Known in nano-electronic a solid state quantum - size systems motivate they creating using the surface electrons over helium. The quantum carry of quasi-one-dimensional surface electrons (Q1D-SEs) over superfluid helium is considered here. Research performed according to an electron phase diagram in gas phase at conditions far from electron quantum melting. A substrate is dense row of the light-guide segments where the SEs channels are formed over helium in the fiber gaps. The electrostatic model of substrate profile demonstrates modulation of potential in the electric field leading to possibility of the fibers tops charge thereby improving the Q1D-SEs channels quality. The experiments carried out by a low - frequency electron transport method at the temperature from 1.5 K to 0.5 K and the concentrations of electrons were up to 109cm-2. The SE move transverse to channel satisfies conditions of quantization in sense both the temperature and the relaxation time of electron in a system. According to an experiment lower some temperature the SE conductivity is ladder-like. The differential of SE conductivity is peak-like corresponding qualitatively to an electron states density. The distance between peaks accords to the 1D energy spectrums in some approach. Possibility of apply the 1D-SEs quantized energy levels (vibration levels) as quantum bits of quantum computer is considered.