  The classical shadows protocol is an efficient strategy for estimating
properties of an unknown state $\rho$ using a small number of state copies and
measurements. In its original form, it involves twirling the state with
unitaries from some ensemble and measuring the twirled state in a fixed basis.
It was recently shown that for computing local properties, optimal sample
complexity (copies of the state required) is remarkably achieved for unitaries
drawn from shallow depth circuits composed of local entangling gates, as
opposed to purely local (zero depth) or global twirling (infinite depth)
ensembles. Here we consider the sample complexity as a function of the depth of
the circuit, in the presence of noise. We find that this noise has important
implications for determining the optimal twirling ensemble. Under fairly
general conditions, we i) show that any single-site noise can be accounted for
using a depolarizing noise channel with an appropriate damping parameter $f$;
ii) compute thresholds $f_{\text{th}}$ at which optimal twirling reduces to
local twirling for arbitrary operators and iii) $n^{\text{th}}$ order Renyi
entropies ($n \ge 2$); and iv) provide a meaningful upper bound
$t_{\text{max}}$ on the optimal circuit depth for any finite noise strength
$f$, which applies to all operators and entanglement entropy measurements.
These thresholds strongly constrain the search for optimal strategies to
implement shadow tomography and can be easily tailored to the experimental
system at hand.
