  Classical shadows are a powerful method for learning many properties of
quantum states in a sample-efficient manner, by making use of randomized
measurements. Here we study the sample complexity of learning the expectation
value of Pauli operators via ``shallow shadows'', a recently-proposed version
of classical shadows in which the randomization step is effected by a local
unitary circuit of variable depth $t$. We show that the shadow norm (the
quantity controlling the sample complexity) is expressed in terms of properties
of the Heisenberg time evolution of operators under the randomizing
(``twirling'') circuit -- namely the evolution of the weight distribution
characterizing the number of sites on which an operator acts nontrivially. For
spatially-contiguous Pauli operators of weight $k$, this entails a competition
between two processes: operator spreading (whereby the support of an operator
grows over time, increasing its weight) and operator relaxation (whereby the
bulk of the operator develops an equilibrium density of identity operators,
decreasing its weight). From this simple picture we derive (i) an upper bound
on the shadow norm which, for depth $t\sim \log(k)$, guarantees an exponential
gain in sample complexity over the $t=0$ protocol in any spatial dimension, and
(ii) quantitative results in one dimension within a mean-field approximation,
including a universal subleading correction to the optimal depth, found to be
in excellent agreement with infinite matrix product state numerical
simulations. Our work connects fundamental ideas in quantum many-body dynamics
to applications in quantum information science, and paves the way to
highly-optimized protocols for learning different properties of quantum states.
