Figure 1

(a) Circuit for a one-spin gate. Time proceeds from left to right. The Heisenberg interaction couples spins $i$ and $j$, carried by the long horizontal lines. Large rectangular boxes with $\pi$, also denoted by $S{W}_{ij}$ for “swap,” represent $U_{ij}\left(\pi \right)$. The action of ${\widetilde{SW}}_{ij}$ is also a swap, as explained in the text. Small boxes with angles ${\theta }_i,{\theta }_j$ or ${\phi }_i,{\phi }_j$ inside denote a global magnetic field inducing evolution according to Eq. (2), and small boxes with $-{\theta }_i,-{\theta }_j$ (or $-{\phi }_i,-{\phi }_j$) denote $V^{\alpha }({\theta }_i,{\theta }_j{)}^{†}$ [or $V^{\alpha }({\phi }_i,{\phi }_j{)}^{†}$]. Consider an arbitrary single spin rotation $\exp (-2i\theta S_i^z)$, where the rotation angle $\theta \equiv {\theta }_i-{\theta }_j$ is constructed from global rectangular pulses of amplitude $A$ and duration $T$. Neighboring spins experience different amplitudes: $A_i=A{a}_i$, where the $a_i$ are device constants. Typically, $A$ will be limited by physical considerations, as in the device of (b), so that $T$ controls the pulse area. Thus, $T=\theta /A(a_i-a_j)$, and the rotation angles appearing in the circuit are given by $-{\theta }_i=-A{a}_{i}T$. The parameters ${\phi }_i$ and ${\phi }_j$ are defined through ${\theta }_i-{\theta }_j={\phi }_i+{\phi }_j$, so that ${\phi }_k=A{a}_k{T}^{\prime }$, where $T^{\prime }=(a_i-a_j)/(a_i+a_j)T$. The magnetic field direction $\alpha$ for each gate is indicated by $X$ or $Z$ between the boxes. Dark boxes with $\alpha =X$ or $Z$ between them denote $V^{\alpha }(\theta ,\theta +\pi )$ with $\theta$ arbitrary, while dark boxes with $X^{†}$ or $Z^{†}$ denote $V^{\alpha }(\theta ,\theta +\pi {)}^{†}$. The circuit yields the gate $e^{-i2({\theta }_i-{\theta }_j)S_i^z}\otimes I_j$ ($I$ is the identity operation). Vertical light dotted lines serve as a reminder that the global magnetic field acts on all spins, not just spins $i$ and $j$. However, our pulse sequence is constructed in such a way that the action on all other spins cancels out at the end of the sequence. This follows from its symmetry and the fact that the Heisenberg interaction couples only spins $i$ and $j$. The circuit uses eleven elementary steps, four of which are Heisenberg interactions and seven of which are global magnetic field pulses. (b) Zigzag spin arrangement, with symmetrically positioned wires. The two current orientations, $I_1=I_2$ and $I_1=-I_2$, enable the complete control of the global field gradients in the $x$ and $z$ directions.Reuse & Permissions