A frequency beam splitter (FBS) with the split ratio of 0.5 or 1 can be used as the frequency-mode Hadamard gate (FHG) for frequency-encoded photonic qubits or as the quantum frequency converter (QFC) for frequency up or down conversion of photons. Previous works revealed that all kinds of the FHG or QFC operating at the single-photon level had overall efficiency or output-to-input ratio around 50% or less. In this work, our FHG and QFC are made with the four-wave mixing process based on the dual-$Lambda$ electromagnetically induced transparency scheme. We achieved an overall efficiency of 90$pm$4% in the FGH and that of 84% in the QFC using coherent-state single photons, both of which are the best up-to-date records. To test the fidelity of the FBS, we propose a novel scheme of Hong-Ou-Mandel interference (HOMI) for quantum process tomography. The fidelity indicated by the HOMIs $g^{(2)}$ measurement of the FHG is 0.99$pm$0.01. Such low-loss high-fidelity FHG and QFC or FBS with the tunable split ratio can lead to useful operations or devices in long-distance quantum communication.
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