  Memristors are resistive elements retaining information of their past
dynamics. They have garnered substantial interest due to their potential for
representing a paradigm change in electronics, information processing and
unconventional computing. Given the advent of quantum technologies, a design
for a quantum memristor with superconducting circuits may be envisaged. Along
these lines, we introduce such a quantum device whose memristive behavior
arises from quasiparticle-induced tunneling when supercurrents are cancelled.
For realistic parameters, we find that the relevant hysteretic behavior may be
observed using current state-of-the-art measurements of the phase-driven
tunneling current. Finally, we develop suitable methods to quantify memory
retention in the system.
