  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 HOMI's $g^{(2)}$ 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.
