  The linear constraint of secret key rate capacity is overcome by the
tiwn-field quantum key distribution (QKD). However, the complex phase-locking
and phase-tracking technique requirements throttle the real-life applications
of twin-field protocol. The asynchronous measurement-device-independent (AMDI)
QKD or called mode-pairing QKD protocol can relax the technical requirements
and keep the similar performance of twin-field protocol. Here, we propose an
AMDI-QKD protocol with a nonclassical light source by changing the
phase-randomized weak coherent state to a phase-randomized coherent-state
superposition in the signal state time window. Simulation results show that our
proposed hybrid source protocol significantly enhances the key rate of the
AMDI-QKD protocol, while exhibiting robustness to imperfect modulation of
nonclassical light sources.
