Quantum Complexity of Integration

It is known that quantum computers yield a speed-up for certain discrete problems. Here we want to know whether quantum computers are useful for continuous problems. We study the computation of the integral of functions from the classical Holder classes Fk, ?d on 0, 1]d and define ? by ?=(k+?)/d. The known optimal orders for the complexity of deterministic and (general) randomized methods are comp(Fk, ?d, ?)???1/? and comprandom(Fk, ?d, ?)???2/(1+2?). For a quantum computer we prove compquantquery(Fk, ?d, ?)???1/(1+?) and compquant(Fk, ?d, ?)?C??1/(1+?)(log??1)1/(1+?). For restricted Monte Carlo (only coin tossing instead of general random numbers) we prove compcoin(Fk, ?d, ?)?C??2/(1+2?)(log??1)1/(1+2?). To summarize the results one can say that ?there is an exponential speed-up of quantum algorithms over deterministic (classical) algorithms, if ? is small; ?there is a (roughly) quadratic speed-up of quantum algorithms over randomized classical methods, if ? is small.

[1]  Michele Mosca,et al.  Quantum Computer Algorithms , 2003 .

[2]  U. Vazirani.  Quantum Algorithms , 2001, LATIN.

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

[4]  Colin P. Williams,et al.  Fast Quantum Algorithms for Numerical Integrals and Stochastic Processes , 1999, quant-ph/9908083.

[5]  A. Werschulz,et al.  Complexity and information , 1999, Lezioni Lincee.

[6]  G. Brassard,et al.  Quantum Counting , 1998, ICALP.

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

[8]  Lov K. Grover.  A framework for fast quantum mechanical algorithms , 1997, STOC '98.

[9]  R. Cleve,et al.  Quantum algorithms revisited , 1997, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

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

[11]  G. Brassard,et al.  Tight bounds on quantum searching , 1996, quant-ph/9605034.

[12]  Erich Novak,et al.  The Real Number Model in Numerical Analysis , 1995, J. Complex..

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

[14]  H. Woxniakowski.  Information-Based Complexity , 1988 .

[15]  Henryk Wozniakowski,et al.  Information-based complexity , 1987, Nature.

[16]  Carl DeVito,et al.  Functional Analysis , 1978 .

[17]  S. Heinrich.  Random Approximation in Numerical Analysis , 1994 .

[18]  Henryk Wozniakowski,et al.  The Monte Carlo Algorithm With a Pseudorandom Generator , 1992 .

[19]  Erich Novak,et al.  Eingeschränkte Monte Carlo-Verfahren zur Numerischen Integration , 1985 .