In 1982, Fredkin and Toffoli introduced conservative logic, whereby they observed that “it is ideally possible to build sequential circuits with zero internal power dissipation.” They introduced a Switch gate which, with its inverse, constitutes a universal set of gates. We propose a dual-rail computing architecture wherein reciprocal pairs of magnetic ballistic solitons are routed on long Josephson junctions. Asynchronous Switch and Merge (Switch’s inverse) can be implemented by coupling long junctions. These two primitive gates can be combined to synthesize reversible gates such as CNOT, Fredkin, or Toffoli gates, or irreversible gates like AND and OR. Data pulses travel ballistically, powered only by their own kinetic energy. Gates are mostly elastic, so energy dissipation is very low. Dissipated energy can be replenished with strategically placed AC clock biases, as is done in various other superconducting digital technologies. Simulations promise good operating margins at speeds up to 4GHz with very low power dissipation. We name our logic family Conservative Reciprocal Asynchronous Ballistic Solitons (CRABS).