  Given a loop or more generally 1-cycle $r$ of size L on a closed
two-dimensional manifold or surface, represented by a triangulated mesh, a
question in computational topology asks whether or not it is homologous to
zero. We frame and tackle this problem in the quantum setting. Given an oracle
that one can use to query the inclusion of edges on a closed curve, we design a
quantum algorithm for such a homology detection with a constant running time,
with respect to the size or the number of edges on the loop $r$, requiring only
a single usage of oracle. In contrast, classical algorithm requires $\Omega(L)$
oracle usage, followed by a linear time processing and can be improved to
logarithmic by using a parallel algorithm. Our quantum algorithm can be
extended to check whether two closed loops belong to the same homology class.
Furthermore, it can be applied to a specific problem in the homotopy detection,
namely, checking whether two curves are \textit{not} homotopically equivalent
on a closed two-dimensional manifold.
