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We present a numerical investigation of the implementation of quantum logic gates through time-frequency resolved coherent anti-Stokes Raman scattering (TFRCARS) in iodine vapour. A specific scheme is given whereby two qubits are encoded in the tensor product space of vibrational and rotational molecular eigenstates. Single-qubit and controlled-logic gates are applied to test the viability of this encoding scheme. Possible experimental constraints are investigated, including the necessary precision in the timing of successive CARS pulses and the minimum resolution of spectral components within a given pulse. It is found that these requirements can be satisfied by current technology. The two-qubit Grover search is performed to demonstrate the implementation of a sequence of quantum operations. The results of our simulations suggest that TFRCARS is a promising experimental system for simple quantum information processing applications.