  Monte Carlo simulations are a powerful tool for elucidating the properties of
complex systems across many disciplines. Not requiring any a priori knowledge,
they are particularly well suited for exploring new phenomena. However, when
applied to fermionic quantum systems, quantum Monte Carlo (QMC) algorithms
suffer from the so-called "negative sign problem", which causes the
computational effort to grow exponentially with problem size. Here we
demonstrate that the fermion sign problem originates in topological properties
of the configurations. In particular, we show that in the widely used auxiliary
field approaches the negative sign of a configuration is a geometric phase that
is the imaginary time counterpart of the Aharonov-Anandan phase, and reduces to
a Berry phase in the adiabatic limit. This provides an intriguing connection
between QMC simulations and classification of topological states. Our results
shed clarify the controversially debated origin of the sign problem in
fermionic lattice models.
