3 research outputs found
On the completeness of quantum computation models
The notion of computability is stable (i.e. independent of the choice of an
indexing) over infinite-dimensional vector spaces provided they have a finite
"tensorial dimension". Such vector spaces with a finite tensorial dimension
permit to define an absolute notion of completeness for quantum computation
models and give a precise meaning to the Church-Turing thesis in the framework
of quantum theory. (Extra keywords: quantum programming languages, denotational
semantics, universality.)Comment: 15 pages, LaTe
The physical Church-Turing thesis and the principles of quantum theory
Notoriously, quantum computation shatters complexity theory, but is innocuous
to computability theory. Yet several works have shown how quantum theory as it
stands could breach the physical Church-Turing thesis. We draw a clear line as
to when this is the case, in a way that is inspired by Gandy. Gandy formulates
postulates about physics, such as homogeneity of space and time, bounded
density and velocity of information --- and proves that the physical
Church-Turing thesis is a consequence of these postulates. We provide a quantum
version of the theorem. Thus this approach exhibits a formal non-trivial
interplay between theoretical physics symmetries and computability assumptions.Comment: 14 pages, LaTe
Causal graph dynamics
We extend the theory of Cellular Automata to arbitrary, time-varying graphs.
In other words we formalize, and prove theorems about, the intuitive idea of a
labelled graph which evolves in time - but under the natural constraint that
information can only ever be transmitted at a bounded speed, with respect to
the distance given by the graph. The notion of translation-invariance is also
generalized. The definition we provide for these "causal graph dynamics" is
simple and axiomatic. The theorems we provide also show that it is robust. For
instance, causal graph dynamics are stable under composition and under
restriction to radius one. In the finite case some fundamental facts of
Cellular Automata theory carry through: causal graph dynamics admit a
characterization as continuous functions, and they are stable under inversion.
The provided examples suggest a wide range of applications of this mathematical
object, from complex systems science to theoretical physics. KEYWORDS:
Dynamical networks, Boolean networks, Generative networks automata, Cayley
cellular automata, Graph Automata, Graph rewriting automata, Parallel graph
transformations, Amalgamated graph transformations, Time-varying graphs, Regge
calculus, Local, No-signalling.Comment: 25 pages, 9 figures, LaTeX, v2: Minor presentation improvements, v3:
Typos corrected, figure adde