  Shallow quantum circuits are believed to be the most promising candidates for
achieving early practical quantum advantage - this has motivated the
development of a broad range of error mitigation techniques whose performance
generally improves when the quantum state is well approximated by a global
depolarising (white) noise model. While it has been crucial for demonstrating
quantum supremacy that random circuits scramble local noise into global white
noise - a property that has been proved rigorously - we investigate to what
degree practical shallow quantum circuits scramble local noise into global
white noise. We define two key metrics as (a) density matrix eigenvalue
uniformity and (b) commutator norm. While the former determines the distance
from white noise, the latter determines the performance of purification based
error mitigation. We derive analytical approximate bounds on their scaling and
find in most cases they nicely match numerical results. On the other hand, we
simulate a broad class of practical quantum circuits and find that white noise
is in certain cases a bad approximation posing significant limitations on the
performance of some of the simpler error mitigation schemes. On a positive
note, we find in all cases that the commutator norm is sufficiently small
guaranteeing a very good performance of purification-based error mitigation.
Lastly, we identify techniques that may decrease both metrics, such as
increasing the dimensionality of the dynamical Lie algebra by gate insertions
or randomised compiling.
