  We use the quantum Monte Carlo (QMC) techniques to calculate the static
structure function $S(q)$ of a one-component ion lattice and use it to
calculate the thermal conductivity $\kappa$ of high-density solid matter
expected in the neutron star crust. By making detailed comparisons with the
results for the thermal conductivity obtained using standard techniques based
on the one-phonon approximation (OPA) valid at low temperature, and the
multi-phonon harmonic approximation expected to be valid over a wide range of
temperatures, we asses the temperature regime where $S(q)$ from QMC can be used
directly to calculate $\kappa$. We also compare the QMC results to those
obtained using classical Monte Carlo to quantitatively asses the magnitude of
the quantum corrections. We find that quantum effects became relevant for the
calculation of $\kappa$ at temperature $T \lesssim 0.3 ~\Omega_\mathrm{P}$,
where $\Omega_\mathrm{P}$ is the ion plasma frequency. At $T \simeq 0.1
~\Omega_\mathrm{P}$ the quantum effects suppress $\kappa$ by about $30\%$. The
comparison with the results of the OPA indicates that dynamical information
beyond the static structure is needed when $T \lesssim 0.1~ \Omega_\mathrm{P}$.
These quantitative comparisons help to establish QMC as a viable technique to
calculate $\kappa$ at moderate temperatures in the range
$T=0.1-1~\Omega_\mathrm{P}$ of relevance to the study of accreting neutron
stars. This finding is especially important because QMC is the only viable
technique so far for calculating $\kappa$ in multi-component systems at
low-temperatures.
