  Most continuous mathematical formulations arising in science and engineering
can only be solved numerically and therefore approximately. We shall always
assume that we're dealing with a numerical approximation to the solution. There
are two major motivations for studying quantum algorithms and complexity for
continuous problems. 1. Are quantum computers more powerful than classical
computers for important scientific problems? How much more powerful? 2. Many
important scientific and engineering problems have continuous formulations. To
answer the first question we must know the classical computational complexity
of the problem. Knowing the classical complexity of a continuous problem we
obtain the quantum computation speedup if we know the quantum complexity. If we
know an upper bound on the quantum complexity through the cost of a particular
quantum algorithm then we can obtain a lower bound on the quantum speedup.
Regarding the second motivation, in this article we'll report on
high-dimensional integration, path integration, Feynman path integration, the
smallest eigenvalue of a differential equation, approximation, partial
differential equations, ordinary differential equations and gradient
estimation. We'll also briefly report on the simulation of quantum systems on a
quantum computer.
