Anodic zirconia nanotubes are a promising functional medium for the formation of non-volatile resistive memory cells. The current–voltage characteristics in the region of low conductance of the fabricated Zr/ZrO2/Au memristive structures were studied in this work. For the first time, an analysis was made of the reversible mechanisms of formation and destruction of single quantum conductors based on oxygen vacancies, which participate in processes of multiple resistive switching between low- and high-resistance states in the nanotubular dioxide layer. An equivalent electrical circuit of a parallel resistor connection was proposed and discussed to describe the observed memristive behavior of the studied layered structures.