  An electron-on-helium qubit is a promising physical platform for quantum
information technologies. Among all the "blueprints" for the qubit realization,
a hybrid Rydberg-spin qubit seems to be a promising one towards quantum
computing using electron spins. The main technological challenge on the way to
such qubits is a detection of fA range image current induced by Rydberg
transition of a single electron. To address this problem we aim to use a tank
LC-circuit in conjunction with a high impedance and low power dissipation
cryogenic amplifier. Here, we report our progress towards realization of a
resonant image current detector with a home-made cryogenic amplifier based on
FHX13LG HEMT. We present a detailed characterization of the transistor at room
and cryogenic temperatures, as well as details of the amplifier design and
performance. At the power dissipation level of amplifier well below
100~${\mu}$W the measured voltage and current noise level is 0.6~nV/$\sqrt{Hz}$
and below 1.5~fA/$\sqrt{Hz}$, respectively. Based on the actual image current
measurements of the Rydberg transition in a many-electron system on liquid
helium, we estimate SNR=8 with the measurement bandwidth 1 Hz for the detection
of a single-electron transition, providing the noise level at the output is
solely determined by the noise of the amplifier.
