Realization of arbitrary gates in holonomic quantum computation

Among the many proposals for the realization of a quantum computer, holonomic quantum computation is distinguished from the rest as it is geometrical in nature and thus expected to be robust against decoherence. Here we analyze the realization of various quantum gates by solving the inverse problem: Given a unitary matrix, we develop a formalism by which we find loops in the parameter space generating this matrix as a holonomy. We demonstrate that such a one-qubit gate as the Hadamard gate and such two-qubit gates as the controlled-NOT gate and the SWAP gate, and the discrete Fourier transformation can be obtained with a single loop.