Randomized and Quantum Algorithms Yield a Speed-Up for Initial-Value Problems.pdf

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Randomized and Quantum Algorithms Yield a Speed-Up for Initial-Value Problems.pdf

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Randomized and Quantum Algorithms Yield a Speed-Up for Initial-Value Problems

a r X i v : q u a n t - p h / 0 3 1 1 1 4 8 v 1 2 1 N o v 2 0 0 3 RANDOMIZED AND QUANTUM ALGORITHMS YIELD A SPEED-UP FOR INITIAL-VALUE PROBLEMS 1 Boles law Kacewicz 2 Abstract Quantum algorithms and complexity have recently been studied not only for discrete, but also for some numerical problems. Most attention has been paid so far to the integration problem, for which a speed-up is shown by quantum computers with respect to deterministic and randomized algorithms on a classical computer. In this paper we deal with the randomized and quantum complexity of initial-value problems. For this nonlinear problem, we show that both randomized and quantum algorithms yield a speed-up over deterministic algorithms. Upper bounds on the complexity in the randomized and quantum setting are shown by constructing algorithms with a suitable cost, where the construction is based on integral information. Lower bounds result from the respective bounds for the integration problem. 1 This research was partly supported by AGH grant No. 10.420.03 2Department of Applied Mathematics, AGH University of Science and Technology, Al. Mickiewicza 30, paw. A3/A4, III p., pok. 301, 30-059 Cracow, Poland kacewicz@.pl, tel. +48(12)617 3996, fax +48(12)617 3165 1 Introduction Potential advantages of quantum computing over deterministic or classical random- ized algorithms have been extensively studied by many authors for discrete problems, starting from Shor’s paper on factorization of integers [14] and Grover’s algorithm for searching databases [2]. Recently, a progress has also been achieved in quantum so- lution of numerical problems. The first paper dealing with the quantum complexity of a continuous problem was the work of Novak [13], who established matching upper and lower bounds on the quantum complexity of integration of functions from Ho?lder classes, based on the results on complexity of summation of real numbers from [1] and [11]. A general model of quantum computing for continuous pro