of (weighted) generating sets, its spectral gap (lowest
nontrivial eigenvalue), is actually equal to the spectral gap of a matrix associated to a -dimensional permutation representation of
. This result can be viewed as an extension to of an analogous
result valid for the symmetric group, known as `Aldous' spectral gap
conjecture', proven in 2010 by Caputo, Liggett and Richthammer.Comment: Version 1 (v1) contains a mistake. The main result is proved here
under a less general hypothesis than in v1. Main result of v1 is left as a
conjectur
Ramanujan Coverings of Graphs
Let be a finite connected graph, and let be the spectral radius of
its universal cover. For example, if is -regular then
. We show that for every , there is an -covering
(a.k.a. an -lift) of where all the new eigenvalues are bounded from
above by . It follows that a bipartite Ramanujan graph has a Ramanujan
-covering for every . This generalizes the case due to Marcus,
Spielman and Srivastava (2013).
Every -covering of corresponds to a labeling of the edges of by
elements of the symmetric group . We generalize this notion to labeling
the edges by elements of various groups and present a broader scenario where
Ramanujan coverings are guaranteed to exist.
In particular, this shows the existence of richer families of bipartite
Ramanujan graphs than was known before. Inspired by Marcus-Spielman-Srivastava,
a crucial component of our proof is the existence of interlacing families of
polynomials for complex reflection groups. The core argument of this component
is taken from a recent paper of them (2015).
Another important ingredient of our proof is a new generalization of the
matching polynomial of a graph. We define the -th matching polynomial of
to be the average matching polynomial of all -coverings of . We show this
polynomial shares many properties with the original matching polynomial. For
example, it is real rooted with all its roots inside .Comment: 38 pages, 4 figures, journal version (minor changes from previous
arXiv version). Shortened version appeared in STOC 201
Symmetric Interconnection Networks from Cubic Crystal Lattices
Torus networks of moderate degree have been widely used in the supercomputer
industry. Tori are superb when used for executing applications that require
near-neighbor communications. Nevertheless, they are not so good when dealing
with global communications. Hence, typical 3D implementations have evolved to
5D networks, among other reasons, to reduce network distances. Most of these
big systems are mixed-radix tori which are not the best option for minimizing
distances and efficiently using network resources. This paper is focused on
improving the topological properties of these networks.
By using integral matrices to deal with Cayley graphs over Abelian groups, we
have been able to propose and analyze a family of high-dimensional grid-based
interconnection networks. As they are built over -dimensional grids that
induce a regular tiling of the space, these topologies have been denoted
\textsl{lattice graphs}. We will focus on cubic crystal lattices for modeling
symmetric 3D networks. Other higher dimensional networks can be composed over
these graphs, as illustrated in this research. Easy network partitioning can
also take advantage of this network composition operation. Minimal routing
algorithms are also provided for these new topologies. Finally, some practical
issues such as implementability and preliminary performance evaluations have
been addressed
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