Book contents
- Frontmatter
- Contents
- Turing's legacy: developments from Turing's ideas in logic
- Computability and analysis: the legacy of Alan Turing
- Alan Turing and the other theory of computation (expanded)
- Turing in Quantumland
- Computability theory, algorithmic randomness and Turing's anticipation
- Computable model theory
- Towards common-sense reasoning via conditional simulation: legacies of Turing in Artificial Intelligence
- Mathematics in the age of the Turing machine
- Turing and the development of computational complexity
- Turing machines to word problems
- Musings on Turing's Thesis
- Higher generalizations of the Turing Model
- Step by recursive step: Church's analysis of effective calculability
- Turing and the discovery of computability
- Transfinite machine models
- References
Turing in Quantumland
Published online by Cambridge University Press: 05 June 2014
Edited by
Book contents
- Frontmatter
- Contents
- Turing's legacy: developments from Turing's ideas in logic
- Computability and analysis: the legacy of Alan Turing
- Alan Turing and the other theory of computation (expanded)
- Turing in Quantumland
- Computability theory, algorithmic randomness and Turing's anticipation
- Computable model theory
- Towards common-sense reasoning via conditional simulation: legacies of Turing in Artificial Intelligence
- Mathematics in the age of the Turing machine
- Turing and the development of computational complexity
- Turing machines to word problems
- Musings on Turing's Thesis
- Higher generalizations of the Turing Model
- Step by recursive step: Church's analysis of effective calculability
- Turing and the discovery of computability
- Transfinite machine models
- References
Summary
Abstract. We revisit the notion of a quantum Turing-machine, whose design is based on the laws of quantum mechanics. It turns out that such a machine is not more powerful, in the sense of computability, than the machine originally constructed by Turing. Quantum Turing-machines do not violate the Church–Turing thesis. The benefit of quantum computing lies in efficiency. Quantum computers appear to be more efficient, in time, than classical Turing-machines, however its exact additional computational power is unclear, as this question ties in with deep open problems in complexity theory. We will sketch where BQP, the quantum analogue of the complexity class P, resides in the realm of complexity classes.
§1. Introduction. A decade before Turing developed his theory of computing, physicists struggled with the advent of quantum mechanics. During the famous 5th Solvay Conference in 1927 it was clear that a new era of physics had surfaced. Its strange features like superposition and entanglement still lead to heated discussions and much confusion. However strange and counter-intuitive, the theory has never been refuted by experiments that are performed daily and in great numbers throughout laboratories around the world. Time after time the predictions of quantum mechanics are in full agreement with experiment.
Shortly after the advent of quantum mechanics, Church, Turing and Post developed the notion of computability [Chu36, Tur36, Pos36]. Less than 10 years later these formal ideas would be put to practice resulting in the ENIAC, the first general purpose machine.
- Type
- Chapter
- Information
- Turing's LegacyDevelopments from Turing's Ideas in Logic, pp. 70 - 89Publisher: Cambridge University PressPrint publication year: 2014
References
[AarlO] , BQP and the polynomial hierarchy, STOC, 2010, pp. 141-150.
[Aarll] , A counterexample to the generalized linial-nisan conjecture, CoRR abs/1110.6126,2011.
[Adl78] , Two theorems on random polynomial time, FOCS, 1978, pp. 75-83.
[ADH97] , , and , Quantum computability, SIAM Journal on Computing, vol. 26 (1997), no. 5, pp. 1524-1540.Google Scholar
[AKS04] , , and , PRIMES Is in P, Annals of Mathematics. Second Series, vol. 160 (2004), no. 2, pp. 781-793.Google Scholar
[AB09] and , Computational complexity—a modern approach, Cambridge University Press, 2009.
[BBC+01] , , , , and , Quantum lower bounds by polynomials, Journal of the ACM, vol. 48 (2001), no. 4, pp. 778-797.Google Scholar
[BB84] and , Quantum cryptography: Public key distribution and coin tossing, Proceedings of IEEE international conference on Computers, Systems, and Signal Processing, IEEE, 1984, pp. 175-179.
[Ben89] , Time/space trade-offs for reversible computation, SIAM Journal on Computing, vol. 18 (1989), no. 4, pp. 766-776.Google Scholar
[BS98] and , Quantum information theory, IEEE Transactions on Information Theory, vol. 44 (1998), no. 6, pp. 2724-2742.Google Scholar
[BV97] and , Quantum complexity theory, SIAM Journal on Computing, vol. 26 (1997), no. 5, pp. 1411-1473.Google Scholar
[BCMdW10] , , , and , Nonlocality and communication complexity, Reviews of Modern Physics, vol. 82 (2010), no. 1, pp. 665-698.Google Scholar
[BdW02] and , Complexity measures and decision tree complexity: a survey, Theoretical Computer Science, vol. 288 (2002), no. 1, pp. 21-43.Google Scholar
[Chu36] , An unsolvable problem of elementary number theory, American Journal of Mathematics, (1936).Google Scholar
[Coo71] , The complexity of theorem-proving procedures, STOC, 1971, pp. 151-158.
[Deu85] , Quantum theory, the Church–Turing principle and the universal quantum computer, Proceedings of the Royal Society of London. Series A, (1985).
[DJ92] and , Rapid solution ofproblems by quantum computation, Proceedings of the Royal Society of London. Series A, vol. 493 (1992), no. 1907, pp. 553-558.Google Scholar
[DdW11] and , Quantum proofs for classical theorems, Theory of Computing, vol. 2 (2011), pp. 1-54.Google Scholar
[EFMP11] , , , and , Invited review article: Single-photon sources and detectors, Review of Scientific Instruments, vol. 82 (2011), no. 7, p. 25.Google Scholar
[FU10] and , Pseudorandom generators and the BQP vs. PH problem, manuscript, 2010.
[Fey82] , Simulating physics with computers, International Journal of Theoretical Physics, vol. 21 (1982), no. 6–7, pp. 467-488.Google Scholar
[GJ79] and , Computers and intractability: A guide to the theory of NP-completeness, W. H. Freeman, New York, 1979.
[Gro96] , A fast quantum mechanical algorithm for database search, Proceedings of 28th ACM STOC, 1996, quant-ph/9605043, pp. 212-219.
[KL80] and , Some connections between nonuniform and uniform complexity classes, STOC, 1980, pp. 302-309.
[Lan61] , Irreversibility and heat generation in the computing process, IBM Journal of Research and Development, vol. 5 (1961), pp. 183-191.Google Scholar
[Lev73] , Universal search problems, Problems of Information Transmission, vol. 9 (1973), no. 3, pp. 265-266.Google Scholar
[Moo65] , Cramming more components onto integrated circuits, Electronics, vol. 38 (1965), no. 8.Google Scholar
[NC00] and , Quantum computation and quantum information, Cambridge University Press, 2000.
[NW94] and , Hardness vs randomness, Journal of Computer and System Sciences, vol. 49 (1994), no. 2, pp. 149-167.Google Scholar
[Pos36] , Finite combinatory processes-formulation 1, The Journal of Symbolic Logic, vol. 1 (1936), no. 3, pp. 103-105.Google Scholar
[RSA78] , , and , A method for obtaining digital signatures and public-key cryptosystems, Communications of the ACM, vol. 21 (1978), no. 2, pp. 120-126.Google Scholar
[RGF+12] , , , , and , A depolarizer as a possible precise sunstone for viking navigation by polarized skylight, Proceedings of the Royal Society A, vol. 468 (2012), no. 2139, pp. 671-684.Google Scholar
[Sch35] , Die gegenwärtige Situation in der Quantenmechanik, Naturwissenschaften, vol. 23 (1935), no. 48, pp. 807-812.Google Scholar
[Sho94] , Algorithms for quantum computation: Discrete logarithms and factoring, Proceedings of the 35th annual symposium on the Foundations of Computer Science (Los Alamitos, CA), IEEE, 1994, pp. 124-134.
[Sim97] , On the power of quantum computation, SIAM Journal on Computing, vol. 26 (1997), no. 5, pp. 1474-1483.Google Scholar
[Sin00] , The code book. The science of secrecy from ancient Egypt to quantum cryptography, Harper Collins Publishers, 2000.
[Sip83] , A complexity theoretic approach to randomness, STOC, 1983, pp. 330-335.
[Tur36] , On computable numbers, with an application to the Entscheidungsproblem, Proceedings of the London Mathematical Society, vol. 2 (1936), no. 42, pp. 230-265, addendum 1937.Google Scholar
[vEB90] , Machine models and simulation, Handbook of theoretical computer science, Volume A: Algorithms and complexity (A), 1990, pp. 1-66.
[Zac86] , Probabilistic quantifiers, adversaries, and complexity classes: An overview, Structure in complexity theory conference, 1986, pp. 383-400.Google Scholar