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Quantum Algorithm For Hilberts Tenth Problem
 Int.J.Theor.Phys
, 2003
"... We explore in the framework of Quantum Computation the notion of 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 noncomp ..."
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Cited by 62 (10 self)
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We explore in the framework of Quantum Computation the notion of 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 ChurchTuring 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. 1
Computing the noncomputable
 Contemporary Physics
"... We explore in the framework of Quantum Computation the notion of computability, which holds a central position in Mathematics and Theoretical Computer Science. A quantum algorithm that exploits the quantum adiabatic which is equivalent to the Turing halting problem and known to be mathematically non ..."
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Cited by 30 (7 self)
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We explore in the framework of Quantum Computation the notion of computability, which holds a central position in Mathematics and Theoretical Computer Science. A quantum algorithm that exploits the quantum adiabatic which is equivalent to the Turing halting problem and known to be mathematically noncomputable. Generalised quantum algorithms are also considered for some other mathematical noncomputables in the same and of different noncomputability classes. The key element of all these algorithms is the measurability of both the values of physical observables and of the quantummechanical probability distributions for these values. It is argued that computability, and thus the limits of Mathematics, ought to be determined not
Hypercomputability of quantum adiabatic processes: facts versus prejudices
 http://arxiv.org/quantph/0504101
, 2005
"... Abstract. We give an overview of a quantum adiabatic algorithm for Hilbert’s tenth problem, including some discussions on its fundamental aspects and the emphasis on the probabilistic correctness of its findings. For the purpose of illustration, the numerical simulation results of some simple Diopha ..."
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Cited by 12 (3 self)
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Abstract. We give an overview of a quantum adiabatic algorithm for Hilbert’s tenth problem, including some discussions on its fundamental aspects and the emphasis on the probabilistic correctness of its findings. For the purpose of illustration, the numerical simulation results of some simple Diophantine equations are presented. We also discuss some prejudicial misunderstandings as well as some plausible difficulties faced by the algorithm in its physical implementations. “To believe otherwise is merely to cling to a prejudice which only gives rise to further prejudices... ” 1
On the identification of the ground state based on occupation probabilities: An investigation of Smith’s apparent counterexample
 Journal of Applied Mathematics and Computation
, 2005
"... Abstract. We study a set of truncated matrices, given by Smith [8], in connection to an identification criterion for the ground state in our proposed quantum adiabatic algorithm for Hilbert’s tenth problem. We identify the origin of the trouble for this truncated example and show that for a suitable ..."
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Cited by 9 (0 self)
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Abstract. We study a set of truncated matrices, given by Smith [8], in connection to an identification criterion for the ground state in our proposed quantum adiabatic algorithm for Hilbert’s tenth problem. We identify the origin of the trouble for this truncated example and show that for a suitable choice of some parameter it can always be removed. We also argue that it is only an artefact of the truncation of the underlying Hilbert spaces, through showing its sensitivity to different boundary conditions available for such a truncation. It is maintained that the criterion, in general, should be applicable provided certain conditions are satisfied. We also point out that, apart from this one, other criteria serving the same identification purpose may also be available. In a proposal of a quantum adiabatic algorithm for Hilbert’s tenth problem [5], we employ an adiabatic process with a timedependent Hamiltonian (1) H(t) = (1 − t/T)HI + (t/T)HP. Here t is time and this Hamiltonian metamorphoses from HI when t = 0 to HP when t = T. The final Hamiltonian HP encodes the Diophantine equation in consideration, while the initial HI is universal and independent of the Diophantine equation, except only on its number of variables K. The process is captured by the Schrödinger equation (2) ∂tψ(t)〉
On the existence of truly autonomic computing systems and the link with quantum computing, arXiv: cs.LO/0411094
"... A theoretical model of truly autonomic computing systems (ACS), with infinitely many constraints, is proposed. An argument similar to Turing’s for the unsolvability of the halting problem, which is permitted in classical logic, shows that such systems cannot exist. Turing’s argument fails in the rec ..."
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Cited by 3 (2 self)
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A theoretical model of truly autonomic computing systems (ACS), with infinitely many constraints, is proposed. An argument similar to Turing’s for the unsolvability of the halting problem, which is permitted in classical logic, shows that such systems cannot exist. Turing’s argument fails in the recently proposed nonAristotelian finitary logic (NAFL), which permits the existence of ACS. NAFL also justifies quantum superposition and entanglement, which are essential ingredients of quantum algorithms, and resolves the EinsteinPodolskyRosen (EPR) paradox in favour of quantum mechanics and nonlocality. NAFL requires that the autonomic manager (AM) must be conceptually and architecturally distinct from the managed element, in order for the ACS to exist as a nonselfreferential entity. Such a scenario is possible if the AM uses quantum algorithms and is protected from all problems by (unbreakable) quantum encryption, while the managed element remains classical. NAFL supports such a link between autonomic and quantum computing, with the AM existing as a metamathematical entity. NAFL also allows quantum algorithms to access truly random elements and thereby supports nonstandard models of quantum (hyper) computation that permit infinite parallelism. 1.
2005 A possible hypercomputational quantum algorithm Quantum
 SPIE) Proc. SPIE 5815 219–26
, 2005
"... The term ‘hypermachine ’ denotes any data processing device (theoretical or that can be implemented) capable of carrying out tasks that cannot be performed by a Turing machine. We present a possible quantum algorithm for a classically noncomputable decision problem, Hilbert’s tenth problem; more sp ..."
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Cited by 3 (3 self)
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The term ‘hypermachine ’ denotes any data processing device (theoretical or that can be implemented) capable of carrying out tasks that cannot be performed by a Turing machine. We present a possible quantum algorithm for a classically noncomputable decision problem, Hilbert’s tenth problem; more specifically, we present a possible hypercomputation model based on quantum computation. Our algorithm is inspired by the one proposed by Tien D. Kieu, but we have selected the infinite square well instead of the (onedimensional) simple harmonic oscillator as the underlying physical system. Our model exploits the quantum adiabatic process and the characteristics of the representation of the dynamical Lie algebra su(1, 1) associated to the infinite square well.
Hilbert’s incompleteness, Chaitin’s Ω number and quantum physics, Los Alamos preprint archive http://arXiv:quantph/0111062
, 2001
"... To explore the limitation of a class of quantum algorithms originally proposed for the Hilbert’s tenth problem, we consider two further classes of mathematically nondecidable problems, those of a modified version of the Hilbert’s tenth problem and of the computation of the Chaitin’s Ω number, which ..."
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Cited by 2 (1 self)
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To explore the limitation of a class of quantum algorithms originally proposed for the Hilbert’s tenth problem, we consider two further classes of mathematically nondecidable problems, those of a modified version of the Hilbert’s tenth problem and of the computation of the Chaitin’s Ω number, which is a representation of the Gödel’s Incompletness theorem. Some interesting connection to Quantum Field Theory is pointed out, but a direct generalisation of the quantum algorithms cannot satisfy, among others, the requirement of finite energy.
MATHEMATICAL COMPUTABILITY QUESTIONS FOR SOME CLASSES OF LINEAR AND NONLINEAR DIFFERENTIAL EQUATIONS ORIGINATED FROM HILBERT’S TENTH PROBLEM
, 2005
"... Abstract. Inspired by Quantum Mechanics, we reformulate Hilbert’s tenth problem in the domain of integer arithmetics into problems involving either a set of infinitelycoupled nonlinear differential equations or a class of linear Schrödinger equations with some appropriate timedependent Hamiltonian ..."
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Cited by 2 (2 self)
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Abstract. Inspired by Quantum Mechanics, we reformulate Hilbert’s tenth problem in the domain of integer arithmetics into problems involving either a set of infinitelycoupled nonlinear differential equations or a class of linear Schrödinger equations with some appropriate timedependent Hamiltonians. We then raise the questions whether these two classes of differential equations are computable or not in some computation models of computable analysis. These are nontrivial and important questions given that: (i) not all computation models of computable analysis are equivalent, unlike the case with classical recursion theory; (ii) and not all models necessarily and inevitably reduce computability of real functions to discrete computations on Turing machines. However unlikely the positive answers to our computability questions, their existence should deserve special attention and be satisfactorily settled since such positive answers may also have interesting logical consequence back in the classical recursion theory for the ChurchTuring thesis.
Quantum Hypercomputation—Hype or Computation?
, 2007
"... A recent attempt to compute a (recursion–theoretic) non–computable function using the quantum adiabatic algorithm is criticized and found wanting. Quantum algorithms may outperform classical algorithms in some cases, but so far they retain the classical (recursion–theoretic) notion of computability. ..."
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Cited by 1 (0 self)
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A recent attempt to compute a (recursion–theoretic) non–computable function using the quantum adiabatic algorithm is criticized and found wanting. Quantum algorithms may outperform classical algorithms in some cases, but so far they retain the classical (recursion–theoretic) notion of computability. A speculation is then offered as to where the putative power of quantum computers may come from.