Multi-Qubit Dyadic Phase Fixing for Fault-Tolerant Quantum Compilation

Fault-tolerant quantum computing requires translating application-level quantum circuits into the Clifford+$T$ gate set, where the $T$ gate is the dominant resource cost. Phase kickback is an ancilla-based technique that can dramatically reduce $T$-count for rotations with dyadic angles, but has previously been limited to highly structured circuit families. We present Dyadic Phase Fixing (DPF), a general multi-qubit synthesis tool that extends phase kickback to general quantum circuits. DPF uses numerical unitary synthesis to greedily extract dyadic angle rotations from any input circuit. Combined with a decision matrix to automatically size the final phase gradient register, our end-to-end workflow achieves up to 70% reduction in $T$-count compared to \texttt{gridsynth} and up to 60% compared to Repeat-Until-Success synthesis on a diverse set of benchmarks. We map these compiled circuits to a surface-code architecture to evaluate space-time volume, demonstrating up to a 60\% reduction in this metric as well. However, for some circuits and mapping strategies the two metrics diverge significantly, demonstrating that $T$-count alone is a useful but incomplete proxy for fault-tolerant program costs.