Graphene quantum dots for valley-based quantum computing: A feasibility study

The rise of graphene opens a door to qubit implementation, as discussed in the recent proposal of valley pair qubits in double quantum dots of gapped graphene (G. Y. Wu, N.-Y. Lue, and L. Chang, arXiv:1104.0443). The work here presents the comprehensive theory underlying the proposal. It discusses the interaction of electrons with external magnetic and electric fields in such structures. Specifically, it examines a strong, unique mechanism, i.e., the analog of the first-order relativistic effect in gapped graphene. This mechanism is state mixing free and allows, together with electrically tunable exchange coupling, a fast, all-electric manipulation of qubits via electric gates, in the time scale of ns. The work also looks into the issue of fault tolerance in a typical case, yielding at 10 K a long qubit coherence time [∼$O$(ms)].

Figure 1

The DQD qubit structure. The QDs are electrostatically defined, for example, by back gates (not shown in the figure). Gate $V_C$ is used to tune the potential barrier. Gates $V_L$ and $V_R$ are applied for dc and ac μ${}_v$ tuning.

Figure 2

One-electron energy levels in a QD placed in the tilted magnetic field $B_{\mathrm{total}}$ (${\mathbf{B}}_{\mathrm{total}}={\mathbf{B}}_{\mathrm{plane}}+{\mathbf{B}}_{\mathrm{normal}}$). Conduction and valence bands of gapped graphene are also shown. The Zeeman-type interaction for spin and valley DOFs, $H_Z=-g^{*}\sigma {\mu }_B|B_{\mathrm{total}}|+{\tau }_v{\mu }_v\left|B_{\mathrm{normal}}\right|$, splits the quadplet $|{\tau }_v=\pm 1,\sigma =\pm 1/2\rangle$. For $g^{*}{\mu }_B|\sigma B_{\mathrm{total}}|>{\mu }_v\left|B_{\mathrm{normal}}\right|$, the splitting leaves $|{\tau }_v=\pm 1,\sigma =\pm 1/2\rangle$ as the lower valley doublet.

Figure 3

Two-electron energy levels in the DQD. $|z_{T\pm }\rangle$ are split away from $\left\{|z_s\rangle ,|z_{T0}\rangle \right\}$, by $\pm \left({\mu }_{vL}+{\mu }_{vR}\right)\left|B_{\mathrm{normal}}\right|$, respectively. $|z_s\rangle$ and $|z_{T0}\rangle$ are split in energy by $J$, while $|x_{-}\rangle$ and $|x_{+}\rangle$ by $2\left({\mu }_{vL}-{\mu }_{vR}\right)\left|B_{\mathrm{normal}}\right|$.

Figure 4

The time evolution of a qubit state, as governed by $H_{\mathrm{eff}}$, consists of a rotation ${\widehat{R}}_x$(θ${}_x$) about the $x$ axis of the Bloch sphere, and a rotation ${\widehat{R}}_z$(θ${}_z$) about the $z$ axis. θ${}_x$ and θ${}_z$ are the respective angles of rotation.

Figure 5

In the ac mode, the initial qubit state, e.g., $|z_s\rangle$, may be manipulated in the alternating sequence ${}_x\left({\theta }_x^{\left(\mathrm{ac}\right)}\right){\to }_z({\theta }_z=\pi ){\to }_x\left(-{\theta }_x^{\left(\mathrm{ac}\right)}\right){\to }_z({\theta }_z=\pi )\to \cdots {}_z\left({\theta }_z^{\left(\mathrm{target}\right)}+\pi /2\right)$ into a target state (${\theta }_z^{\left(\mathrm{target}\right)}$ is the target state longitude).