Fig. 1. (Color online) The exchange interaction $J$ is the energy difference between the antisymmetric orbital wave function and the symmetric orbital wave function of two electrons.

Fig. 2. (Color online) (a) Quartic potential as illustrated in Eq. (2) with $\hslash {\omega }_0=3\text{meV}$ and $a=19\text{nm}$. The potential is used to simulate the coupling of two electrons locating in two harmonic wells centered at $\left(-a\mathrm{,0}\right)$ and $(a\mathrm{,0})$. The effective Bohr radius of harmonic well is $a_{\text{B}}=\sqrt{\hslash /m{\omega }_0}$. (b) The exchange coupling strength $J$ between two spins as a function of the inter-dot spacing $d=a/a_{\text{B}}$ with $\hslash {\omega }_0=3\text{meV}$, $a_{\text{B}}=19\text{nm}$ and $c=2.4$ [See Eq. (6)].

Fig. 3. (Color online) (a) Double quantum dots in GaAs/AlGaAs heterostructure. (b) Two dimensional stability diagram of double quantum dots. The purple dashed-line is the position where $\Delta \epsilon$ is zero, the yellow dashed-arrow indicates the direction along which $\Delta \epsilon$ increases.

Fig. 4. (Color online) (a) Energy of $|{\psi }_{\text{A}}\rangle$ and $|{\psi }_{\text{B}}\rangle$ versus the detuning energy $\Delta \epsilon$. (b) The electrostatic potential $U$ of quantum dots in Fig. 3(a) along the X-axis at Y ~ $0.08\mu \text{m}$. The barrier height $E_{\text{B}}$ between two quantum dots can be manipulated by the barrier gate voltage $V_{\text{b}}$. (c) The barrier height $E_{\text{B}}$ is negatively proportional to the barrier gate voltage $V_{\text{b}}$. The black straight line is a linear fit.

Fig. 5. (Color online) (a) Infinitely deep double-well model. $a$ is the width of the well, $L$ the width of the barrier between two wells, and $E_{\text{B}}$ the barrier height between two wells. For simplicity, the potential out of the double wells is set infinitely high due to Coulomb blockade effect. (b) The tunnel coupling $t_{\text{c}}$ as a function of barrier height $E_{\text{B}}$. The data points are calculated based on the device shown in Fig. 3(a). The curve is a fit to Eq. (22).

Fig. 6. (a) The exchange energy $J$ as a function of the barrier gate voltage $V_{\text{b}}$ as described in Eqs. (18) and (22). (b) The exchange energy $J$ as a function of detuning energy $\Delta \epsilon$ as described in Eq. (18).

Fig. 7. (Color online) (a) The construction of S−T₀ qubit. The exchange interaction $J$ between quantum dots, and a magnetic field gradient $\Delta B$ is required to build a S−T₀ qubit with double quantum dots. (b) States with two electron spins. The states with color mapping of orange and cyan color are the computational states of S−T₀ qubit. The gray color mapped states are the leakage states. (c) Two states of $|S\rangle$ and $|T_0\rangle$ form a two-level subspace. (d) Exchange coupling $J$ and a longitudinal magnetic-field gradient $\Delta B$ provide two orthogonal manipulation axes for S−T₀ qubit operation.

Fig. 8. (Color online) (a) The construction of exchange-only qubit. Only the exchange interaction $J_{12}$ and $J_{23}$ between quantum dots are required to build the exchange-only qubit with linearly coupled triple quantum dots. (b) States with three electron spins. The states with color mapping of orange and cyan color are the computational states of exchange-only qubit. The gray color mapped states are the leakage states. (c) Two states of $|D\rangle$ and $|D^{\prime }\rangle$ form a two-level subspace. $|D\rangle$ is the mixture of $|D_{+1/2}\rangle$ and $|D_{-1/2}\rangle$, $|D^{\prime }\rangle$ is the mixture of $|D_{+1/2}\text{'}\rangle$ and $|D_{-1/2}\text{'}\rangle$. (d) The exchange coupling parameters of $J_{12}$ and $J_{23}$ provide two independent manipulation axes of the exchange-only qubit manipulation.

Fig. 9. (Color online) The spin configuration of $|D^{\prime }\rangle$ and $|D\rangle$. $S_1$ is the spin quantum number of electron in the first quantum dot. $S_{23}$ is the total spin quantum number of two electrons, which makes the second and the third quantum dot singly occupied.

Fig. 10. (Color online) (a) The quantum circuit of SWAP gate. (b) Double quantum dots with one single electron in each dot. $S_1$($S_2$) is the spin of electron in left (right) quantum dot. The exchange interaction $J$ is required to obtain the SWAP gate in specific duration.

Fig. 11. (Color online) (a) The quantum circuit of CNOT gate. (b) Double quantum dots with one single electron in each dot. $S_1$($S_2$) is the spin of electron in left (right) quantum dot. The exchange interaction $J$ between two quantum dots, and a magnetic field gradient $\Delta B$ are required to achieve the CNOT gate operation. (c) Schematic energy level diagram adopted from Table 5. (d) The spin of electron in the left quantum dot is flipped with absorbing a photo from an alternating magnetic field at frequency of $f_{|{\psi }_{\text{R}}\rangle =|\uparrow \rangle }^{\text{L}}$ when the spin of electron in the right quantum dot is $|\uparrow \rangle$.

Fig. 12. (Color online) (a) Spin-coherent transport through adiabatic passage. (b) Long-range coupler of spins to transfer an arbitrary spin from the left most Alice to the right most Bob^[91]. (c) A representation of triple quantum dots for simulation of interaction-driven Mott metal−insulator transition. (d) Energy spectrum of low spin state and ferromagnetic state as a function of tunnel coupling strength, where $S$ is the total spin number of the three electrons.

Table 0. The eigenstates and corresponding eigenenergies of the Hamiltonian ${\widehat{H}}_{\text{CNOT}}\mathrm{.}$

Table 0. The eigenstates and corresponding eigenenergies of Hamiltonian ${\widehat{H}}_{\text{E}-\text{O}}^{\text{1}}$ and ${\widehat{H}}_{\text{E}-\text{O}}^{\text{2}}$^[41].

Table 0. The eigenstates and corresponding eigenenergies of Hamiltonians ${\widehat{H}}_{\text{S}-{\text{T}}_{\text{0}}}^1$ and ${\widehat{H}}_{\text{S}-{\text{T}}_{\text{0}}}^2$ with $\Delta B\equiv B_2-B_1$.

Table 0. The eigenstates and corresponding eigenenergies of Hamiltonian ${\widehat{H}}_{\text{SWAP}}\mathrm{.}$

Table 0. The eigenstates and the corresponding eigenenergy of the Hamiltonian (Eq. (19)).