  We provide practical and powerful schemes for learning many properties of an
unknown n-qubit quantum state using a sparing number of copies of the state.
Specifically, we present a depth-modulated randomized measurement scheme that
interpolates between two known classical shadows schemes based on random Pauli
measurements and random Clifford measurements. These can be seen within our
scheme as the special cases of zero and infinite depth, respectively. We focus
on the regime where depth scales logarithmically in n and provide evidence that
this retains the desirable properties of both extremal schemes whilst, in
contrast to the random Clifford scheme, also being experimentally feasible. We
present methods for two key tasks; estimating expectation values of certain
observables from generated classical shadows and, computing upper bounds on the
depth-modulated shadow norm, thus providing rigorous guarantees on the accuracy
of the output estimates. We consider observables that can be written as a
linear combination of poly(n) Paulis and observables that can be written as a
low bond dimension matrix product operator. For the former class of observables
both tasks are solved efficiently in n. For the latter class, we do not
guarantee efficiency but present a method that works in practice; by
variationally computing a heralded approximate inverses of a tensor network
that can then be used for efficiently executing both these tasks.
