Long-range interaction between charge and spin qubits in quantum dots

We analyze and give estimates for the long-distance coupling via floating metallic gates between different types of spin qubits in quantum dots made of different commonly used materials. In particular, we consider the hybrid, the singlet-triplet, and the spin-$\frac{1}{2}$ qubits, and the pairwise coupling between each type of these qubits with another hybrid qubit in GaAs, InAs, Si, and ${\mathrm{Si}}_{0.9}{\mathrm{Ge}}_{0.1}$. We show that hybrid qubits can be capacitively coupled strongly enough to implement two-qubit gates, as long as the distance of the dots from the metallic gates is small enough. Thus, hybrid qubits are good candidates for scalable implementations of quantum computing in semiconducting nanostructures.

Figure 1

States of the double dot hybrid qubit [7, 8, 9, 10]. The logical state $|0\rangle$ is defined as $|0\rangle =|S\rangle |\downarrow \rangle$, where $|S\rangle$ is the singlet of the two electrons in the ground state of the first dot and $|\downarrow \rangle$ denotes the electron with spin down in the second dot. The logical state $|1\rangle$ is $|1\rangle =\sqrt{1/3}|T_0\rangle |\downarrow \rangle -\sqrt{2/3}|T_{-}\rangle |\uparrow \rangle$, where $|T_0\rangle$ $\left(|T_{-}\rangle \right)$ is the two-electron triplet state in the first dot with spin projection zero $\left(-1\right)$ along the quantization axis, and $|\uparrow \rangle$ denotes the electron with spin up in the second dot. Our diagram for $|1\rangle$ is a simplified sketch of $|T_0\rangle |\downarrow \rangle$. The state $|E\rangle =|\downarrow \rangle |S\rangle$ depicted on the right is used as an intermediate state to prepare the logical states. We note that each of these three states can be written in the simple form “orbital part $\times$ spin part” because additional corrections are negligible in our calculations of the coupling strengths.

Figure 2

(a) Setup consisting of a floating metallic gate (blue) and two DQDs (red dots). (b) Sketch of the confinement potential of a DQD. Here, $a_0$ is an in-plane distance between the floating gate disk center and the DQD, $d$ is the vertical distance between the floating gate disk center and the DQD, $R$ is the radius of the floating gate disk, $L$ is the length of the section connecting the disks, $R_w$ is the radius of this section (modeled as a wire), $\lambda$ is the length of the DQD, and $l$ is the distance between the DQD minima.

Figure 3

Charge distributions in the logical states of the DQD forming the hybrid qubit. In the point charge approximation, we model the charge distribution via delta functions localized at the positions $(x$ coordinates) shown by dotted lines, with the corresponding charges written in multiples of the elementary charge $e^{-}=-e$. In the numerical integration we use for the charge distributions the corresponding eigenfunctions of the parabolic potential.