Fast quantum algorithms for numerical integrals and stochastic processes

We discuss quantum algorithms that calculate numerical integrals and descriptive statistics of stochastic processes. With either of two distinct approaches, one obtains an exponential speed increase in comparison to the fastest known classical deterministic algorithms and a quadratic speed increase in comparison to classical Monte Carlo (probabilistic) methods. We derive a simpler and slightly faster version of Grover's mean algorithm, demonstrate how to apply quantum counting to the problem, develop some variations of these algorithms, and show how both (apparently quite different) approaches can be understood from the same unified framework. Finally, we discuss how the exponential speed increase appears to (but does not) violate results obtained via the method of polynomials, from which it is known that a bounded-error quantum algorithm for computing a total function can be only polynomially more efficient than the fastest deterministic classical algorithm.