An Efficient and Scalable Clocking Assignment Algorithm for Multi-Threaded Multi-Phase Single Flux Quantum Circuits

A key distinguishing feature of single flux quantum (SFQ) circuits is that each logic gate is clocked. This feature forces the introduction of path-balancing flip-flops to ensure proper synchronization of inputs at each gate. This paper proposes a polynomial time complexity approximation algorithm for clocking assignments that minimizes the insertion of path balancing buffers for multi-threaded multi-phase clocking of SFQ circuits. Existing SFQ multi-phase clocking solutions have been shown to effectively reduce the number of required buffers inserted while maintaining high throughput, however, the associated clock assignment algorithms have exponential complexity and can have prohibitively long runtimes for large circuits, limiting the scalability of this approach. Our proposed algorithm is based on a linear program (LP) that leads to solutions that are experimentally on average within 5% of the optimum and helps accelerate convergence towards the optimal integer linear program (ILP) based solution. The improved LP and ILP runtimes permit multi-phase clocking schemes to scale to larger SFQ circuits than previous state of the art clocking assignment methods. We further extend the existing algorithm to support fanout sharing of the added buffers, saving, on average, an additional 10% of the inserted DFFs. Compared to traditional full path balancing (FPB) methods across 10 benchmarks, our enhanced LP saves 79.9%, 87.8%, and 91.2% of the inserted buffers for 2, 3, and 4 clock phases respectively. Finally, we extend this approach to the generation of circuits that completely mitigate potential hold-time violations at the cost of either adding on average less than 10% more buffers (for designs with 3 or more clock phases) or, more generally, adding a clock phase and thereby reducing throughput.