Hyperfine-induced dephasing in three-electron spin qubits

We calculate the pure dephasing time of three-electron exchange-only qubits due to interaction with the nuclear hyperfine field. Within the $S=S_z=1/2$ spin subspace, we derive formulas for the dephasing time in the $(1,1,1)$ charge region and in the neighboring charge sectors coupled by tunneling. The nuclear field and the tunneling are taken into account in a second order approximation. The analytical solutions accurately reproduce the numerical evaluation of the full problem, and in comparison with existing experimental data, we find that the dephasing times are longer but on the same timescale as for single spins. Our analysis also applies to the resonant exchange (RX), always-on exchange-only (AEON) and hybrid qubits.