7,430 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
Quantum Algorithm for Hilbert's Tenth Problem
We explore in the framework of Quantum Computation the notion of {\em
Computability}, which holds a central position in Mathematics and Theoretical
Computer Science. A quantum algorithm for Hilbert's tenth problem, which is
equivalent to the Turing halting problem and is known to be mathematically
noncomputable, is proposed where quantum continuous variables and quantum
adiabatic evolution are employed. If this algorithm could be physically
implemented, as much as it is valid in principle--that is, if certain
hamiltonian and its ground state can be physically constructed according to the
proposal--quantum computability would surpass classical computability as
delimited by the Church-Turing thesis. It is thus argued that computability,
and with it the limits of Mathematics, ought to be determined not solely by
Mathematics itself but also by Physical Principles
Computability of probability measures and Martin-Lof randomness over metric spaces
In this paper we investigate algorithmic randomness on more general spaces
than the Cantor space, namely computable metric spaces. To do this, we first
develop a unified framework allowing computations with probability measures. We
show that any computable metric space with a computable probability measure is
isomorphic to the Cantor space in a computable and measure-theoretic sense. We
show that any computable metric space admits a universal uniform randomness
test (without further assumption).Comment: 29 page
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