The decoherence of exchange-only qubits in triple quantum dots

We study decoherence of a three-electron-spin qubit in a linear triple quantum dot (TQD) by hyperfine interaction. The qubit is encoded in the ($S=1/2$, $S_{z}=1/2$) subspace, and can be fully controlled electrically via exchange interactions $J_{12}$ and $J_{23}$ between the electron spins. We clarify how hyperfine interaction dephases the qubit by constructing effective Hamiltonians and presenting estimates of free evolution and Hahn echo decay for such qubit in a GaAs TQD. When the three electron spins are uniformly coupled, i.e., $J_{12}=J_{23}$, the two states of our qubit are the eigenstates. We find that the qubit decoherence is of order of single-spin decoherence ($T^{*}_{2}\sim 10$ ns, $T_{2}$ on the scale of $\mu$s). On the other hand, a difference between $J_{12}$ and $J_{23}$ requires one to diagonalize the qubit space to obtain an appropriate eigenbasis. Alternatively, the qubit can be viewed as undergoing a rotation. We find that the decoherence rates in the new basis are not significantly modified when comparing them with those in the $J_{12}=J_{23}$ case.