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  • Print publication year: 2014
  • Online publication date: June 2014

Turing in Quantumland

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.

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[AarlO] Scott, Aaronson, BQP and the polynomial hierarchy, STOC, 2010, pp. 141-150.
[Aarll] Scott, Aaronson, A counterexample to the generalized linial-nisan conjecture, CoRR abs/1110.6126,2011.
[Adl78] Leonard M., Adleman, Two theorems on random polynomial time, FOCS, 1978, pp. 75-83.
[ADH97] Leonard M., Adleman, Jonathan, De Marrais, and Ming-Deh A., Huang, Quantum computability, SIAM Journal on Computing, vol. 26 (1997), no. 5, pp. 1524-1540.
[AKS04] Manindra, Agrawal, Neeraj, Kayal, and Nitin, Saxena, PRIMES Is in P, Annals of Mathematics. Second Series, vol. 160 (2004), no. 2, pp. 781-793.
[AB09] Sanjeev, Arora and Boaz, Barak, Computational complexity—a modern approach, Cambridge University Press, 2009.
[BBC+01] Robert, Beals, Harry, Buhrman, Richard, Cleve, Michele, Mosca, and Ronald, de Wolf, Quantum lower bounds by polynomials, Journal of the ACM, vol. 48 (2001), no. 4, pp. 778-797.
[BB84] C. H., Bennett and G., Brassard, 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] Charles H., Bennett, Time/space trade-offs for reversible computation, SIAM Journal on Computing, vol. 18 (1989), no. 4, pp. 766-776.
[BS98] Charles H., Bennett and Peter W., Shor, Quantum information theory, IEEE Transactions on Information Theory, vol. 44 (1998), no. 6, pp. 2724-2742.
[BV97] Ethan, Bernstein and Umesh V., Vazirani, Quantum complexity theory, SIAM Journal on Computing, vol. 26 (1997), no. 5, pp. 1411-1473.
[BCMdW10] Harry, Buhrman, Richard, Cleve, Serge, Massar, and Ronald, de Wolf, Nonlocality and communication complexity, Reviews of Modern Physics, vol. 82 (2010), no. 1, pp. 665-698.
[BdW02] Harry, Buhrman and Ronald, de Wolf, Complexity measures and decision tree complexity: a survey, Theoretical Computer Science, vol. 288 (2002), no. 1, pp. 21-43.
[Chu36] Alonzo, Church, An unsolvable problem of elementary number theory, American Journal of Mathematics, (1936).
[Coo71] Stephen A., Cook, The complexity of theorem-proving procedures, STOC, 1971, pp. 151-158.
[Deu85] David, Deutsch, Quantum theory, the Church–Turing principle and the universal quantum computer, Proceedings of the Royal Society of London. Series A, (1985).
[DJ92] David, Deutsch and Richard, Jozsa, Rapid solution ofproblems by quantum computation, Proceedings of the Royal Society of London. Series A, vol. 493 (1992), no. 1907, pp. 553-558.
[DdW11] Andrew, Drucker and Ronald, de Wolf, Quantum proofs for classical theorems, Theory of Computing, vol. 2 (2011), pp. 1-54.
[EFMP11] M. D., Eisaman, J., Fan, A., Migdall, and S. V., Polyakov, Invited review article: Single-photon sources and detectors, Review of Scientific Instruments, vol. 82 (2011), no. 7, p. 25.
[FU10] Bill, Fefferman and Chris, Umans, Pseudorandom generators and the BQP vs. PH problem, manuscript, 2010.
[Fey82] Richard, Feynman, Simulating physics with computers, International Journal of Theoretical Physics, vol. 21 (1982), no. 6–7, pp. 467-488.
[GJ79] M. R., Garey and D. S., Johnson, Computers and intractability: A guide to the theory of NP-completeness, W. H. Freeman, New York, 1979.
[Gro96] L. K., Grover, A fast quantum mechanical algorithm for database search, Proceedings of 28th ACM STOC, 1996, quant-ph/9605043, pp. 212-219.
[KL80] Richard M., Karp and Richard J., Lipton, Some connections between nonuniform and uniform complexity classes, STOC, 1980, pp. 302-309.
[Lan61] Rolf, Landauer, Irreversibility and heat generation in the computing process, IBM Journal of Research and Development, vol. 5 (1961), pp. 183-191.
[Lev73] Leonid, Levin, Universal search problems, Problems of Information Transmission, vol. 9 (1973), no. 3, pp. 265-266.
[Moo65] Gordon, Moore, Cramming more components onto integrated circuits, Electronics, vol. 38 (1965), no. 8.
[NC00] Michael A., Nielsen and Isaac L., Chuang, Quantum computation and quantum information, Cambridge University Press, 2000.
[NW94] Noam, Nisan and Avi, Wigderson, Hardness vs randomness, Journal of Computer and System Sciences, vol. 49 (1994), no. 2, pp. 149-167.
[Pos36] Emil, Post, Finite combinatory processes-formulation 1, The Journal of Symbolic Logic, vol. 1 (1936), no. 3, pp. 103-105.
[RSA78] R. L., Rivest, A., Shamir, and L., Adleman, A method for obtaining digital signatures and public-key cryptosystems, Communications of the ACM, vol. 21 (1978), no. 2, pp. 120-126.
[RGF+12] Guy, Ropars, Gabriel, Gorre, Albert, le Floch, Jay, Enoch, and Vasudevan, Lakshminarayanan, 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.
[Sch35] E., Schrödinger, Die gegenwärtige Situation in der Quantenmechanik, Naturwissenschaften, vol. 23 (1935), no. 48, pp. 807-812.
[Sho94] P. W., Shor, 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] Daniel R., Simon, On the power of quantum computation, SIAM Journal on Computing, vol. 26 (1997), no. 5, pp. 1474-1483.
[Sin00] Simon, Singh, The code book. The science of secrecy from ancient Egypt to quantum cryptography, Harper Collins Publishers, 2000.
[Sip83] Michael, Sipser, A complexity theoretic approach to randomness, STOC, 1983, pp. 330-335.
[Tur36] Alan, Turing, 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.
[vEB90] Peter Van Emde, Boas, Machine models and simulation, Handbook of theoretical computer science, Volume A: Algorithms and complexity (A), 1990, pp. 1-66.
[Zac86] Stathis, Zachos, Probabilistic quantifiers, adversaries, and complexity classes: An overview, Structure in complexity theory conference, 1986, pp. 383-400.