  We present a robust shadow estimation protocol for wide classes of shallow
measurement circuits that mitigates noise as long as the effective measurement
map is locally unitarily invariant. This is in practice an excellent
approximation, encompassing for instance the case of ideal single-qubit
Clifford gates composing the first circuit layer of an otherwise arbitrary
circuit architecture and even non-Markovian, gate-dependent noise in the rest
of the circuit. We argue that for approximately local noise the measurement
channel has an efficient matrix-product (tensor-train) representation, and show
how to estimate this directly from experimental data using tensor-network
tools, eliminating the need for analytical or numeric calculations. We
illustrate the relevance of our method with both numerics and
proof-of-principle experiments on an IBM Q device. Numerically, we show that,
while unmitigated shallow shadows with noisy circuits become more biased as the
depth increases, robust ones become more accurate for relevant parameter
regimes. Experimentally, we observe major bias reductions in two simple
fidelity estimation tasks using 5-qubit circuits with up to 2 layers of
entangling gates using the mitigated variant, of close to an order of magnitude
for $10^4$ measurement shots, e.g. Under the practical constraints of current
and near-term noisy quantum devices, our method maximally realizes the
potential of shadow estimation with global rotations.
