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We exhibit quantum cluster algebra structures on quantum Grassmannians$K_q[Gr(2,n)]$ and their quantum Schubert cells, as well as on $K_q[Gr(3,6)]$,$K_q[Gr(3,7)]$ and $K_q[Gr(3,8)]$. These cases are precisely those where thequantum cluster algebra is of finite type and the structures we describequantize those found by Scott for the classical situation.

Quantum cluster algebra structures on quantum Grassmannians and their quantum Schubert cells: the finite-type cases