## Computing the non-computable

Venue: | Contemporary Physics |

Citations: | 31 - 7 self |

### BibTeX

@ARTICLE{Kieu_computingthe,

author = {Tien D Kieu},

title = {Computing the non-computable},

journal = {Contemporary Physics},

year = {},

pages = {2003}

}

### Years of Citing Articles

### OpenURL

### Abstract

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 quantum-mechanical probability distributions for these values. It is argued that computability, and thus the limits of Mathematics, ought to be determined not

### Citations

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(Show Context)
Citation Context ...al paths to extract the information desired. These characteristics have been exploited to reduce the computational complexity of some problems. So far there are only few quantum algorithms discovered =-=[36, 19]-=-; most notable is Shor’s factorisation algorithm which employs Quantum Fourier Transformation. QFT is the only known quantum algorithm that could offer an exponential increase in computational speed, ... |

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Citation Context ...nfinite because each machine can be mapped into a unique integer. On the other hand, the whole class of functions from natural numbers to natural numbers can be shown by the Cantor diagonal arguments =-=[35]-=- to be uncountably infinite (in fact, this class of functions has the same cardinality as the set of reals). A pictorial and heuristic way to visualise this fact is where each mapping from natural num... |

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A decision method for elementary algebra and geometry
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Citation Context ...ems –because the unsolvability of the Hilbert’s tenth problem is only established in the framework of integer arithmetic and in Turing computability, not necessarily in Mathematics in general. Tarski =-=[39]-=- has shown that the question about the existence of real solutions of polynomials over the reals is, in fact, decidable. It seems appropriate to end here with a quotation from the man whose famous Inc... |

584 |
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(Show Context)
Citation Context ...water waves, can also have superposition but cannot provide a better computation model.) The second and most important power of quantum computation is thought to have its root in quantum entanglement =-=[4]-=-, which has no counterpart in the classical world (even though it might be expensively simulated by classical means). Quantum entanglement provides the extra dimensions in information storage and proc... |

491 | Quantum complexity theory
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(Show Context)
Citation Context ...g machine”, begs the question whether it could be extended with quantum principles. Initial efforts seem to confirm that quantum computability is no more than classical and mathematical computability =-=[3]-=-. However, more recent indications may prove otherwise [22, 8]. We will come back to this computability notion in a section below. All of the above have inevitably led to the recent convergence of qua... |

426 | Simulating physics with computers
- Feynman
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(Show Context)
Citation Context ...ose given by quantum physics. However, the best present day computers, as they are of classical nature, cannot even in principle simulate quantum systems efficiently. Feynman pointed out in 1982 that =-=[16]-=-, see also [2], only quantum mechanical systems may be able to simulate other quantum systems more efficiently. Furthermore, according to Moore’s law, the exponential rate of miniaturisation of micro-... |

337 | Algorithmic information theory
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(Show Context)
Citation Context ... only have bounded universal quantifiers as opposed to arbitrarily unbounded universal quantifiers of Arithmetic at large. Chaitin, approaching from the perspectives of Algorithmic Information Theory =-=[6, 7]-=-, has shown that there exist many unprovable statements in Arithmetic simply because they have irreducible algorithmic contents, measurable in bits, that are more than the complexity, also measurable ... |

321 | Quantum mechanics helps in searching for a needle in a haystack,” Phys
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(Show Context)
Citation Context ...al paths to extract the information desired. These characteristics have been exploited to reduce the computational complexity of some problems. So far there are only few quantum algorithms discovered =-=[36, 19]-=-; most notable is Shor’s factorisation algorithm which employs Quantum Fourier Transformation. QFT is the only known quantum algorithm that could offer an exponential increase in computational speed, ... |

251 |
Quantum Optics
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- 1995
(Show Context)
Citation Context ...r states are not the states of travelling optical modes generated by idealised lasers [13], which have an indefinite number of photons. For the description of these modes, we need the coherent states =-=[44]-=-, |α〉, which are the eigenstates of a and are labeled by the complex number α, With the relation to the number states, a|α〉 = α|α〉. (18) |α〉 |α|2 − = e 2 ∞∑ n=0 α n √ n! |n〉, |α|2 − = e 2 e αa† |0〉, (... |

151 |
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(Show Context)
Citation Context ...ubroutine. This will be the case if qh can only accept integers while quantum algorithms, with proper definitions, cannot in general be themselves encoded as integers. In fact, the no-cloning theorem =-=[45]-=- of quantum mechanics does restrict the type of operations available to quantum algorithms. In essence, the way we will break the self-referential reasoning here by the differentiation between quantum... |

149 |
Hilbert’s Tenth Problem
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(Show Context)
Citation Context ...utation and Turing machines which were to be introduced some 40 years later. Eventually, the Hilbert’s tenth problem was finally shown to be undecidable in 1970 through a crucial step by Matiyasevich =-=[27, 9]-=-: The Hilbert’s tenth problem could be solved if and only if the Turing halting problem could also be solved. The two are simply equivalent. Consequently, as we have a proof that the Turing’s is not s... |

149 |
A single quantum cannot be cloned
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(Show Context)
Citation Context ...ubroutine. This will be the case if qh can only accept integers while quantum algorithms, with proper definitions, cannot in general be themselves encoded as integers. In fact, the no-cloning theorem =-=[42]-=- of quantum mechanics does restrict the type of operations available to quantum algorithms. In essence, the way we will break the self-referential reasoning here by the differentiation between quantum... |

120 |
Computability and Unsolvability
- Davis
- 1983
(Show Context)
Citation Context ...utation and Turing machines which were to be introduced some 40 years later. Eventually, the Hilbert’s tenth problem was finally shown to be undecidable in 1970 through a crucial step by Matiyasevich =-=[27, 9]-=-: The Hilbert’s tenth problem could be solved if and only if the Turing halting problem could also be solved. The two are simply equivalent. Consequently, as we have a proof that the Turing’s is not s... |

70 | NonTuring computations via MalamentHogarth space-times Int.J.Theor.Phys
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- 2002
(Show Context)
Citation Context ...computation. Our study is an illustration of “Information is physical”. Very recently there also exist in the literature some efforts to explore mathematical computability in the framework of Physics =-=[37, 14, 8]-=-. Inspired by Quantum Mechanics, we have then reformulated the question of solution existence of a Diophantine equation into the question of certain properties contained in an infinitely coupled set o... |

60 | Quantum algorithm for the Hilbert’s Tenth Problem
- Kieu
- 2003
(Show Context)
Citation Context ... with quantum principles. Initial efforts seem to confirm that quantum computability is no more than classical and mathematical computability [3]. However, more recent indications may prove otherwise =-=[22, 8]-=-. We will come back to this computability notion in a section below. All of the above have inevitably led to the recent convergence of quantum physics, mathematics, and computing and information proce... |

48 |
Recursive Functions
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(Show Context)
Citation Context ... 0 then the Turing machine, which corresponds to g, would halt upon the input corresponding to (x1, · · ·, xn). If no such y exists, the Turing machine in question would not halt. Gödel, as quoted in =-=[33]-=-, has shown that the µ-minimising operation (3) can always be represented as some arithmetic statement with a set of identities between multi-variate polynomials over the integers, together with a fin... |

35 |
The Church-Turing thesis as a guiding principle for physics
- Svozil
- 1998
(Show Context)
Citation Context ...Russell resolved the set theory paradox (to do with “The set of all sets which are not members of themselves”) by the introduction of classes as distinct from sets. (For other lines of arguments, see =-=[38, 37, 8]-=- for example.) To investigate the decidability of the Turing halting problem in the framework of quantum computability, we will need to isolate the point which causes the classical undecidability. Wit... |

23 |
Quantum computation over continuous variables, Phys
- Lloyd, Braunstein
- 1999
(Show Context)
Citation Context ...ntum computation. Its advantage over the infinitely many qubits which would otherwise be required is obvious. One way to construct any suitable Hamiltonian so desired is through the technique of ref. =-=[27]-=-. We consider the hermitean operators, where j is the index for the unknowns of the Diophantine equation, Xj = 1 √ 2 (aj + a † j), Pj = i √ 2 (aj − a † j), (29) [Pj, Xk] = iδjk. Together with the avai... |

19 |
Computable functions, quantum measurements, and quantum dynamics. Phys
- Nielsen
- 1997
(Show Context)
Citation Context ...t do in a finite number of steps. Note that it was the general Hamiltonian computation that was discussed by Benioff and Feynman [2, 16] in the conception days of quantum computation. Indeed, Nielsen =-=[32]-=- has also found no logical contradiction in applying the most general quantum mechanical principles to the computation of the classical noncomputable, unless certain Hermitean operators cannot somehow... |

16 | A reformulation of Hilbert’s tenth problem through quantum mechanics
- Kieu
- 2004
(Show Context)
Citation Context ...ithms above. Mathematically, all we need to do is to sort out the instantaneous ground state |g〉 among the infinitely many eigenvectors of H(S) in (25); but this is a hard task. The trick we will use =-=[24]-=-, as inspired by quantum adiabatic processes, is to tag the state |g〉 by some other known state |gI〉 which is the ground state of some other operator HI and can be smoothly connected to |g〉 through so... |

11 |
Private communication
- Nielsen
- 2005
(Show Context)
Citation Context ...ions should be controlled and reduced to a degree that is smaller than the size of the smallest energy gap such as not to cause transitions out of the adiabatic process. This may present a difficulty =-=[34]-=- for 27our algorithm, but one difficulty of technical nature rather than of principle. Even though these fluctuations in the energy levels are ever present and cannot be reduced to zero, there is no ... |

8 |
Algorithmic Information Theory, Cambridge University Press Chaitin, G.J
- Chaitin
- 1987
(Show Context)
Citation Context ... only have bounded universal quantifiers as opposed to arbitrarily unbounded universal quantifiers of Arithmetic at large. Chaitin, approaching from the perspectives of Algorithmic Information Theory =-=[6, 7]-=-, has shown that there exist many unprovable statements in Arithmetic simply because they have irreducible algorithmic contents, measurable in bits, that are more than the complexity, also measurable ... |

8 |
Alan Turing: the Enigma, Burnett books and
- Hodges
- 1983
(Show Context)
Citation Context ... Turing machines The concept of computation is at the heart of Mathematics and hard sciences but it was only made precise by Alan Turing relatively recently, at the beginning of the twentieth century =-=[20]-=-. With the introduction of theoretical and mathematically well-defined machines, Turing was able to capture the essence of computation processes and algorithms. There are a few other models of computa... |

6 |
private communication
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- 1979
(Show Context)
Citation Context ...tate having more photons! Only in the final act of measuring photon numbers, the more-photon state would transfer more energy in the measuring device than the less-photon state. A fundamental problem =-=[12]-=- is that the hamiltonians which we need to be simulated in the optical apparati are only effective hamiltonians in that their descriptions are only valid for certain range of number of photons. When t... |

6 |
Hypercomputation is experimentally irrefutable
- Stannett
- 2001
(Show Context)
Citation Context ...Russell resolved the set theory paradox (to do with “The set of all sets which are not members of themselves”) by the introduction of classes as distinct from sets. (For other lines of arguments, see =-=[38, 37, 8]-=- for example.) To investigate the decidability of the Turing halting problem in the framework of quantum computability, we will need to isolate the point which causes the classical undecidability. Wit... |

4 |
The computer as a physical system
- Benioff
- 1980
(Show Context)
Citation Context ...antum physics. However, the best present day computers, as they are of classical nature, cannot even in principle simulate quantum systems efficiently. Feynman pointed out in 1982 that [16], see also =-=[2]-=-, only quantum mechanical systems may be able to simulate other quantum systems more efficiently. Furthermore, according to Moore’s law, the exponential rate of miniaturisation of micro-electronic sem... |

4 | Gödel’s Incompleteness Theorem - Uspensky - 1987 |

2 |
Quantum computation by adiabatic evolution, Archive quant-ph/0001106
- Farhi, Goldstone, et al.
- 2000
(Show Context)
Citation Context ... but is the same as classical computability. However, it is not the only model available. 4.2 Quantum Adiabatic Computation Among the alternative models for quantum computation is the recent proposal =-=[15]-=- to employ quantum adiabatic processes for computation. The idea is to encode the solution of some problem to be solved into the ground state, |g〉, of some suitable hamiltonian, HP. But as it is easie... |

2 |
Quantum annealing in the transverse Ising model,” Phys
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- 1998
(Show Context)
Citation Context ...nomial hamiltonian constructed, we need to obtain its ground state. Any approach that allows us to access the ground state will suffice. One way is perhaps to use that of quantum annealing or cooling =-=[21]-=-. Another way is to employ the quantum computation method of adiabatic evolution [15]. 5.2 Quantum adiabatic approach In the adiabatic approach, one starts, for example, with a hamiltonian HI, K∑ HI =... |

1 |
statement delivered to the
- Gödel
- 1951
(Show Context)
Citation Context ...appropriate to end here with a quotation from the man whose famous Incompleteness result has often been misquoted as spelling the end for computability/provability in Arithmetic. In Gödel’s own words =-=[18]-=-: “... On the other hand, on the basis of what has been proved so far, it remains possible that there may exist (and even be empirically discoverable) a theorem-proving machine which in fact is equiva... |

1 |
Elements of the Theory of Computation (Prentice
- Lewis, Papadimitriou
- 1981
(Show Context)
Citation Context ...h the introduction of theoretical and mathematically well-defined machines, Turing was able to capture the essence of computation processes and algorithms. There are a few other models of computation =-=[25]-=- but they all, except possibly the quantum model to be considered later on, can be shown to be equivalent to that given by Turing machines. Turing machines are equipped with an infinite one-dimensiona... |

1 |
Quantum principles and mathematical computability, Archive quant-ph/0205093
- Kieu
- 2002
(Show Context)
Citation Context ...preparing the same quantum state over and over again for subsequent measurements! This quantum mechanically implied infinity seems to be both needed for and consistent with the finitely measured, see =-=[25]-=- and references therein for further discussion. 3.3 Coherent states One of the simplest and most widely applicable problems in Quantum Mechanics is that of the (one-dimensional) Simple Harmonic Oscill... |