  We employ matrix product state simulations to study energy transport within
the non-integrable regime of the one-dimensional $\mathbb{Z}_3$ chiral clock
model. To induce a non-equilibrium steady state throughout the system, we
consider open system dynamics with boundary driving featuring jump operators
with adjustable temperature and footprint in the system. Given a steady state,
we diagnose the effective local temperature by minimizing the trace distance
between the true local state and the local state of a uniform thermal ensemble.
Via a scaling analysis, we extract the transport coefficients of the model at
relatively high temperatures above both its gapless and gapped low-temperature
phases. In the medium-to-high temperature regime we consider, diffusive
transport is observed regardless of the low-temperature physics. We calculate
the temperature dependence of the energy diffusion constant as a function of
model parameters, including in the regime where the model is quantum critical
at the low temperature. Notably, even within the gapless regime, an analysis
based on power series expansion implies that intermediate-temperature transport
can be accessed within a relatively confined setup. Although we are not yet
able to reach temperatures where quantum critical scaling would be observed,
our approach is able to access the transport properties of the model over a
broad range of temperatures and parameters. We conclude by discussing the
limitations of our method and potential extensions that could expand its scope,
for example, to even lower temperatures.
