Continuous quantum computation: Difference between revisions

Revision as of 09:10, 7 January 2015

Revision as of 04:00, 6 April 2015

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* Many scientific problems have continuous mathematical formulations. Examples of such formulations are

** [[Path Integral Formulation|Path integration]]

** [[Feynman-KacFeynman–Kac]] path integration

** [[Schrödinger equation]]

* In their standard monograph Nielsen and Chuang state "Of particular interest is a decisive answer to the problem whether quantum computers are more powerful than classical computers." To answer this question one must know the classical and quantum computational complexities

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==Applications==

Besides path integration there have been numerous recent papers studying algorithms and quantum speedups for continuous problems. These include matrix eigenvalues, phase estimation, the Sturm-LiouvilleSturm–Liouville eigenvalue problem, Feynman-KacFeynman–Kac path integration, initial value problems, function approximation and high-dimensional integration. See the papers listed below and the references therein.

*Bessen, A. J. (2005), A lower bound for phase estimation, Physical Review A, 71(4), 042313. Also http://arXiv.org/quant-ph/0412008.

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*Heinrich, S. (2004), Quantum Approximation II. Sobolev Embeddings, J. Complexity, 20, 27–45. Also http://arXiv.org/quant-ph/0305031.

*Jaksch, P. and Papageorgiou, A. (2003), Eigenvector approximation leading to exponential speedup of quantum eigenvalue calculation, Phys. Rev. Lett., 91, 257902. Also http://arXiv.org/quant-ph/0308016.

*Kacewicz, B. Z. (2005), Randomized and quantum solution of initial value problems, J. Complexity, 21, 740-756740–756.

*Kwas, M., Complexity of multivariate Feynman-KacFeynman–Kac Path Integration in Randomized and Quantum settings, 2004. Also http://arXiv.org/quant-ph/0410134.

*Novak, E. (2001), Quantum complexity of integration, J. Complexity, 17, 2–16. Also http://arXiv.org/quant-ph/0008124.

*Novak, E., Sloan, I. H., and Wozniakowski, H., Tractability of Approximation for Weighted orobov Spaces on Classical and Quantum Computers, J. Foundations of Computational Mathematics, 4, 121-156, 2004. Also http://arXiv.org/quant-ph/0206023

*Papageorgiou, A. and Wo´zniakowski, H. (2005), Classical and Quantum Complexity of the Sturm-LiouvilleSturm–Liouville Eigenvalue Problem, Quantum Information Processing, 4(2), 87–127. Also http://arXiv.org/quant-ph/0502054.

*Papageorgiou, A. and Wo´zniakowski, H. (2007), The Sturm-LiouvilleSturm–Liouville Eigenvalue Problem and NP-Complete Problems in the Quantum Setting with Queries, Quantum Information Processing, 6(2), 101-120101–120. Also http://arXiv.org/quant-ph/0504194.

*Traub, J. F. and Wo´zniakowski, H. (2002), Path integration on a quantum computer, Quantum Information Processing, 1(5), 365–388, 2002. Also http://arXiv.org/quant-ph/0109113.

*Woźniakowski, H. (2006), The Quantum Setting with Randomized Queries for Continuous Problems, Quantum Information Processing, 5(2), 83–130. Also http://arXiv.org/quant-ph/0601196.