Hybrid Spin and Valley Quantum Computing with Singlet-Triplet Qubits

The valley degree of freedom in the electronic band structure of silicon, graphene, and other materials is often considered to be an obstacle for quantum computing (QC) based on electron spins in quantum dots. Here we show that control over the valley state opens new possibilities for quantum information processing. Combining qubits encoded in the singlet-triplet subspace of spin and valley states allows for universal QC using a universal two-qubit gate directly provided by the exchange interaction. We show how spin and valley qubits can be separated in order to allow for single-qubit rotations.

Figure 1

Two quantum registers for QC with spin and valley singlet-triplet qubits. The single (yellow) circles denote dots with spin DOF only (valley degeneracy lifted) hosting auxiliary spins, whereas the double (orange) circles denote dots with both spin and valley DOFs storing one spin and one valley $S\text{−}{T}_0$ qubit. The lines within the DQDs represent the exchange interaction (a single line for spin-only dots, solid and dotted lines for spin and valley), which can realize single-qubit gates when the spin qubit is transferred to the spin-only DQD and a two-qubit gate when spin and valley qubits are in the same DQD. The arrows between dots from neighboring DQDs represent swap operations, interchanging the spin (solid arrow) or valley (dotted arrow) state between these dots. Register (b) has more auxiliary dots; quantum gate sequences are shorter than in (a).

Figure 2

Fidelity $F$ of the time evolution operator $U_{\text{eff}}(\tau )$ describing the exchange interaction between a spin-only dot and a spin-valley dot with respect to the spin swap gate according to Eq. (3). The parameters are chosen to be $U=1\mathrm{meV}$, $h=0.1\mathrm{meV}$, and $t=6\mu \mathrm{eV}$. (a),(b) $F$ maximized over the gate time $\tau$ (scale bar). The detuning is chosen to be $ϵ=\sqrt{U^2-h^2}+\mathrm{\Delta }ϵ\approx 0.995\mathrm{meV}+\mathrm{\Delta }ϵ$ and the phase in the hopping matrix elements, $\phi =\pi /2+\mathrm{\Delta }\phi$ (a) and $\phi =\arccos \frac{3}{4}+\mathrm{\Delta }\phi$ (b). (c),(d) Contour plots of maximal fidelity in dependence of $ϵ$ and $\phi$. The maximum over $\tau$ is taken numerically by searching around the first (c) and fourth (d) local maximum. At $\mathrm{\Delta }\phi =0$, $\mathrm{\Delta }ϵ=0$, $F=1$ can be realized. A shift in the valley hopping phase, $\mathrm{\Delta }\phi$, can be compensated by adjusting $ϵ$. (e),(f) Averaged fidelity for a Gaussian distribution of $ϵ$ with variance ${\sigma }_{ϵ}^2=⟨{ϵ}^2⟩-⟨ϵ{⟩}^2$. The maximization over time is done for the value $ϵ=⟨ϵ⟩$ at the first (e) and fourth (f) local maximum.

Figure 3

Fidelity $F$ for a broad range of the detuning $ϵ$ and the phase $\phi$ in the hopping matrix elements, using the same parameters $U$, $h$, and $t$ as in Fig. 2. (a),(b) $F$ maximized over the gate time $\tau$. We consider the first (a) and fourth (b) local maximum of $F$ as a function of $\tau$. The black squares indicate the positions in parameter space for the time-dependent plots (c),(d), namely, $ϵ=1.35\mathrm{meV}$ for the solid lines and $ϵ=0.65\mathrm{meV}$ for the dotted lines in (c),(d), $\phi =1.9$ (solid line), $\phi =1.333$ (dotted line) in (c), and $\phi =0.972$ (solid line), $\phi =0.581$ (dotted line) in (d). The horizontal dotted lines in (a),(b) represent $ϵ=\sqrt{U^2-h^2}$, and the vertical dotted lines belong to values of $\phi$ being $\pi /2$ (a),(b) and $\arccos \frac{3}{4}$, $\pi -\arccos \frac{3}{4}$ (b). Note that the Schrieffer-Wolff transformation breaks down close to $ϵ=U\pm h$. (e),(f) Averaged fidelity for a Gaussian distribution of $ϵ$ with ${\sigma }_{ϵ}^2=⟨{ϵ}^2⟩-⟨ϵ{⟩}^2$. The maximum in time is determined for $ϵ=⟨ϵ⟩$ at the first (e) and fourth (f) local maximum. The fidelity $F$ is more robust at the first local maximum.