  Hong-Ou-Mandel (HOM) interference is a compelling quantum phenomenon that
demonstrates the nonclassical nature of single photons. In this study, we
investigate an electromagnetically induced transparency-based double-$\Lambda$
four-wave mixing system from the perspective of quantized light fields. The
system can be used to realize efficient HOM interference in the frequency
domain. By using the reduced density operator theory, we demonstrate that,
although the double-$\Lambda$ medium does not exhibit phase-dependent
properties for the closed-loop case of two incident single photons,
frequency-domain HOM two-photon interference occurs. For experimentally
achievable optical depth conditions, our theory indicates that this
double-$\Lambda$ scheme can perform high-fidelity Hadamard gate operations on
frequency-encoded single-photon qubits, and thereby generate HOM two-photon
NOON states with a fidelity greater than 0.99. Furthermore, we demonstrate that
this scheme can be used to realize arbitrary single-qubit gates and two-qubit
SWAP gates by simply controlling the laser detuning and phase, exhibiting its
multifunctional properties and providing a new route to scalable optical
quantum computing.
