  Controllable quantized conductance states of TiN/Ti/HfO$_x$/TiN memristors
are realized with great precision through a pulse-mode reset procedure,
assisted with analytical differentiation of the condition of the set procedure,
which involves critical monitoring of the measured bias voltage. An intriguing
training effect that leads to faster switching of the states is also observed
during the operation. Detailed analyses on the low- and high-resistance states
under different compliance currents reveal a complete picture of the structural
evolution and dynamic behaviors of the conductive filament in the HfO$_x$
layer. This study provides a closer inspection on the quantum-level
manipulation of nanoscale atomic configurations in the memristors, which helps
to develop essential knowledge about the design and fabrication of the future
memristor-based quantum devices and neuromorphic computing devices.
