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The nonadiabatic holonomic quantum computation based on the geometric phaseis robust against the built-in noise and decoherence. In this work, wetheoretically propose a scheme to realize nonadiabatic holonomic quantum gatesin a surface electron system, which is a promising two-dimensional platform forquantum computation. The holonomic gate is realized by a three-level structurethat combines the Rydberg states and spin states via an inhomogeneous magneticfield. After a cyclic evolution, the computation bases pick up differentgeometric phases and thus perform a geometric gate. Only the electron with spinup experiences the geometric gate, while the electron with spin down isdecoupled from the state-selective driving fields. The arbitrary controlled-Ugate encoded on the Rydberg states and spin states can then be realized. Thefidelity of the output state exceeds 0.99 with experimentally achievableparameters.

Universal quantum gates by nonadiabatic holonomic evolution for the surface electron