## Computational Power of Infinite Quantum Parallelism (2005)

Venue: | pp.2057–2071 in International Journal of Theoretical Physics vol.44:11 |

Citations: | 2 - 1 self |

### BibTeX

@INPROCEEDINGS{Ziegler05computationalpower,

author = {Martin Ziegler},

title = {Computational Power of Infinite Quantum Parallelism},

booktitle = {pp.2057–2071 in International Journal of Theoretical Physics vol.44:11},

year = {2005}

}

### OpenURL

### Abstract

Recent works have independently suggested that quantum mechanics might permit procedures that fundamentally transcend the power of Turing Machines as well as of ‘standard ’ Quantum Computers. These approaches rely on and indicate that quantum mechanics seems to support some infinite variant of classical parallel computing. We compare this new one with other attempts towards hypercomputation by separating (1) its computing capabilities from (2) realizability issues. The first are shown to coincide with recursive enumerability; the second are considered in analogy to ‘existence’ in mathematical logic. KEY WORDS: Hypercomputation; quantum mechanics; recursion theory; infinite parallelism.

### Citations

4144 |
Introduction to Automata Theory, Languages, and Computation
- Hopcroft, Ullman
- 1979
(Show Context)
Citation Context ...etical Computer Science known as uniformity. (iv) A TM M0 is capable of generating, upon input of k, (the encoding of) Mk. Here, encoding refers to a ‘blueprint’ of Mk; formally: to its Gödel Number (=-=Hopcroft et al., 2001-=-, Section 9.1.2). 2.3. Computational Power of Infinite Turing Concurrency This section reveals that the Definition 2.3. (i)–(iv) indeed yields an interesting non-trivial way of hypercomputation. More ... |

1308 | On computable numbers, with an application to the Entscheidungsproblem - Turing - 1936 |

506 |
Recursively Enumerable Sets and Degrees
- Soare
(Show Context)
Citation Context ...field of Computability or Recursion Theory (Odifreddi, 1989). Its goal is to distinguish computable from uncomputable problems and to classify the latter according to their degree of uncomputability (=-=Soare, 1987-=-). For example, the following celebrated result of Matiyasevich has settled Hilbert’s Tenth Problem in the negative by proving it equivalent to H : Theorem 1.2. (Matiyasevich (1970)) On the one hand, ... |

395 |
On a Theory of Computation and Complexity over the Real Numbers: NP-Completeness
- BLUM, SHUB, et al.
- 1989
(Show Context)
Citation Context ...uring, 1939) and now core of Recursion Theory (Odifreddi, 1989; Soare, 1987), correspond to TMs with initial memory inscription, that is, they remove Condition (b); Blum, Shub, and Smale’s R-Machine (=-=Blum et al., 1989-=-, in particular Section 1, Example 6) abolishes Condition (c) by allowing each cell to store a real number; while Infinite Time Machines due to Hamkins et al. (2000) lift Condition (d). The proposal, ... |

324 | Classical recursion theory
- Odifreddi
- 1999
(Show Context)
Citation Context ...er M does not halt; for, simply simulating M step-by-step, M0 can easily identify the case when M does terminate. Turing’s result initiated the flourishing field of Computability or Recursion Theory (=-=Odifreddi, 1989-=-). Its goal is to distinguish computable from uncomputable problems and to classify the latter according to their degree of uncomputability (Soare, 1987). For example, the following celebrated result ... |

156 |
Quantum computing
- Gruska
- 1999
(Show Context)
Citation Context ...th problems that cannot be solved computationally at all, neither quickly nor slowly. For the very same reason, (at least ‘standard’) Quantum Computers are still no more powerful than an ordinary TM (=-=Gruska, 1999-=-, p. 3, footnote 1). 1.1. Church-Turing Hypothesis It should be stressed that both Halting and Hilbert’s Tenth Problem are desirable to be solved for very practical reasons. The first for instance ari... |

133 |
Systems of logic based on ordinals
- Turing
- 1939
(Show Context)
Citation Context ...em leads to several well-known models of hypercomputation; see, e.g., Ord (2002, Section 3) or Copeland (2002, Section 2). Oracle machines for instance, subject of Turing’s dissertation in Princeton (=-=Turing, 1939-=-) and now core of Recursion Theory (Odifreddi, 1989; Soare, 1987), correspond to TMs with initial memory inscription, that is, they remove Condition (b); Blum, Shub, and Smale’s R-Machine (Blum et al.... |

126 | The computer as a physical system: A microscopic quantum mechanical hamiltonian model of computers as represented by turing machines - Benioff - 1980 |

93 | The consistency of the axiom of choice and of the generalized continuum hypothesis - Gödel - 1938 |

83 | Infinite Time Turing Machines - Hamkins, Lewis - 2000 |

75 | Theory of Computational Complexity - Du, Ko - 2000 |

70 | NonTuring computations via MalamentHogarth space-times Int.J.Theor.Phys - Etesi, Nemeti - 2002 |

60 | Quantum algorithm for the Hilbert’s Tenth Problem
- Kieu
- 2003
(Show Context)
Citation Context ... Mechanical Hypercomputation Recently, several new approaches have been suggested for solving either the Halting Problem (Adamyan et al., 2004; Calude et al., 2001, 2002b) or Hilbert’s Tenth Problem (=-=Kieu, 2003-=-a,b). They exploit quantum mechanics and thus form a nice counterpart to previous approaches based on General Relativity (Etesi et al., 2002; Hogarth, 1992; Shagrir et al., 2003) as the other pillar o... |

49 | Does General Relativity Allow an Observer to View an Eternity in a Finite Time
- Hogarth
- 1992
(Show Context)
Citation Context ...t al., 2001, 2002b) or Hilbert’s Tenth Problem (Kieu, 2003a,b). They exploit quantum mechanics and thus form a nice counterpart to previous approaches based on General Relativity (Etesi et al., 2002; =-=Hogarth, 1992-=-; Shagrir et al., 2003) as the other pillar of 2 Refer to Remark 1.4.sComputational Power of Infinite Quantum Parallelism 2063 non-classical physics. Recalling that ‘standard’ Quantum Computing does n... |

41 | Enumerable sets are Diophantine - MATIYASEVICH - 1970 |

35 | 2003) Computation beyond Turing Machines - Wegner, Goldin - 2003 |

35 | Hypercomputation: computing more than the Turing machine, Honours Thesis
- Ord
- 2002
(Show Context)
Citation Context ...selves have put forward a claim as bold as the way ‘their’ hypothesis is often (mis-)interpreted (Copeland, 1997). Instead, the literature contains and discusses a rich variety of related hypotheses (=-=Ord, 2002-=-, Section 2.2). 1.2. Hypercomputation Anyway, the question remains open whether there might exist a computingdevicemorepowerfulthantheTM or not. To get an idea how such a Hypercomputer might look, the... |

31 | Computing the noncomputable
- Kieu
- 2003
(Show Context)
Citation Context ... Mechanical Hypercomputation Recently, several new approaches have been suggested for solving either the Halting Problem (Adamyan et al., 2004; Calude et al., 2001, 2002b) or Hilbert’s Tenth Problem (=-=Kieu, 2003-=-a,b). They exploit quantum mechanics and thus form a nice counterpart to previous approaches based on General Relativity (Etesi et al., 2002; Hogarth, 1992; Shagrir et al., 2003) as the other pillar o... |

27 | Alan turing’s forgotten ideas in computer science - Copeland, Proudfoot - 1999 |

25 | The fundamental physical limits of computation - Bennett, Landauer - 1985 |

22 | Computability and physical theories - Geroch, Hartle - 1986 |

18 | Transcending the limits of Turing computability
- Adamyan, Calude, et al.
(Show Context)
Citation Context ...ecute an infinite number of steps and thus to lift Condition (d). 1.3. Quantum Mechanical Hypercomputation Recently, several new approaches have been suggested for solving either the Halting Problem (=-=Adamyan et al., 2004-=-; Calude et al., 2001, 2002b) or Hilbert’s Tenth Problem (Kieu, 2003a,b). They exploit quantum mechanics and thus form a nice counterpart to previous approaches based on General Relativity (Etesi et a... |

15 | Computations via experiments with kinematic systems - Beggs, Tucker - 2004 |

12 | The broad conception of Computation
- Copeland
- 1997
(Show Context)
Citation Context ...t the fundamental capabilities of computation (Bennett, 1985). In fact, neither Church nor Turing themselves have put forward a claim as bold as the way ‘their’ hypothesis is often (mis-)interpreted (=-=Copeland, 1997-=-). Instead, the literature contains and discusses a rich variety of related hypotheses (Ord, 2002, Section 2.2). 1.2. Hypercomputation Anyway, the question remains open whether there might exist a com... |

10 |
Existence of bases implies the axiom of choice, p
- Blass
- 1984
(Show Context)
Citation Context ...d to a contradiction. For example, the claim “For every vector space there exists a basis”isofkind(C) as are many principles in Functional Analysis: Each of them is equivalent to the Axiom of Choice (=-=Blass, 1984-=-) and thus does not lead to a contradiction to conservative set theory (C) but cannot be deduced from it (B) as has first been proven by K. Gödel in Gödel (1940) and later strengthened by P. J. Cohen.... |

9 | Hypercomputation - Copeland - 2002 |

5 | quantum measurements, and Turing’s barrier - Calude, Pavlov - 2002 |

5 | Nondeterminism fairness and a fundamental analogy - Spaan, Torenvliet, et al. - 1989 |

2 | Super recursive algorithm and hypercomputation - Burgin, Klinger - 2004 |

2 | Reflections on quantum computing
- Calude, Dinneen, et al.
- 2001
(Show Context)
Citation Context ...er of steps and thus to lift Condition (d). 1.3. Quantum Mechanical Hypercomputation Recently, several new approaches have been suggested for solving either the Halting Problem (Adamyan et al., 2004; =-=Calude et al., 2001-=-, 2002b) or Hilbert’s Tenth Problem (Kieu, 2003a,b). They exploit quantum mechanics and thus form a nice counterpart to previous approaches based on General Relativity (Etesi et al., 2002; Hogarth, 19... |

1 | Computational Power of Infinite Quantum Parallelism 2071 - Shagrir, Pitowsky - 2003 |