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Halting probability amplitude of quantum computers
 Journal of Universal Computer Science
, 1995
"... The classical halting probability to quantum computations. introduced by Chaitin is generalized Chaitin's [1,2,3] is a magic number. It is a measure for arbitrary programs to take a nite number of execution steps and then halt. It contains the solution for all halting problems, and hence to que ..."
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The classical halting probability to quantum computations. introduced by Chaitin is generalized Chaitin's [1,2,3] is a magic number. It is a measure for arbitrary programs to take a nite number of execution steps and then halt. It contains the solution for all halting problems, and hence to questions codable into halting problems, such asFermat's theorem. It contains the solution for the question of whether or not a particular exponential Diophantine equation has in nitely many ora nite number of solutions. And, since is provable \algorithmically incompressible," it is MartinLof/Chaitin/Solovay random. Therefore, is both: a mathematicians \fair coin, " and a formalist's nightmare. Here, is generalized to quantum computations. Consider a (not necessarily universal) quantum computer C and its ith program pi, which, at time t 2 Z, can be described by a quantum state [4, 5,6,7,
Set Theory and Physics
 FOUNDATIONS OF PHYSICS, VOL. 25, NO. 11
, 1995
"... Inasmuch as physical theories are formalizable, set theory provides a framework for theoretical physics. Four speculations about the relevance of set theoretical modeling for physics are presented: the role of transcendental set theory (i) hr chaos theory, (ii) for paradoxical decompositions of soli ..."
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Inasmuch as physical theories are formalizable, set theory provides a framework for theoretical physics. Four speculations about the relevance of set theoretical modeling for physics are presented: the role of transcendental set theory (i) hr chaos theory, (ii) for paradoxical decompositions of solid threedimensional objects, (iii) in the theory of effective computability (ChurchTurhrg thesis) related to the possible "solution of supertasks," and (iv) for weak solutions. Several approaches to set theory and their advantages and disadvatages for" physical applications are discussed: Cantorian "naive" (i.e., nonaxiomatic) set theory, contructivism, and operationalism, hr the arrthor's ophrion, an attitude of "suspended attention" (a term borrowed from psychoanalysis) seems most promising for progress. Physical and set theoretical entities must be operationalized wherever possible. At the same thne, physicists shouM be open to "bizarre" or "mindboggling" new formalisms, which treed not be operationalizable or testable at the thne of their " creation, but which may successfully lead to novel fields of phenomenology and technology.
How real are virtual realities, how virtual is reality? The constructive reinterpretation of physical undecidability
"... constructive reinterpretation of physical undecidability ..."
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constructive reinterpretation of physical undecidability
Speedup in Quantum Computation is Associated With Attenuation of Processing Probability
"... Quantum coherence allows the computation of an arbitrary number of distinct computational paths in parallel. Based on quantum parallelism it has been conjectured that exponential or even larger speedups of computations are possible. Here it is shown that, although in principle correct, any speed ..."
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Quantum coherence allows the computation of an arbitrary number of distinct computational paths in parallel. Based on quantum parallelism it has been conjectured that exponential or even larger speedups of computations are possible. Here it is shown that, although in principle correct, any speedup is accompanied by an associated attenuation of detection rates. Thus, on the average, no effective speedup is obtained relative to classical (nondeterministic) devices. 1 Recent findings in quantum complexity theory suggest an exponential speedup of discrete logarithms and factoring [1] and the travelling salesman problem [2] with respect to classical complexity. (The best classical estimate for the computing time for factoring is e cn 1/3 , where n is the number of bits in the number to be factored and c is a constant; the travelling salesman problem is NPcomplete). At the heart of these types of speedups is quantum parallelism. Roughly stated, quantum parallelism assures that a ...
Quantum computation and complexity theory (Course given at the Institut für Informationssysteme, Abteilung für Datenbanken and Expertsysteme
 University of Technology, Vienna.) LANL Eprint archive HEPTH 9412047
, 1994
"... The Hilbert space formalism of quantum mechanics is reviewed with emphasis on applications to quantum computing. Standard interferomeric techniques are used to construct a physical device capable of universal quantum computation. Some consequences for recursion theory and complexity theory are discu ..."
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The Hilbert space formalism of quantum mechanics is reviewed with emphasis on applications to quantum computing. Standard interferomeric techniques are used to construct a physical device capable of universal quantum computation. Some consequences for recursion theory and complexity theory are discussed. qct.tex
Undecidability Everywhere?
, 1996
"... We discuss the question of if and how undecidability might be translatable into physics, in particular with respect to prediction and description, as well as to complementarity games. 1 1 Physics after the incompleteness theorems There is incompleteness in mathematics [22, 63, 65, 13, 9, 12, 51 ..."
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We discuss the question of if and how undecidability might be translatable into physics, in particular with respect to prediction and description, as well as to complementarity games. 1 1 Physics after the incompleteness theorems There is incompleteness in mathematics [22, 63, 65, 13, 9, 12, 51]. That means that there does not exist any reasonable (consistent) finite formal system from which all mathematical truth is derivable. And there exists a "huge" number [11] of mathematical assertions (e.g., the continuum hypothesis, the axiom of choice) which are independent of any particular formal system. That is, they as well as their negations are compatible with the formal system. Can such formal incompleteness be translated into physics or the natural sciences in general? Is there some question about the nature of things which is provable unknowable for rational thought? Is it conceivable that the natural phenomena, even if they occur deterministically, do not allow their complete d...
THE BERRY PARADOX
, 1994
"... videotaped; this is an edited transcript. It also incorporates remarks made at the Limits to Scientific Knowledge meeting held at the Santa ..."
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videotaped; this is an edited transcript. It also incorporates remarks made at the Limits to Scientific Knowledge meeting held at the Santa
unknown title
, 2008
"... Consistent use of paradoxes in deriving constraints on the dynamics of physical systems and of nogotheorems ..."
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Consistent use of paradoxes in deriving constraints on the dynamics of physical systems and of nogotheorems
Quantum algorithmic information theory
, 2008
"... The agenda of quantum algorithmic information theory, ordered ‘topdown, ’ is the quantum halting amplitude, followed by the quantum algorithmic information content, which in turn requires the theory of quantum computation. The fundamental atoms processed by quantum computation are the quantum bits ..."
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The agenda of quantum algorithmic information theory, ordered ‘topdown, ’ is the quantum halting amplitude, followed by the quantum algorithmic information content, which in turn requires the theory of quantum computation. The fundamental atoms processed by quantum computation are the quantum bits which are dealt with in quantum information theory. The theory of quantum computation will be based upon a model of universal quantum computer whose elementary unit is a twoport interferometer capable of arbitrary U(2) transformations. Basic to all these considerations is quantum theory, in particular Hilbert space quantum mechanics.