  Classical shadows are a versatile tool to probe many-body quantum systems,
consisting of a combination of randomised measurements and classical
post-processing computations. In a recently introduced version of the protocol,
the randomization step is performed via unitary circuits of variable depth $t$,
defining the so-called shallow shadows. For sufficiently large $t$, this
approach allows one to get around the use of non-local unitaries to probe
global properties such as the fidelity with respect to a target state or the
purity. Still, shallow shadows involve the inversion of a many-body map, the
measurement channel, which requires non-trivial computations in the
post-processing step, thus limiting its applicability when the number of qubits
$N$ is large. In this work, we put forward a simple approximate post-processing
scheme where the infinite-depth inverse channel is applied to the finite-depth
classical shadows and study its performance for fidelity and purity estimation.
The scheme allows for different circuit connectivity, as we illustrate for
geometrically local circuits in one and two spatial dimensions and
geometrically non-local circuits made of two-qubit gates. For the fidelity, we
find that the resulting estimator coincides with a known linear cross-entropy,
achieving an arbitrary small approximation error $\delta$ at depth $t=O(\log
(N/\delta))$ (independent of the circuit connectivity). For the purity, we show
that the estimator becomes accurate at a depth $O(N)$. In addition, at those
depths, the variances of both the fidelity and purity estimators display the
same scaling with $N$ as in the case of global random unitaries. We establish
these bounds by analytic arguments and extensive numerical computations in
several cases of interest. Our work extends the applicability of shallow
shadows to large system sizes and general circuit connectivity.
