  In this paper, we present a general framework for quantum many-body
simulations called the operator learning renormalization group (OLRG). Inspired
by machine learning perspectives, OLRG is a generalization of Wilson's
numerical renormalization group and White's density matrix renormalization
group, which recursively builds a simulatable system to approximate a target
system of the same number of sites via operator maps. OLRG uses a loss function
to minimize the error of a target property directly by learning the operator
map in lieu of a state ansatz. This loss function is designed by a scaling
consistency condition that also provides a provable bound for real-time
evolution. We implement two versions of the operator maps for classical and
quantum simulations. The former, which we call the Operator Matrix Map, can be
implemented via neural networks on classical computers. The latter, which we
call the Hamiltonian Expression Map, generates device pulse sequences to
leverage the capabilities of quantum computing hardware. We illustrate the
performance of both maps for calculating time-dependent quantities in the
quantum Ising model Hamiltonian.
