  Non-standard interactions in the propagation of neutrinos in matter can lead
to significant deviations from expectations within the standard neutrino
oscillation framework and atmospheric neutrino detectors have been considered
to set constraints. However, most previous works have focused on relatively
low-energy atmospheric neutrino data. Here, we consider the one-year
high-energy through-going muon data in IceCube, which has been already used to
search for light sterile neutrinos, to constrain new interactions in the
$\mu\tau$-sector. In our analysis we include several systematic uncertainties
on both, the atmospheric neutrino flux and on the detector properties, which
are accounted for via nuisance parameters. After considering different primary
cosmic-ray spectra and hadronic interaction models, we obtain the most
stringent bound on the off-diagonal $\varepsilon_{\mu \tau}$ parameter to date,
with the 90\% credible interval given by $-6.0 \times 10^{-3} <
\varepsilon_{\mu \tau} < 5.4 \times 10^{-3}$. In addition, we also estimate the
expected sensitivity after 10~years of collected data in IceCube and study the
precision at which non-standard parameters could be determined for the case of
$\varepsilon_{\mu \tau}$ near its current bound.
