  The complex band structure, large spin-orbit induced band splitting, and
heavy effective mass of two-dimensional (2D) hole systems hosted in GaAs
quantum wells render them rich platforms to study many-body physics and
ballistic transport phenomena. Here we report ultra-high-quality (001) GaAs 2D
hole systems, fabricated using molecular beam epitaxy and modulation doping,
with mobility values as high as $5.8\times10^6$ cm$^2$/Vs at a hole density of
$p=1.3\times10^{11}$ /cm$^2$, implying a mean-free path of $\simeq27$ $\mu$m.
In the low-temperature magnetoresistance trace of this sample, we observe
high-order fractional quantum Hall states up to the Landau level filling
$\nu=12/25$ near $\nu=1/2$. Furthermore, we see a deep minimum develop at
$\nu=1/5$ in the magnetoresistance of a sample with a much lower hole density
of $p=4.0\times10^{10}$ /cm$^2$ where we measure a mobility of $3.6\times10^6$
cm$^2$/Vs. These improvements in sample quality were achieved by reduction of
residual impurities both in the GaAs channel and the AlGaAs barrier material,
as well as optimization in design of the sample structure.
