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We present a design for a superconducting nanowire binary shift register, which stores digital states in the form of circulating supercurrents in high-kinetic-inductance loops. Adjacent superconducting loops are connected with nanocryotrons, three-terminal electrothermal switches, and fed with an alternating two-phase clock to synchronously transfer the digital state between the loops. A two-loop serial-input shift register was fabricated with thin-film NbN and a bit error rate of less than 10−4 was achieved, when operated at a maximum clock frequency of
83
MHz and in an out-of-plane magnetic field of up to
6
mT. A shift register based on this technology offers an integrated solution for low-power readout of superconducting nanowire single photon detector arrays and is capable of interfacing directly with room-temperature electronics and operating unshielded in high magnetic field environments.