Runtime Compilation for Quantum Circuits with PQF-Enabled $R_z$ Synthesis

In Fault-Tolerant Quantum Computing (FTQC) using surface codes, reducing $T$-count is critical for efficient computation due to the high cost of $T$-state distillation. Probabilistic Quantum Circuit with Fallback (PQF) synthesis significantly lowers $T$-counts compared to standard protocols like Gridsynth, but introduces high circuit dynamism through extra gates inserted at runtime. This dynamism invalidates static compile-time routing, as these new $T$ gates require dynamic lattice surgery routing to establish non-intersecting ancilla paths to $T$-state factories while minimizing total execution time. To address this, we introduce $\text{R}^2$-PQF, a Runtime Routing protocol for PQF-synthesized quantum circuits. $\text{R}^2$-PQF dynamically routes gates layer-by-layer, computing paths during the execution of the previous logical timestep. To satisfy the strict computational window of $\sim t_{lattice\_surgery} + t_{reaction}$ and non-intersectionality constraints, we implement a Wavefront Breadth-First Search (W-BFS) and synthesise it on a 500MHz FPGA and 2 GHz ASIC. For a clock cycle of 2GHz, we achieve a median logical time-step of $1.51 μs$ for a 300 qubit Ising Hamiltonian circuit, demonstrating that $\text{R}^2$-PQF successfully resolves the vast majority of paths (with rare exceptions) within the execution window and enables the $T$-count benefits of PQF without latency penalties. Compared to a baseline stalling-based solution (the only no-runtime-recomputation solution), $\text{R}^2$-PQF speeds up circuit execution by $1.34\times$ on average and up to $1.76 \times$.