  We present the Lyman-$\alpha$ flux power spectrum measurements of the XQ-100
sample of quasar spectra obtained in the context of the European Southern
Observatory Large Programme "Quasars and their absorption lines: a legacy
survey of the high redshift universe with VLT/XSHOOTER". Using $100$ quasar
spectra with medium resolution and signal-to-noise ratio we measure the power
spectrum over a range of redshifts $z = 3 - 4.2$ and over a range of scales $k
= 0.003 - 0.06\,\mathrm{s\,km^{-1}}$. The results agree well with the
measurements of the one-dimensional power spectrum found in the literature. The
data analysis used in this paper is based on the Fourier transform and has been
tested on synthetic data. Systematic and statistical uncertainties of our
measurements are estimated, with a total error (statistical and systematic)
comparable to the one of the BOSS data in the overlapping range of scales, and
smaller by more than $50\%$ for higher redshift bins ($z>3.6$) and small scales
($k > 0.01\,\mathrm{s\,km^{-1}}$). The XQ-100 data set has the unique feature
of having signal-to-noise ratios and resolution intermediate between the two
data sets that are typically used to perform cosmological studies, i.e. BOSS
and high-resolution spectra (e.g. UVES/VLT or HIRES). More importantly, the
measured flux power spectra span the high redshift regime which is usually more
constraining for structure formation models.
