Protection of Exponential Operation using Stabilizer Codes in the Early Fault Tolerance Era

Quantum error correction offers a promising path to suppress errors in quantum processors, but the resources required to protect logical operations from noise, especially non-Clifford operations, pose a substantial challenge to achieve practical quantum advantage in the early fault-tolerant quantum computing (EFTQC) era. In this work, we develop a systematic scheme to encode exponential maps of the form $\exp(-i\theta P)$ into stabilizer codes with simple circuit structures and low qubit overhead. We provide encoded circuits with small first-order logical error rate after postselection for the [[n, n-2, 2]] quantum error-detecting codes and the [[5, 1, 3]], [[7, 1, 3]], and [[15, 7, 3]] quantum error-correcting codes. Detailed analysis shows that under the level of physical noise of current devices, our encoding scheme is 4--7 times less noisy than the unencoded operation, while at most 3% of runs need to be discarded.