  In this work, we consider a fundamental task in quantum many-body physics -
finding and learning ground states of quantum Hamiltonians and their
properties. Recent works have studied the task of predicting the ground state
expectation value of sums of geometrically local observables by learning from
data. For short-range gapped Hamiltonians, a sample complexity that is
logarithmic in the number of qubits and quasipolynomial in the error was
obtained. Here we extend these results beyond the local requirements on both
Hamiltonians and observables, motivated by the relevance of long-range
interactions in molecular and atomic systems. For interactions decaying as a
power law with exponent greater than twice the dimension of the system, we
recover the same efficient logarithmic scaling with respect to the number of
qubits, but the dependence on the error worsens to exponential. Further, we
show that learning algorithms equivariant under the automorphism group of the
interaction hypergraph achieve a sample complexity reduction, leading in
particular to a constant number of samples for learning sums of local
observables in systems with periodic boundary conditions. We demonstrate the
efficient scaling in practice by learning from DMRG simulations of $1$D
long-range and disordered systems with up to $128$ qubits. Finally, we provide
an analysis of the concentration of expectation values of global observables
stemming from the central limit theorem, resulting in increased prediction
accuracy.
