Randomized and quantum algorithms yield a speed-up for initial-value problems

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 and approximation problems, for which a speed-up is shown in many important cases 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 settings 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]  S. Heinrich,et al.  Optimal Summation and Integration by Deterministic, Randomized, and Quantum Algorithms , 2002 .

[2]  Lov K. Grover.  A fast quantum mechanical algorithm for database search , 1996, STOC '96.

[3]  Fred J. Hickernell,et al.  Monte Carlo and Quasi-Monte Carlo Methods 2000 , 2002 .

[4]  Peter W. Shor,et al.  Algorithms for quantum computation: discrete logarithms and factoring , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[5]  Stefan Heinrich.  Quantum approximation I. Embeddings of finite-dimensional Lp spaces , 2004, J. Complex..

[6]  G. Brassard,et al.  Quantum Amplitude Amplification and Estimation , 2000, quant-ph/0005055.

[7]  E. Novak.  Deterministic and Stochastic Error Bounds in Numerical Analysis , 1988 .

[8]  Boleslaw Z. Kacewicz.  On sequential and parallel solution of initial value problems , 1990, J. Complex..

[9]  Stefan Heinrich.  Quantum Summation with an Application to Integration , 2002, J. Complex..

[10]  Felix Wu,et al.  The quantum query complexity of approximating the median and related statistics , 1998, STOC '99.

[11]  Boleslaw Z. Kacewicz.  Minimum asymptotic error of algorithms for solving ODE , 1988, J. Complex..

[12]  Stefan Heinrich.  Quantum approximation II. Sobolev embeddings , 2004, J. Complex..

[13]  B. Kacewicz.  How to increase the order to get minimal-error algorithms for systems of ODE , 1984 .

[14]  Erich Novak.  Quantum Complexity of Integration , 2001, J. Complex..

[15]  Henryk Wozniakowski,et al.  Path Integration on a Quantum Computer , 2002, Quantum Inf. Process..