  Standard rulers such as the baryon acoustic oscillation (BAO) scale serve as
workhorses for precision tests of cosmology, enabling distance measurements
that probe the geometry and expansion history of our Universe. Aside from BAO
measurements from the cosmic microwave background (CMB), most standard ruler
techniques operate at relatively low redshifts and depend on biased tracers of
the matter density field. In a companion paper, we explored the scientific
reach of nulling estimators, where CMB lensing convergence maps are
cross-correlated with linear combinations of similar maps from line intensity
mapping (LIM) to precisely null out the low-redshift contributions to CMB
lensing. We showed that nulling estimators can be used to constrain the high
redshift matter power spectrum and showed that this spectrum exhibits
discernible BAO features. Here we propose using these features as a standard
ruler at high redshifts that does not rely on biased tracers. Forecasting such
a measurement at $z \sim 5$, we find that next-generation instruments will be
able to constrain the BAO scale to percent-level precision at $7.2 \%$, while
our futuristic observing scenario can constrain the BAO scale to $4\%$
precision. This constitutes a fundamentally new kind of BAO measurement during
early epochs in our cosmic history.
