1,016 research outputs found
The structure of causal sets
More often than not, recently popular structuralist interpretations of
physical theories leave the central concept of a structure insufficiently
precisified. The incipient causal sets approach to quantum gravity offers a
paradigmatic case of a physical theory predestined to be interpreted in
structuralist terms. It is shown how employing structuralism lends itself to a
natural interpretation of the physical meaning of causal sets theory.
Conversely, the conceptually exceptionally clear case of causal sets is used as
a foil to illustrate how a mathematically informed rigorous conceptualization
of structure serves to identify structures in physical theories. Furthermore, a
number of technical issues infesting structuralist interpretations of physical
theories such as difficulties with grounding the identity of the places of
highly symmetrical physical structures in their relational profile and what may
resolve these difficulties can be vividly illustrated with causal sets.Comment: 19 pages, 4 figure
A matrix representation of graphs and its spectrum as a graph invariant
We use the line digraph construction to associate an orthogonal matrix with
each graph. From this orthogonal matrix, we derive two further matrices. The
spectrum of each of these three matrices is considered as a graph invariant.
For the first two cases, we compute the spectrum explicitly and show that it is
determined by the spectrum of the adjacency matrix of the original graph. We
then show by computation that the isomorphism classes of many known families of
strongly regular graphs (up to 64 vertices) are characterized by the spectrum
of this matrix. We conjecture that this is always the case for strongly regular
graphs and we show that the conjecture is not valid for general graphs. We
verify that the smallest regular graphs which are not distinguished with our
method are on 14 vertices.Comment: 14 page
The Road to Quantum Computational Supremacy
We present an idiosyncratic view of the race for quantum computational
supremacy. Google's approach and IBM challenge are examined. An unexpected
side-effect of the race is the significant progress in designing fast classical
algorithms. Quantum supremacy, if achieved, won't make classical computing
obsolete.Comment: 15 pages, 1 figur
A remark on zeta functions of finite graphs via quantum walks
From the viewpoint of quantum walks, the Ihara zeta function of a finite
graph can be said to be closely related to its evolution matrix. In this note
we introduce another kind of zeta function of a graph, which is closely related
to, as to say, the square of the evolution matrix of a quantum walk. Then we
give to such a function two types of determinant expressions and derive from it
some geometric properties of a finite graph. As an application, we illustrate
the distribution of poles of this function comparing with those of the usual
Ihara zeta function.Comment: 14 pages, 1 figur
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