Genuine geometric quantum gates induced by non-cyclic geodesic evolution of computational basis

To reach the error threshold required to successfully perform error-correcting algorithms in quantum computers, geometric quantum gates have been considered because of their natural resilience against noise. Non-cyclic geometric gates have been proposed to reduce the run time of conventional geometric gates, to further guard against decoherence. However, while these proposed gates remove the dynamical phase from the computational basis, they do not in general remove it from the eigenstates of the time evolution operator. For a non-cyclic gate to genuinely be considered geometric the dynamical phase should be removed from both the computational basis and the eigenstates. Here, a scheme for finding genuine non-cyclic geometric gates is proposed. The gates are designed to evolve the computational basis along non-cyclic paths, consisting of two geodesic segments, chosen such that the dynamical phase is removed from the eigenstates. The gates found with this scheme did not have shorter runtimes than cyclic gates, but it was possible to implement any gate with this scheme. The findings are important for the understanding of how general quantum computations can be implemented with geometric gates