Intrinsic errors in transporting a single-spin qubit through a double quantum dot

Coherent spatial transport or shuttling of a single electron spin through semiconductor nanostructures is an important ingredient in many spintronic and quantum computing applications. In this work we analyze the possible errors in solid-state quantum computation due to leakage in transporting a single-spin qubit through a semiconductor double quantum dot. In particular, we consider three possible sources of leakage errors associated with such transport: finite ramping times, spin-dependent tunneling rates between quantum dots induced by finite spin-orbit couplings, and the presence of multiple valley states. In each case we present quantitative estimates of the leakage errors, and discuss how they can be minimized. The emphasis of this work is on how to deal with the errors intrinsic to the ideal semiconductor structure, such as leakage due to spin-orbit couplings, rather than on errors due to defects or noise sources. In particular, we show that in order to minimize leakage errors induced by spin-dependent tunnelings, it is necessary to apply pulses to perform certain carefully designed spin rotations. We further develop a formalism that allows one to systematically derive constraints on the pulse shapes and present a few examples to highlight the advantage of such an approach.

Figure 1

Illustration of the double quantum dot potential $V_{\text{DQD}}$ defined in Eq. (2). The dimensionless separation between the two dots is $\lambda =2$, and $\epsilon$ is the quantum dot detuning parameter.

Figure 2

(a) The wave function before (dashed line) and after (red line) the ramping process. The ramping time in this plot is ${\omega }_0{T}_I=10$. (b) Ramping errors as a function of ramping time $T_I$, with $s$ accounting for possible nonlinearity in the ramping process [see Eq. (11)]. In this figure the initial detuning is ${\epsilon }_0=4a^2$, and the dimensionless quantum dot separation is $\lambda =2$.

Figure 3

(a) The overall error ${\eta }_{\text{overall}}$ as a function of the time adjustment $\mathrm{\Delta }{T}_{II}$. (b) The optimal time adjustment $\mathrm{\Delta }{T}_{II}^{*}$ as a function of the ramping time $T_I$. The red dot marks the $T_I$ used in (a). (c) Overview of the overall error as a function of $T_I$ and $\mathrm{\Delta }{T}_{II}$. The color bar shows the overall error in a log scale, i.e., ${log}_{10}{\eta }_{\text{overall}}$. We again use ${\epsilon }_0=4$ and $\lambda =2$ in this figure.

Figure 4

The overall error of transporting a spin-up FD state in the presence of finite spin-orbit couplings. Panels (a) and (b) correspond to the overall error without and with a $\pi$-pulse applied in stage II, respectively. The two pulse shapes considered in (b) are defined in Eqs. (16) and (36), and they are both resonant with the Zeeman splitting. The parameter $\zeta$ captures the difference between the tunneling rates of spin-up and spin-down electrons [see Eq. (15)]. The time adjustment $\mathrm{\Delta }{T}_{II}$ is measured with respect to the operation time $T_{II}^{\left(0\right)}$ in a pristine system $(\zeta =0)$, and $\hslash {\omega }_0\simeq 1\text{meV}$.

Figure 5

(a) Comparison of the square $\pi$ pulse ${\mathrm{\Omega }}_{\text{sq}}\left(\tau \right)$ in Eq. (16) (dashed line) and the $sech$-shaped $\pi$ pulse ${\mathrm{\Omega }}_{\text{sech}}\left(\tau \right)$ in Eq. (36) (solid line). Here ${\mathrm{\Omega }}_{\text{sech}}\left(\tau \right)$ has $a=20$. The inset shows a comparison of the corresponding $\mathrm{\Phi }\left(\tau \right)$ for the two pulses. (b) Comparison of the integral kernel in Eq. (33) for ${\mathrm{\Omega }}_{\text{sq}}\left(\tau \right)$ and ${\mathrm{\Omega }}_{\text{sech}}\left(\tau \right)$, respectively.

Figure 6

(a) Error in transporting an arbitrary spin state $|{\chi }_{\alpha }\rangle$. This plot does not include ramping errors. The SOC parameter is set to $\zeta =0.01$, and no pulse $\mathrm{\Omega }\left(t\right)$ is applied during the transport. (b) The transport error when a $2\pi$ pulse is applied to help preserve the spin state. The blue and black solid lines are the same as those in (a). The two pulses ${\mathrm{\Omega }}_{\text{sq}}^{\left(2\pi \right)}$ and ${\mathrm{\Omega }}_{\text{sech}}^{\left(2\pi \right)}$ are defined in Eqs. (47) and (50), respectively, and the ${\mathrm{\Omega }}_{\text{sech}}^{\left(2\pi \right)}$ pulse has a parameter of $a=20$. Also, we set the phase of all initial spin states to be zero ($\gamma =0$). Finally we have $\hslash {\omega }_0\simeq 1\text{meV}$.

Figure 7

(a) Illustration of the ${\mathrm{\Omega }}_{\text{sin}}^{\left(2\pi \right)}\left(\tau \right)$ pulse in Eq. (49). (b) A plot of the corresponding $cos\left[2\mathrm{\Phi }\right(\tau \left)\right]$ (solid line) and $sin\left[2\mathrm{\Phi }\right(\tau \left)\right]$ (dashed line) for the pulse in (a).

Figure 8

Comparison between $\pi$ pulses and $2\pi$ pulses. The solid lines correspond to $2\pi$ pulses, while the dashed lines correspond to their $\pi$-pulse counterparts. Panels (a) and (b) correspond to the square pulse and $sech$-shaped pulse, respectively.

Figure 9

Illustration of two different $s_x$ rotations. The initial spin state is $\alpha =\pi /4$ and $\gamma =\pi /2$, which is marked by the red arrow. The angle of the two rotations are $2{\theta }_1=\pi /2$ and $2{\theta }_2=2\pi /3$, respectively.

Figure 10

Illustration of the valley and Zeeman splitting in a single-spin qubit in silicon. The red dot labeled by $B_c$ marks the “hot spot” where the Zeeman splitting equals the valley splitting.

Figure 11

Leakage to the other valley eigenstate when the SOC strength ${\mathrm{\Delta }}_a$ is (a) $0.1\mu \text{eV}$ and (b) $0.5\mu \text{eV}$, respectively. The detuning of the pulse is $\delta =0.1\text{meV}$, and the valley splitting is $E_{\text{VS}}=0.33\text{meV}$. The pulse duration is set to $T_0=\pi \hslash /10\delta \simeq 2.0\text{ps}$.