5,637 research outputs found
Quantum Turing Machines Computations and Measurements
Contrary to the classical case, the relation between quantum programming
languages and quantum Turing Machines (QTM) has not being fully investigated.
In particular, there are features of QTMs that have not been exploited, a
notable example being the intrinsic infinite nature of any quantum computation.
In this paper we propose a definition of QTM, which extends and unifies the
notions of Deutsch and Bernstein and Vazirani. In particular, we allow both
arbitrary quantum input, and meaningful superpositions of computations, where
some of them are "terminated" with an "output", while others are not. For some
infinite computations an "output" is obtained as a limit of finite portions of
the computation. We propose a natural and robust observation protocol for our
QTMs, that does not modify the probability of the possible outcomes of the
machines. Finally, we use QTMs to define a class of quantum computable
functions---any such function is a mapping from a general quantum state to a
probability distribution of natural numbers. We expect that our class of
functions, when restricted to classical input-output, will be not different
from the set of the recursive functions.Comment: arXiv admin note: substantial text overlap with arXiv:1504.02817 To
appear on MDPI Applied Sciences, 202
Probabilistic Computability and Choice
We study the computational power of randomized computations on infinite
objects, such as real numbers. In particular, we introduce the concept of a Las
Vegas computable multi-valued function, which is a function that can be
computed on a probabilistic Turing machine that receives a random binary
sequence as auxiliary input. The machine can take advantage of this random
sequence, but it always has to produce a correct result or to stop the
computation after finite time if the random advice is not successful. With
positive probability the random advice has to be successful. We characterize
the class of Las Vegas computable functions in the Weihrauch lattice with the
help of probabilistic choice principles and Weak Weak K\H{o}nig's Lemma. Among
other things we prove an Independent Choice Theorem that implies that Las Vegas
computable functions are closed under composition. In a case study we show that
Nash equilibria are Las Vegas computable, while zeros of continuous functions
with sign changes cannot be computed on Las Vegas machines. However, we show
that the latter problem admits randomized algorithms with weaker failure
recognition mechanisms. The last mentioned results can be interpreted such that
the Intermediate Value Theorem is reducible to the jump of Weak Weak
K\H{o}nig's Lemma, but not to Weak Weak K\H{o}nig's Lemma itself. These
examples also demonstrate that Las Vegas computable functions form a proper
superclass of the class of computable functions and a proper subclass of the
class of non-deterministically computable functions. We also study the impact
of specific lower bounds on the success probabilities, which leads to a strict
hierarchy of classes. In particular, the classical technique of probability
amplification fails for computations on infinite objects. We also investigate
the dependency on the underlying probability space.Comment: Information and Computation (accepted for publication
Zeno machines and hypercomputation
This paper reviews the Church-Turing Thesis (or rather, theses) with
reference to their origin and application and considers some models of
"hypercomputation", concentrating on perhaps the most straight-forward option:
Zeno machines (Turing machines with accelerating clock). The halting problem is
briefly discussed in a general context and the suggestion that it is an
inevitable companion of any reasonable computational model is emphasised. It is
hinted that claims to have "broken the Turing barrier" could be toned down and
that the important and well-founded role of Turing computability in the
mathematical sciences stands unchallenged.Comment: 11 pages. First submitted in December 2004, substantially revised in
July and in November 2005. To appear in Theoretical Computer Scienc
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