  The power spectrum of cosmic microwave background lensing is a powerful tool
for constraining fundamental physics such as the sum of neutrino masses and the
dark energy equation of state. Current lensing measurements primarily come from
distortions to the microwave background temperature field, but the polarization
lensing signal will dominate upcoming experiments with greater sensitivity.
Cosmic birefringence refers to the rotation of the linear polarization
direction of microwave photons propagating from the last scattering surface to
us, which can be induced by parity-violating physics such as axion-like dark
matter or primordial magnetic fields. We find that, for an upcoming CMB-S4-like
experiment, if there exists the scale-invariant anisotropic birefringence with
an amplitude corresponding to the current $95\%$ upper bound, the measured
lensing power spectrum could be biased by up to a factor of few at small
scales, $L\gtrsim 1000$. We show that the bias scales linearly with the
amplitude of the scale-invariant birefringence spectrum. The signal-to-noise of
the contribution from anisotropic birefringence is larger than unity even if
the birefringence amplitude decreases to $\sim 5\%$ of the current upper bound.
Our results indicate that a measurement and characterization of the anisotropic
birefringence is important for lensing analysis in future low-noise
polarization experiments.
