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It has been known for a long time that the thermodynamic limit kBTln2 on the energy dissipation per logic operation can be overcome by physically and logically reversible circuits. However, explicit experimental demonstration of this is still lacking, and would be highly desirable both in its own right and in view of strong interest in inherently reversible quantum computation. In this work, we suggest a new gate, "negative-inductance SQUID", that is suitable for the experimental demonstration of reversible information processing in Josephson-junction circuits, and present results of its theoretical analysis. We also describe layout-level designs of an individual nSQUID and an 8-cell circular shift register made of nSQUIDs.