In quantum mechanics, the dimension of the Hilbert space grows exponentially with system size. Therefore, a classical calculation of many-body quantum states becomes practically impossible for a small number of particles. Richard Feynman was the first to suggest a different paradigm to overcome this difficulty, a quantum computational machine (“quantum computer”). The quest to build a quantum computer has been going on for more than 20 years, but so far, no single experimental platform has emerged as technologically superior. I present our suggestion for a new platform based on ultracold   fermionic atoms held in an optical microtrap. In our scheme, quantum information can be stored in the internal states of these atoms or in vibrational states of the trap. Single qubit gates are implemented by coupling the atom to an external field, and a universal two-qubit  gate is implemented by a novel protocol that takes advantage of our ability to precisely control the tunnelling energy and the interaction energy (by using Feshbach resonance) between two atoms at two adjacent traps. I present numerical simulations of the qubits and gates and report on our progress in the lab towards testing our ideas in real life.