  The lensing power spectra for gravitational potential, astrometric shift, and
convergence perturbations are powerful probes to investigate dark matter
structures on small scales. We report the first lower and upper bounds of these
lensing power spectra on angular scale $\sim 1$arcsec towards the anomalous
quadruply lensed quasar MG$\,$J0414+0534 at a redshift $z=2.639$. To obtain the
spectra, we conducted observations of MG$\,$J0414+0534 using the Atacama Large
Millimeter/submillimeter Array (ALMA) with high angular resolution (0.02 arcsec
to 0.05 arcsec). We developed a new partially non-parametric method in which
Fourier coefficients of potential perturbation are adjusted to minimize the
difference between linear combinations of weighted mean de-lensed images. Using
positions of radio jet components, extended dust emission on scales $>1\,$kpc,
and mid-infrared flux ratios, the range of measured convergence, astrometric
shift, and potential powers at an angular scale of $\sim 1.1$arcsec
(corresponding to an angular wave number of $l=1.2\times 10^6$ or $\sim 9\,$kpc
in the primary lens plane) within $1\,\sigma$ are
$\varDelta_\kappa=0.021-0.028$, $\varDelta_\alpha =7-9\,$mas, and
$\varDelta_\psi=1.2-1.6\,$$\textrm{mas}^2$, respectively. Our result is
consistent with the predicted abundance of halos in the line of sight and
subhalos in cold dark matter models. Our partially non-parametric lens models
suggest a presence of a clump in the vicinity of object Y, a possible dusty
dwarf galaxy and some small clumps in the vicinity of other lensed quadruple
images. Although much fainter than the previous report, we detected weak
continuum emission possibly from object Y with a peak flux of $\sim 100\,\mu
\textrm{Jy}\, \textrm{beam} ^{-1}$ at the $\sim 4\,\sigma$ level.
