  While most approaches to geometric quantum computation is based on geometric
phase in cyclic evolution, noncyclic geometric gates have been proposed to
increase further the flexibility. While these gates remove the dynamical phase
of the computational basis, they do not in general remove it from the
eigenstates of the time evolution operator, which makes the geometric nature of
the gates ambiguous. Here, we resolve this ambiguity by proposing a scheme for
genuinely noncyclic geometric gates. These gates are obtained by evolving the
computational basis along open paths consisting geodesic segments, and
simultaneously assuring that no dynamical phase is acquired by the eigenstates
of the time evolution operator. While we illustrate the scheme for the simplest
nontrivial case of two geodesic segments starting at each computational basis
state of a single qubit, the scheme can be straightforwardly extended to more
elaborate paths, more qubits, or even qudits.
