Exchange-controlled single-electron-spin rotations in quantum dots

We show theoretically that arbitrary coherent rotations can be performed quickly (with a gating time $\sim 1\mathrm{ns}$) and with high fidelity on the spin of a single confined electron using control of exchange only, without the need for spin-orbit coupling or ac fields. We expect that implementations of this scheme would achieve gate error rates on the order of $\eta \lesssim {10}^{-3}$ in GaAs quantum dots, within reach of several known error-correction protocols.

Figure 1

(Color online) Possible setup to implement the scheme proposed here. Ancillary electron spins at $z_1$ are maintained in a polarized state with a large Zeeman field $b_1^z$ along $z$. Qubit spins at $z_2$ are free to precess in a weaker effective Zeeman field lying in the $x-z$ plane: $\mathbf{\Delta }=(b_2^x,0,b_2^z-J∕2)$. Here, $J$ is the exchange coupling between the qubit and ancillary spins and ${\mathbf{b}}_2$ is the qubit Zeeman field in the absence of exchange. When $b_1^z⪢b_2^z⪢b_2^x$, $z$ rotations are performed if $J\approx 0$ and $x$ rotations are achieved when $J\approx 2b_2^z$.

Figure 2

(Color online) (a) Charge stability diagram indicating the ground-state charge configuration $(N_1,N_2)$ for local dot potentials $V_1,V_2$. In the (1, 1) configuration, the exchange interaction $J\left(ϵ\right)$ can be tuned by shifting the double-dot potential difference $ϵ\sim V_2-V_1$. (b) When the electron spin in dot 1 is polarized, the qubit electron acquires a Zeeman shift given by $t_{12}^2∕ϵ=-J\left(ϵ\right)∕2$ due to virtual hopping processes that are allowed for spin down, but forbidden for spin up due to the Pauli principle. (c) Energy spectrum of the Hamiltonian given in Eq. (1) in the presence of a strong inhomogeneous magnetic field.

Figure 3

(Color online) Error rates for $H$ and $T$ gates. For these plots we have chosen the parameters $t_{12}=100\mu \mathrm{eV}$, $b_1^z=135\mu \mathrm{eV}$, $\overline{b_2^z}=10\mu \mathrm{eV}$, $\overline{b_2^x}=1\mu \mathrm{eV}$, and ${ϵ}_A=-6\mathrm{meV}$. For the Hadamard gate, these values result in ${ϵ}_B=-t_{12}^2∕\overline{b_2^z}=-1000\mu \mathrm{eV}$, a pulse time $t_{\mathrm{p}}=\pi \hslash ∕2\overline{b_2^x}=1\mathrm{ns}$, and a rise and fall time $\tau \approx \pi \hslash ∕4\overline{b_2^z}=50\mathrm{ps}$. Symbols give the results of numerical integration of the time-dependent Schrödinger equation for the Hadamard gate ($H$, circles) and $\pi ∕8$ gate ($T$, squares). Lines give the estimates for gating error from Eq. (10).