When companies like Google claim to have achieved “quantum supremacy”, an emphatic term that indicates a clear advantage in using quantum computers over classical ones, they often base their claims on solving computational riddles that have no real practical application. “In this article, we identify problems that really matter to physics and try to measure how complex they actually are.”

Carleo and his colleagues, who come from several institutions in Europe, America, and Asia, started by assembling a library of ground-state simulations of different physical systems based on different techniques. They focused on “model Hamiltonians,” that are functions that simplify interesting phenomena found in groups of material, rather than specific chemical compounds. A typical example is the Hubbard model, that is used to simulate superconductivity.