Abstract
We give a short introduction to quantum computing and its relation to numerical analysis. We survey recent research on quantum algorithms and quantum complexity theory for two basic numerical problems — high dimensional integration and approximation. Having matching upper and lower complexity bounds for the quantum setting, we are in a position to compare them with those for the classical deterministic and randomized setting, previously obtained in information-based complexity theory. This enables us to assess the possible speedups quantum computation could provide over classical deterministic or randomized algorithms for these numerical problems.
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References
Aharonov, D.: Quantum computation — a review. In: Stauffer, D. (ed.) Annual Review of Computational Physics, vol. VI, World Scientific, Singapore (1998), http://arXiv.org/abs/quant-ph/9812037
Beals, R., Buhrman, H., Cleve, R., Mosca, M., de Wolf, R.: Quantum lower bounds by Polynomials. In: Proceedings of 39th IEEE FOCS, 352–361 (1998), http://arXiv.org/abs/quant-ph/9802049
Boyer, M., Brassard, P., Høyer, P., Tapp, A.: Tight bounds on quantum searching. Fortschritte der Physik 46, 493–505 (1998), http://arXiv.org/abs/quantph/9605034
Brassard, G., Høyer, P., Mosca, M., Tapp, A.: Quantum amplitude amplification and estimation. In: Quantum Computation and Quantum Information: A Millennium Volume. AMS Contemporary Mathematics Series 305 (2002), http://arXiv.org/abs/quant-ph/0005055
Deutsch, D.: Quantum theory, the Church-Turing principle and the universal quantum computer. Proc. R. Soc. Lond., Ser. A 400, 97–117 (1985)
Ekert, A., Hayden, P., Inamori, H.: Basic concepts in quantum computation (2000), http://arXiv.org/abs/quant-ph/0011013
Feynman, R.: Simulating physics with computers. Int. J. Theor. 21, 467–488 (1982)
Grover, L.: A fast quantum mechanical algorithm for database search. In: Proc. 28 Annual ACM Symp. on the Theory of Computing, pp. 212–219. ACM Press, New York (1996), http://arXiv.org/abs/quant-ph/9605043
Gruska, J.: Quantum Computing. McGraw-Hill, New York (1999)
Heinrich, S.: Random approximation in numerical analysis. In: Bierstedt, K.D., Pietsch, W.M.R., Vogt, D. (eds.) Functional Analysis, pp. 123–171. Marcel Dekker, New York (1993)
Heinrich, S.: Quantum summation with an application to integration. J. Complexity 18, 1–50 (2002), http://arXiv.org/abs/quant-ph/0105116
Heinrich, S.: Quantum integration in Sobolev classes. J. Complexity 19, 19–42 (2003), http://arXiv.org/abs/quant-ph/0112153
Heinrich, S.: Quantum Approximation I. Embeddings of Finite Dimensional Lp Spaces, J. Complexity 20, 5–26 (2004), http://arXiv.org/abs/quantph/0305030
Heinrich, S.: Quantum Approximation II. Sobolev Embeddings, J. Complexity 20, 27–45 (2004), http://arXiv.org/abs/quant-ph/0305031
Heinrich, S.: The quantum query complexity of elliptic PDE (in preparation)
Heinrich, S., Novak, E.: On a problem in quantum summation. J. Complexity 19, 1–18 (2003), http://arXiv.org/abs/quant-ph/0109038
Kacewicz, B.: Randomized and quantum algorithms yield a speed-up for initial-value problems. J. Complexity 20, 821–834 (2004), http://arXiv.org/abs/quant-ph/0311148
Kacewicz, B.: Improved bounds on the randomized and quantum complexity of initial-value problems, http://arXiv.org/abs/quant-ph/0405018
Kwas, M.: Complexity of multivariate Feynman-Kac path integration in randomized and quantum settings, http://arXiv.org/abs/quant-ph/0410134
Maiorov, V.E.: Discretization of the problem of diameters. Usp. Mat. Nauk 30(6), 179–180 (1975) (in Russian)
Manin, Y.I.: Computable and uncomputable. Sovetskoye Radio, Moscow (1980) (in Russian)
Manin, Y.I.: Classical computing, quantum computing, and Shor’s factoring algorithm (1999), http://arXiv.org/abs/quant-ph/9903008
Nayak, A., Wu, F.: The quantum query complexity of approximating the median and related statistics. In: STOC, May 1999, pp. 384–393 (1999), http://arXiv.org/abs/quantph/ 9804066
Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)
Novak, E.: Deterministic and Stochastic Error Bounds in Numerical Analysis. Lecture Notes in Mathematics, vol. 1349. Springer, Berlin (1988)
Novak, E.: Quantum complexity of integration. J. Complexity 17, 2–16 (2001), http://arXiv.org/abs/quant-ph/0008124
Novak, E., Sloan, I.H., Woźniakowski, H.: Tractability of approximation for weighted Korobov spaces on classical and quantum computers. Found. Comput. Math. 4, 121–156 (2004), http://arXiv.org/abs/quant-ph/0206023
Papageorgiou, A., Woźniakowski, H.: Classical and quantum complexity of the Sturm-Liouville eigenvalue problem, http://arXiv.org/abs/quant-ph/0502054
Pietsch, A.: Eigenvalues and s-Numbers. Cambridge University Press, Cambridge (1987)
Pittenger, A.O.: Introduction to Quantum Computing Algorithms. Birkhauser, Boston (1999)
Shor, P.W.: Algorithms for quantum computation: Discrete logarithms and factoring. In: Proceedings of the 35th Annual Symposium on Foundations of Computer Science, pp. 124–134. IEEE Computer Society Press, Los Alamitos (1994), http://arXiv.org/abs/quant-ph/9508027
Shor, P.W.: Introduction to quantum algorithms (2000), http://arXiv.org/abs/quant-ph/0005003
Traub, J.F., Woźniakowski, H.: Path integration on a quantum computer. Quantum Information Processing 1 5, 365–388 (2002), http://arXiv.org/abs/quantph/0109113
Traub, J.F., Wasilkowski, G.W., Woźniakowski, H.: Information-Based Complexity. Academic Press, New York (1988)
Wiegand, C.: Quantum complexity of parametric integration. J. Complexity 20, 75–96 (2004), http://arXiv.org/abs/quant-ph/0305103
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Heinrich, S. (2006). Numerical Analysis on a Quantum Computer. In: Lirkov, I., Margenov, S., Waśniewski, J. (eds) Large-Scale Scientific Computing. LSSC 2005. Lecture Notes in Computer Science, vol 3743. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11666806_3
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DOI: https://doi.org/10.1007/11666806_3
Publisher Name: Springer, Berlin, Heidelberg
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