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Reflections on Quantum Computing
, 2000
"... In this rather speculative note three problems pertaining to the power and limits of quantum computing are posed and partially answered: a) when are quantum speedups possible?, b) is fixedpoint computing a better model for quantum computing?, c) can quantum computing trespass the Turing barrier? 1 ..."
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In this rather speculative note three problems pertaining to the power and limits of quantum computing are posed and partially answered: a) when are quantum speedups possible?, b) is fixedpoint computing a better model for quantum computing?, c) can quantum computing trespass the Turing barrier? 1 When are quantum speedups possible? This section discusses the possibility that speedups in quantum computing can be achieved only for problems which have a few or even unique solutions [12]. For instance, this includes the computational complexity class UP [15]. Typical examples are Shor's quantum algorithm for prime factoring [18] and Grover's database search algorithm [13] for a single item satisfying a given condition in an unsorted database (see also Gruska [14]). In quantum complexity, one popular class of problems is BQP,whichisthe set of decision problems that can be solved in polynomial time (on a quantum computer) so that the correct answer is obtained with probability at l...
TIME TRAVEL: A NEW HYPERCOMPUTATIONAL PARADIGM
, 2009
"... Assuming that all objections to time travel are set aside, it is shown that a computational system with closed timelike curves is a powerful hypercomputational tool. Speci cally, such a system allows us to solve four out of five problems recently advanced as counterexamples to the fundamental princi ..."
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Assuming that all objections to time travel are set aside, it is shown that a computational system with closed timelike curves is a powerful hypercomputational tool. Speci cally, such a system allows us to solve four out of five problems recently advanced as counterexamples to the fundamental principle of universality in computation. The fifth counterexample, however, remains unassailable, indicating that universality in computation cannot be achieved, even with the help of such an extraordinary ally as time travel.
Dynamics and stability of the Gödel universe
, 2003
"... We use covariant techniques to describe the properties of the Gödel universe and then consider its linear response to a variety of perturbations. Against matter aggregations, we find that the stability of the Gödel model depends primarily upon the presence of gradients in the centrifugal energy, and ..."
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We use covariant techniques to describe the properties of the Gödel universe and then consider its linear response to a variety of perturbations. Against matter aggregations, we find that the stability of the Gödel model depends primarily upon the presence of gradients in the centrifugal energy, and secondarily on the equation of state of the fluid. The latter dictates the behaviour of the model when dealing with homogeneous perturbations. The vorticity of the perturbed Gödel model is found to evolve as in almostFRW spacetimes, with some additional directional effects due to shape distortions. We also consider gravitationalwave perturbations by investigating the evolution of the magnetic Weyl component. This tensor obeys a simple planewave equation, which argues for the neutral stability of the Gödel model against linear gravitywave distortions. The implications of the background rotation for scalarfield Gödel cosmologies are also discussed. PACS numbers: 0420Cv, 9880Jk 1
The time travel paradox
, 2001
"... We define the time travel paradox in physical terms and prove its existence by constructing an explicit example. We argue further that in the theories — such as general relativity — where the spacetime geometry is subject to nothing but differential equations and initial data no paradoxes arise. 1 ..."
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We define the time travel paradox in physical terms and prove its existence by constructing an explicit example. We argue further that in the theories — such as general relativity — where the spacetime geometry is subject to nothing but differential equations and initial data no paradoxes arise. 1
Bidirectional classical stochastic processes with measurements and feedback
, 2006
"... A measurement on a quantum system is said to cause the “collapse” of the quantum state vector or density matrix. An analogous collapse occurs with measurements on a classical stochastic process. This paper addresses the question of describing the response of a classical stochastic process when there ..."
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A measurement on a quantum system is said to cause the “collapse” of the quantum state vector or density matrix. An analogous collapse occurs with measurements on a classical stochastic process. This paper addresses the question of describing the response of a classical stochastic process when there is feedback from the output of a measurement to the input, and is intended to give a simplified model for quantummechanical processes that occur along a spacelike reaction coordinate. The classical system can be thought of in physical terms as two counterflowing probability streams, which stochastically exchange probability currents in a way that the net probability current, and hence the overall probability, suitably interpreted, is conserved. The proposed formalism extends the mathematics of those stochastic processes describable with linear, singlestep, unidirectional transition probabilities, known as Markov chains and stochastic matrices. It is shown that a certain rearrangement and combination of the input and output of two stochastic matrices of the same order yields another matrix of the same type. Each measurement causes the partial collapse of the probability current distribution in the midst of such a process, giving rise to calculable, but nonMarkov, values for the ensuing modification of the system’s output probability distribution. The paper concludes with an analysis of a simple classical probabilistic version of a socalled grandfather paradox. 1
Gödel Universes in String Theory
, 1998
"... We show that homogeneous Gödel spacetimes need not contain closed timelike curves in lowenergyeffective string theories. We find exact solutions for the Gödel metric in string theory for the full O(α ′) action including both dilaton and axion fields. The results are valid for bosonic, heterotic an ..."
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We show that homogeneous Gödel spacetimes need not contain closed timelike curves in lowenergyeffective string theories. We find exact solutions for the Gödel metric in string theory for the full O(α ′) action including both dilaton and axion fields. The results are valid for bosonic, heterotic and superstrings. To first order in the inverse string tension α ′, these solutions display a simple relation between the angular velocity of the Gödel universe, Ω, and the inverse string tension of the form α ′ = 1/Ω 2 in the absence of the axion field. The generalization of this relationship is also found when the axion field is present. PACS number(s): 98.80.Hw, 04.50.+h, 11.25.Mj, 98.80.Cq 1 I.