  Superconducting computing promises enhanced computational power in both
classical and quantum approaches. Yet, scalable and fast superconducting
memories are not implemented. Here, we propose a fully superconducting memory
cell based on the hysteretic phase-slip transition existing in long aluminum
nanowire Josephson junctions. Embraced by a superconducting ring, the memory
cell codifies the logic state in the direction of the circulating persistent
current, as commonly defined in flux-based superconducting memories. But,
unlike the latter, the hysteresis here is a consequence of the phase-slip
occurring in the long weak link and associated to the topological transition of
its superconducting gap. This disentangle our memory scheme from the
large-inductance constraint, thus enabling its miniaturization. Moreover, the
strong activation energy for phase-slip nucleation provides a robust
topological protection against stochastic phase-slips and magnetic-flux noise.
These properties make the Josephson phase-slip memory a promising solution for
advanced superconducting classical logic architectures or flux qubits.
