  The nonadiabatic holonomic quantum computation based on the geometric phase
is robust against the built-in noise and decoherence. In this work, we
theoretically propose a scheme to realize nonadiabatic holonomic quantum gates
in a surface electron system, which is a promising two-dimensional platform for
quantum computation. The holonomic gate is realized by a three-level structure
that combines the Rydberg states and spin states via an inhomogeneous magnetic
field. After a cyclic evolution, the computation bases pick up different
geometric phases and thus perform a geometric gate. Only the electron with spin
up experiences the geometric gate, while the electron with spin down is
decoupled from the state-selective driving fields. The arbitrary controlled-U
gate encoded on the Rydberg states and spin states can then be realized. The
fidelity of the output state exceeds 0.99 with experimentally achievable
parameters.
