  Non-Hermitian systems can exhibit unconventional spectral singularities
called exceptional points (EPs). Various EP sensors have been fabricated in
recent years, showing strong spectral responses to external signals. Here we
propose how to achieve a nonlinear anti-parity-time ($\mathcal{APT}$) gyroscope
by spinning an optical resonator. We show that, in the absence of any
nonlinearity, the sensitivity or optical mode splitting of the linear device
can be magnified up to 3 orders than that of the conventional device without
EPs. Remarkably, the $\mathcal{APT}$ symmetry can be broken when including the
Kerr nonlinearity of the materials and, as the result, the detection threshold
can be significantly lowered, i.e., much weaker rotations which are well beyond
the ability of a linear gyroscope can now be detected with the nonlinear
device. Our work shows the powerful ability of $\mathcal{APT}$ gyroscopes in
practice to achieve ultrasensitive rotation measurement.
