## Quantum Speed-Up of Computations (2002)

Venue: | Philosophy of Science |

Citations: | 8 - 0 self |

### BibTeX

@INPROCEEDINGS{Pitowsky02quantumspeed-up,

author = {Itamar Pitowsky},

title = {Quantum Speed-Up of Computations},

booktitle = {Philosophy of Science},

year = {2002},

pages = {168--177}

}

### OpenURL

### Abstract

Church-Turing Thesis as saying something about the scope and limitations of physical computing machines. Although this was not the intention of Church or Turing, the Physical Church Turing thesis is interesting in its own right. Consider, for example, Wolfram’s formulation: One can expect in fact that universal computers are as powerful in their computational capabilities as any physically realizable system can be, that they can simulate any physical system...Nophysically implementable procedure could then shortcut a computationally irreducible process. (Wolfram 1985) Wolfram’s thesis consists of two parts: (a) Any physical system can be simulated (to any degree of approximation) by a universal Turing machine (b) Complexity bounds on Turing machine simulations have physical significance. For example, suppose that the computation of the minimum energy of some system of n particles takes at least exponentially (in n) many steps. Then the relaxation time of the actual physical system to its minimum energy state will also take exponential time. An even more extreme formulation of (more or less) the same thesis is due to Aharonov (1998): A probabilistic Turing machine can simulate any reasonable physical device in polynomial cost. She calls this The Modern Church Thesis. Aharonov refers here to probabilistic Turing machines that use random numbers in addition to the usual deterministic table of steps. It seems that such machines are capable to perform certain tasks faster than fully deterministic machines. The most famous randomized algorithm of that kind concerns the decision whether a given natural number is prime. A probabilistic algorithm that decides primality in a number of

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Citation Context ...y add phase factors like e i� 1 to each component. This is a crucial difference which will play an important role in the sequel. Can we simulate such dynamics on a Turing machine? The answer is yes! (=-=Deutsch 1985-=-). If the system starts from a finite superposition of states, we can calculate the dynamics of each element of the superposition separately on one Turing machine, and then add the results with the pr... |

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Citation Context ...takes is exponential in the size of the input. The bigger trouble is that no (essentially) better algorithm is known to exist. This is the most important open problem in theoretical computer science (=-=Gary and Johnson 1979-=-). It seems superficially that quantum parallelism suits this problem. We can easily represent the proposition and the truth assignments by the states of a quantum system. We first encode them as a se... |

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Citation Context ...thm on a physical machine will have an economic, as well as scientific consequences. Unfortunately, the engineering of real quantum computers faces many obstacles which we shall not discuss here (see =-=Ekert and Jozsa 1996-=-, Aharonov 1998). Let L be a natural number. Any natural number a such that 0 � a � k = ∑ k = 0 k 2 where a0, a1,...,aL�1 � 2L�1 L−1 has a binary representation a a {0, 1}. We shall represent the numb... |

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Citation Context ...ters of physics to their computational counterparts: memory capacity and number of computation steps, respectively. There are various ways to do that, leading to different formulations of the thesis (=-=Pitowsky, 1990-=-). For example, one can encode the set of instructions of a universal Turing machine and the state of its infinite tape in the binary development of the position coordinates of a single particle. Cons... |

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Citation Context ...pe in the binary development of the position coordinates of a single particle. Consequently, one can physically ‘realize’ a universal Turing machine as a billiard with hyperbolic mirrors (Moore 1990; =-=Pitowsky 1994-=-). However, the most intuitive connection between abstract Turing machines and physical devices appear in the pioneering work of Gandy. 3 A discrete deterministic mechanical device is a physical syste... |

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Citation Context ... find large prime numbers fast, and it is hard to factor large composite numbers in any reasonable amount of time. Many protocols such as public key and electronic signature are based on these facts (=-=Giblin 1993-=-). Therefore, the discovery that quantum computers can solve FACTORING in polynomial time has had a dramatic effect. 6 The implementation of the algorithm on a physical machine will have an economic, ... |

1 | Quantum Entanglement and Classical Computation”, this volume - MacCallum - 2001 |