  We investigate electron transport through clean open quantum dots (quantum
billiards). We present a semiclassical theory that allows to accurately
reproduce quantum transport calculations. Quantitative agreement is reached for
individual energy and magnetic field dependent elements of the scattering
matrix. Two key ingredients are essential: (i) inclusion of pseudo-paths which
have the topology of linked classical paths resulting from diffraction in
addition to classical paths and (ii) a high-level approximation to diffractive
scattering. Within this framework of the pseudo-path semiclassical
approximation (PSCA), typical shortcomings of semiclassical theories such as
violation of the anti-correlation between reflection and transmission and the
overestimation of conductance fluctuations are overcome. Beyond its predictive
capabilities the PSCA provides deeper insights into the quantum-to-classical
crossover.
