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==An Example: Path Integration==
Path integration has numerous applications including quantum mechanics, quantum chemistry, statistical mechanics, and computational finance. We want to compute an approximation to within error at most <math>\epsilon</math> with probability, say, at least 3/4. Then the following was [http://arXiv.org/quant-ph/0109113 shown] by Traub and Woźniakowski:
* A quantum computer enjoys exponential speedup over the classical worst case and quadratic speedup over the classical randomized case.
* The query complexity is of order <math>\epsilon^{-1}</math>.
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