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In ideal quantum circuits, qubits are tacitly assumed to be uniformlyfabricated and operated by prescribed signals. In reality, however, we mustcope with different control signals to adjust individual qubits, which requiresa large overhead of control circuits. Here, we theoretically investigate howrandom local fields affect cluster states and surface code states whichconstitute the key highly entangled states in quantum computation. We findsimilar behavior of temporal degradation of the fidelity for both clusterstates and surface code states for the number of qubits up to ten. We find thatthe effect of local field fluctuations is greatly mitigated if the magnitude offluctuations can be suppressed below 10 % of the energy gap $\Delta$ for bothcluster states and surface code states. If the magnitude of fluctuationsexceeds $\Delta/2$, the fidelity for both states deteriorates dramatically. Asimple estimation based on the average fidelity up to $t\sim 2\hbar/\Delta$shows that the maximum number of qubits that can be corrected with the 1 %error threshold is less than 31 for surface code states and 27 for clusterstates. This means that the error correction should be carried out during atime shorter than $2\hbar/\Delta$.

Robustness of cluster states and surface code states against random local fields