Leakage and sweet spots in triple-quantum-dot spin qubits: A molecular-orbital study

A triple-quantum-dot system can be operated as either an exchange-only qubit or a resonant-exchange qubit. While it is generally believed that the decisive advantage of the resonant-exchange qubit is the suppression of charge noise because it is operated at a sweet spot, we show that the leakage is also an important factor. Through molecular-orbital-theoretic calculations, we show that when the system is operated in the exchange-only scheme, the leakage to states with double electron occupancy in quantum dots is severe when rotations around the axis ${120}^{\circ }$ from $\widehat{z}$ is performed. While this leakage can be reduced by either shrinking the dots or separating them further, the exchange interactions are also suppressed at the same time, making the gate operations unfavorably slow. When the system is operated as a resonant-exchange qubit, the leakage is three to five orders of magnitude smaller. We have also calculated the optimal detuning point which minimizes the leakage for the resonant-exchange qubit, and have found that although it does not coincide with the double sweet spot for the charge noise, they are rather close. Our results suggest that the resonant-exchange qubit has another advantage, that leakage can be greatly suppressed compared to the exchange-only qubit, and operating at the double sweet spot point should be optimal both for reducing charge noise and suppressing leakage.

Figure 1

(a) Schematic diagram of a lateral triple-quantum-dot system. (b) Schematic of the triple-well confinement potential with the energy of each dot ${\epsilon }_1, {\epsilon }_2$, and ${\epsilon }_3$ indicated. The two detuning values $\varepsilon$ and ${\varepsilon }^m$ are also shown [cf. Eqs. (9a) and (9b)].

Figure 2

Calculated parameters $U_1, U_2$, and $U_{12}$ in the Hamiltonian, Eq. (13c), as functions of $\varepsilon$. Parameters: $\hslash {\omega }_0=8\mathrm{meV}, a=22\mathrm{nm}$, and ${\varepsilon }^m=3.3\mathrm{meV}$.

Figure 3

(a) Calculated parameters $t_{12}$ (black line) and $t_{23}$ [red (gray) line] in the Hamiltonian, Eq. (13b), as functions of $\varepsilon$ for ${\varepsilon }^m=3.3\mathrm{meV}$. (b) Schematic plots of the triple-well confinement potential with detuning $\varepsilon <0, \varepsilon =0$, and $\varepsilon >0$ as indicated. Parameters: $\hslash {\omega }_0=8\mathrm{meV}, a=22\mathrm{nm}$.

Figure 4

(a) Calculated parameters $t_{12}$ (black line) and $t_{23}$ [red (gray) line] in the Hamiltonian, Eq. (13b), as functions of $\varepsilon$ for ${\varepsilon }^m=-2.25\mathrm{meV}$. (b) Schematic plots of the triple-well confinement potential with detuning $\varepsilon <0, \varepsilon =0$, and $\varepsilon >0$ as indicated. Parameters: $\hslash {\omega }_0=8\mathrm{meV}, a=22\mathrm{nm}$.

Figure 5

Calculated energy spectra of the triple-quantum-dot system. The ground state is marked as $\left|\mathrm{GS}\right.〉$, the first excited state as $\left|1\mathrm{ES}\right.〉$, and the second excited state as $\left|Q\right.〉$. The splitting between $\left|\mathrm{GS}\right.〉$ and $\left|1\mathrm{ES}\right.〉$ gives the exchange interactions which, at different $\varepsilon$ values, are denoted as $J_L, J_M$, and $J_R$. The angles near the vertical dashed lines give the direction of the rotating axis if a qubit is operated at the corresponding detuning. Parameters: $\hslash {\omega }_0=8\mathrm{meV}, a=22\mathrm{nm}$, and ${\varepsilon }^m=3.3\mathrm{meV}$. [Note that one should not confuse ${\varepsilon }_M$ with ${\varepsilon }^m$: ${\varepsilon }_M=0$ in this figure is the detuning value of $\varepsilon$ relevant to ${\epsilon }_1$ and ${\epsilon }_3$ at which the exchange only qubit is rotated at an axis ${60}^{\circ }$ apart from $\widehat{z}$, cf. Eq. (9a), while ${\varepsilon }^m$ is another detuning value defined in Eq. (9b).]

Figure 6

The composition of states as functions of the detuning $\varepsilon$. (a) Composition of the ground state $\left|\mathrm{GS}\right.〉$. (b) Composition of the first excited state $\left|1\mathrm{ES}\right.〉$. Parameters: $\hslash {\omega }_0=8\mathrm{meV}, a=22\mathrm{nm}$, and ${\varepsilon }^m=3.3\mathrm{meV}$.

Figure 7

The polar angle (the angle between the rotation axis and $\widehat{z}$) vs detuning $\varepsilon$, when $|0_L\rangle$ and $|1_L\rangle$ are defined as qubit states. Parameters: $\hslash {\omega }_0=8\mathrm{meV}, a=22\mathrm{nm}$, and ${\varepsilon }^m=3.3\mathrm{meV}$.

Figure 8

The leakage parameter $\eta$ for both the ground state $\left|\mathrm{GS}\right.〉$ and the first excited state $\left|1\mathrm{ES}\right.〉$ vs the detuning $\varepsilon$ when the triple-dot system is operated as an EO qubit with $|0_L\rangle$ and $|1_L\rangle$ being qubit states. Parameters: $\hslash {\omega }_0=8\mathrm{meV}, a=22\mathrm{nm}$, and ${\varepsilon }^m=3.3\mathrm{meV}$.

Figure 9

The exchange interaction and leakage as functions of the half interdot distance $a$ and the dot size $\hslash {\omega }_0$. For the exchange interactions, the results of $J_L$ (black solid lines), $J_R$ [red (gray) solid lines] and $J_M$ [blue (gray) dash-dotted lines] are shown. For the leakage, the results for $|0_M\rangle$ (black solid lines), $|0_R\rangle$ (black dashed lines), $|1_M\rangle$ [red (gray) solid lines], and $|1_R\rangle$ [red (gray) dashed lines] are shown. (Note that $\eta =0$ for $|0_L\rangle$ and $|1_L\rangle$. ${\varepsilon }^m$ is fixed on 3.3 meV for all the cases.)

Figure 10

The exchange interaction $J$ as a function of detuning ${\varepsilon }^m$. We can see that the sweet spot is at ${\varepsilon }^m=-2.25$ meV. Parameters: $\hslash {\omega }_0=8\mathrm{meV}, a=22\mathrm{nm}$, and $\varepsilon =0$.

Figure 11

The location of the sweet spot ${\varepsilon }^m$ as a function of the half dot distance $a$ (a) and the dot size $\hslash {\omega }_0$ (b). Parameters: $\hslash {\omega }_0=8\mathrm{meV}, a=22\mathrm{nm}$, and $\varepsilon =0$.

Figure 12

The leakage parameter $\eta$ for a triple-dot system operated as an RX qubit as a function of the detuning values. (a) $\eta$ vs $\varepsilon$ while ${\varepsilon }^m$ is fixed at its sweet spot ${\varepsilon }^m=-2.25$ meV. (b) $\eta$ vs ${\varepsilon }^m$ while $\varepsilon$ is fixed at its sweet spot $\varepsilon =0$. The black lines indicate the results for the $\left|\mathrm{GS}\right.〉$ state while the red (gray) lines indicate the $\left|1\mathrm{ES}\right.〉$ state. Parameters: $\hslash {\omega }_0=8\mathrm{meV}, a=22\mathrm{nm}$.

Figure 13

The leakage parameter $\eta$ for a triple-dot system operated as an RX qubit as functions of the detuning ${\varepsilon }^m$. For (a) and (b), $\varepsilon =2\mathrm{meV}$; for (c) and (d), $\varepsilon =-2\mathrm{meV}$ (away from the sweet spot $\varepsilon =0$) The black lines indicate the results for the $\left|\mathrm{GS}\right.〉$ state while the red (gray) lines indicate the $\left|1\mathrm{ES}\right.〉$ state. Parameters: $\hslash {\omega }_0=8\mathrm{meV}, a=22\mathrm{nm}$.

Figure 14

The leakage parameter $\eta$ for a triple-dot system operated as an RX qubit as functions of the detuning $\varepsilon$. For (a) and (b), ${\varepsilon }^m=0$; for (c) and (d), ${\varepsilon }^m=-4\mathrm{meV}$ (away from the sweet spot ${\varepsilon }^m=-2.25$ meV). The black lines indicate the results for the $\left|\mathrm{GS}\right.〉$ state while the red (gray) lines indicate the $\left|1\mathrm{ES}\right.〉$ state. Parameters: $\hslash {\omega }_0=8\mathrm{meV}, a=22\mathrm{nm}$.