Scalable Asynchronous Single Flux Quantum Up-Down Counter using Josephson Trapping Lines and {\alpha}-Cells

We present a scalable, clockless up-down counter architecture implemented using single-flux quantum (SFQ) logic to enable efficient state management in superconductor digital systems. The proposed design eliminates the reliance on clocked storage elements by introducing the Josephson Trapping Line (JTrL). This bidirectional pulse-trapping structure enables persistent, non-volatile state storage without clocking. The counter integrates $\upalpha$-cells with a splitter (SPL) element to make bidirectional data propagation possible and support multi-fanout connectivity. The design supports increment, decrement, and read operations and includes a control unit that guarantees correct output behavior across all valid state transitions. Circuit-level simulations based on SPICE models demonstrate robust bidirectional functionality across a 3-bit state range [-4 to +4] at an operating frequency of 4 GHz. The proposed counter offers a modular and scalable solution suitable for integration into larger superconducting systems targeting quantum computing, neuromorphic processing, and cryogenic sensing applications.