  Buried-channel semiconductor heterostructures are an archetype material
platform to fabricate gated semiconductor quantum devices. Sharp confinement
potential is obtained by positioning the channel near the surface, however
nearby surface states degrade the electrical properties of the starting
material. In this paper we demonstrate a two-dimensional hole gas of high
mobility ($5\times 10^{5}$ cm$^2$/Vs) in a very shallow strained germanium
channel, which is located only 22 nm below the surface. This high mobility
leads to mean free paths $\approx6 \mu m$, setting new benchmarks for holes in
shallow FET devices. Carriers are confined in an undoped Ge/SiGe
heterostructure with reduced background contamination, sharp interfaces, and
high uniformity. The top-gate of a dopant-less field effect transistor controls
the carrier density in the channel. The high mobility, along with a percolation
density of $1.2\times 10^{11}\text{ cm}^{-2}$, light effective mass (0.09
m$_e$), and high g-factor (up to $7$) highlight the potential of undoped
Ge/SiGe as a low-disorder material platform for hybrid quantum technologies.
