  The utility of effective model spaces in quantum simulations of
non-relativistic quantum many-body systems is explored in the context of the
Lipkin-Meshkov-Glick model of interacting fermions. We introduce an iterative
hybrid-classical-quantum algorithm, Hamiltonian learning variational quantum
eigensolver (HL-VQE), that simultaneously optimizes an effective Hamiltonian,
thereby rearranging entanglement into the effective model space, and the
associated ground-state wavefunction. HL-VQE is found to provide an exponential
improvement in Lipkin-Meshkov-Glick model calculations, compared to a naive
truncation without Hamiltonian learning, throughout a significant fraction of
the Hilbert space. Quantum simulations are performed to demonstrate the HL-VQE
algorithm, using an efficient mapping where the number of qubits scales with
the $\log$ of the size of the effective model space, rather than the particle
number, allowing for the description of large systems with small quantum
circuits. Implementations on IBM's QExperience quantum computers and simulators
for 1- and 2-qubit effective model spaces are shown to provide accurate and
precise results, reproducing classical predictions. This work constitutes a
step in the development of entanglement-driven quantum algorithms for the
description of nuclear systems, that leverages the potential of noisy
intermediate-scale quantum (NISQ) devices.
