Going to try giving an "in English, damnit!" explanation for these since I didn't know two of them when reading this title:
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The Fermi–Hubbard model is a model of how some fermionic particles (e.g. quarks, electrons, neutrinos) at very low temperatures arranged on a 2D lattice can spontaneously choose to localize, or condense their quantum probabilities to a small region.
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The Loschmidt amplitude is the difference between an initial state and a time-evolved one in dynamical quantum phase transitions.
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An ion trap quantum computer is one in which ions are suspended in space using electromagnetic fields.
Essentially, "we used a quantum computer to measure the quantum state difference for the evolution of very cool fermions situated on a 2D lattice."
Take what I've said with a grain of salt; I am not a physicist.