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 km$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$ s. 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 per cent for higher redshift bins ($z$ > 3.6) and small scales ($k$ > 0.01 km$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$ s). 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 that is usually more constraining for structure formation models.