  The exponential and Gaussian functions are among the most fundamental and
important operations, appearing ubiquitously throughout all areas of science,
engineering, and mathematics. Whereas formally, it is well-known that any
function may in principle be realized on a quantum computer, in practice
present-day algorithms tend to be very expensive. In this work, we present
algorithms for evaluating exponential and Gaussian functions efficiently on
quantum computers. The implementations require a (generally) small number of
multiplications, which represent the overall computational bottleneck. For a
specific, realistic NISQ application, the Toffoli count of the exponential
function is found to be reduced from 15,690 down to 912, when compared against
a state-of-the art competing method by H\"aner and coworkers
[arXiv:1805.12445], under the most favorable conditions for each method. For
the corresponding Gaussian function comparison, the Toffoli count is reduced
from 19,090 down to 704. Space requirements are also quite modest, to the
extent that the aforementioned NISQ application can be implemented with as few
as 71 logical qubits. More generally, the methods presented here could also be
equally well applied in a fault-tolerant context, using error-corrected
multiplications, etc.
