  The core accretion model for giant planet formation suggests a two layer
picture for the initial structure of Jovian planets, with heavy elements in a
dense core and a thick H-He envelope. Late planetesimal accretion and core
erosion could potentially enrich the H-He envelope in heavy elements, which is
supported by the three-fold solar metallicity that was measured in Jupiter's
atmosphere by the Galileo entry probe. In order to reproduce the observed
gravitational moments of Jupiter and Saturn, models for their interiors include
heavy elements, $Z$, in various proportions. However, their effect on the
equation of state of the hydrogen-helium mixtures has not been investigated
beyond the ideal mixing approximation. In this article, we report results from
\textit{ab initio} simulations of fully interacting H-He-$Z$ mixtures in order
to characterize their equation of state and to analyze possible consequences
for the interior structure and evolution of giant planets. Considering C, N, O,
Si, Fe, MgO and SiO$_2$, we show that the behavior of heavy elements in H-He
mixtures may still be represented by an ideal mixture if the effective volumes
and internal energies are chosen appropriately. In the case of oxygen, we also
compute the effect on the entropy. We find the resulting changes in the
temperature-pressure profile to be small. A homogeneous distribution of 2\%
oxygen by mass changes the temperature in Jupiter's interior by only 80 K.
