Error in a spin-SWAP gate due to hyperfine interaction in a double quantum dot

We study the SWAP gate for two exchange-coupled electron spins under the influence of hyperfine (hf) interaction in a double quantum dot. A gate error develops during such a gate because hf interaction causes dephasing between any pair of two-spin states. We find that this gate error is initial-state-dependent. For example, an initial state in the $S_z=0$ subspace suffers only from $S-T_0$ dephasing, leading to smaller gate error than in the case of other initial states. We calculate the gate fidelity for typical initial states, and compare the resulting gate errors. We also analyze the effects of inhomogeneous broadening on the gate fidelity in the presence of a random Overhauser field.