  The Berry curvature and quantum metric are the imaginary part and real part,
respectively, of the quantum geometric tensor which characterizes the topology
of quantum states. The former is known to generate a zoo of important
discoveries such as quantum Hall effect and anomalous Hall effect (AHE), while
the consequences of the quantum metric have rarely been probed by transport. In
this work, we observed quantum metric induced nonlinear transport, including
both nonlinear AHE and diode-like nonreciprocal longitudinal response, in thin
films of a topological antiferromagnet, MnBi$_2$Te$_4$. Our observation reveals
that the transverse and longitudinal nonlinear conductivities reverse signs
when reversing the antiferromagnetic order, diminish above the N\'eel
temperature, and are insensitive to disorder scattering, thus verifying their
origin in the band structure topology. They also flip signs between electron
and hole-doped regions, in agreement with theoretical calculations. Our work
provides a pathway to probe the quantum metric through nonlinear transport and
to design magnetic nonlinear devices.
