Next, we place a single spin excitation in one of the eigenstates of the gate. To be specific, for the magnetic field h=h− we place the control spin in state and denote it as . Then the control spin cannot leak out of the gate region and therefore it is stationary. Moreover, if we place a target spin-up at the input port, the resulting state will have the energy which differs significantly from the energies and of the states to which it can couple via a single spin-exchange (assuming Δ≠0, see below and Supplementary Note 2), see Fig. 1c. Therefore, such an initial state will remain stationary and the control spin excitation on the gate will block the transfer of the target spin between the input and output ports, see Fig. 1d. Indeed, the large energy difference will preclude the escape of the gate spin excitation to the output port, while the large energy mismatch will suppress the probability of the second-order exchange of spin excitation between the input and output ports by a factor of <<1 (with Δ=−1, see below and Supplementary Note 2). Exactly the same arguments apply to the initial state with the magnetic field set to h=h+.