  The "sign problem" (SP) is the fundamental limitation to simulations of
strongly correlated materials in condensed matter physics, solving quantum
chromodynamics at finite baryon density, and computational studies of nuclear
matter. As a result, it is part of the reason fields such as ultra-cold atomic
physics are so exciting: they can provide quantum emulators of models that
could not otherwise be solved, due to the SP. For the same reason, it is also
one of the primary motivations behind quantum computation. It is often argued
that the SP is not intrinsic to the physics of particular Hamiltonians, since
the details of how it onsets, and its eventual occurrence, can be altered by
the choice of algorithm or many-particle basis. Despite that, we show that the
SP in determinant quantum Monte Carlo (DQMC) is quantitatively linked to
quantum critical behavior. We demonstrate this via simulations of a number of
fundamental models of condensed matter physics, including the spinful and
spinless Hubbard Hamiltonians on a honeycomb lattice and the ionic Hubbard
Hamiltonian, all of whose critical properties are relatively well understood.
We then propose a reinterpretation of the low average sign for the Hubbard
model on the square lattice when away from half-filling, an important open
problem in condensed matter physics, in terms of the onset of pseudogap
behavior and exotic superconductivity. Our study charts a path for exploiting
the average sign in QMC simulations to understand quantum critical behavior,
rather than solely as an obstacle that prevents quantum simulations of
many-body Hamiltonians at low temperature.
