Various physical computing devices have been explored in quantum mechanical regime for more than two decades to implement the primitive idea of quantum computation^[1−4], which can be seen as superior to classical computation in exponentially faster executing certain computational tasks and simulating complex quantum systems^[5−7]. The electron spin degree of freedom confined in semiconductor quantum dots (QDs) is one of leading candidates to construct physical quantum bits (qubits) for quantum information processors^[8−14]. The fabrication processes of semiconductor QDs can take advantages of modern semiconductor chip engineering^{[12, 15, 16]}. Remarkably, the processes of large-scale and high-quality quantum dot arrays on silicon are fully compatible to an industry-standard CMOS platform^{[17, 18]}. The quantum dot arrays provide convenient building-blocks to achieve large-scale integration of qubits for universal quantum computation^[19]. The precise control of interaction between quantum dots is an essential step in single qubit manipulation, two-qubit gate execution, quantum communication and quantum simulation^[15]. The underlying physical mechanism leading to the interaction between quantum dots is the exchange interaction ($J$) between the confined electron spins in each dot. The exchange interaction, in general, is in short range, strong between confined electrons on the identical dot (intra-dot exchange). The tunnel coupling, however, favors finite exchange interaction taking place in long range between the nearest neighbor dots, and even between remote dots via intermediate dot(s) (inter-dot exchange). The latter is termed as superexchange^{[20, 21]}. We limit our discussion on the inter-dot exchange interaction between the nearest neighbor quantum dots and prefer to refer it simply to exchange interaction if not specifically stated. Coupling more than two quantum dots, in principle, can be achieved by the assistance of the coupling between the nearest neighboring quantum dots. The comprehensive study of the exchange interaction between quantum dots is indispensable to exemplify the electrical method to control the interaction desired in developing the exchange dependent universal quantum computations.

Quantum dots are man-made microstructures and behave in many ways as artificial atoms with reduced energy scales^{[22, 23]}. Therefore, the interactions between quantum dots can be explored using the physical model in study of atom bonding. For two electrons residing in double quantum dots, the method used to study a hydrogen molecule, such as the Heitler−London approach^{[24, 25]} and the Hund−Mulliken approach^[26] can help us understand the exchange interactions between the quantum dots. Furthermore, combining Hubbard and constant interaction models, we can develop electrical methods to control $J$ in a realistic device.

The review concentrates on exploring the underlying mechanism of the exchange interaction between the nearest neighbor quantum dots. The physical mechanisms of exchange interaction and practical method to control the exchange interaction are discussed in Section 2. The role of exchange interaction in constructing qubits, realizing two-qubit gates, performing quantum communication and quantum simulation is intensively surveyed in Section 3. Section 4 is the summary of the review.

The exchange interaction had been discovered independently by Heisenberg^[27] and Dirac^[28] in 1920s. The Pauli exclusion principle states that no two fermions can have the same four electronic quantum numbers^[29]. Thus, the principle exerts a strong influence on each other because a fermion occupying a state excludes the others from it^[30]. This is equivalent to a strong repulsive force which is the source of exchange interaction between fermions^[31]. The repulsion requires that the wave function of indistinguishable fermions is in antisymmetricity, i.e. the wave function changes sign when two fermions are exchanged. The physical effect of the antisymmetric requirement increases the curvature of wavefunctions to prevent the overlap of the states occupied by indistinguishable fermions. As a result, the exchange antisymmetricity increases the expectation value of the distance between two fermions when their wave functions overlap^[32].

Electrons with spin half are fermions. The overall wave function of the system composed of electrons must be antisymmetric. This means if the orbital wave function is symmetric, the spin one must be antisymmetric, vice versa^[33]. The exchange interaction $J$ is, therefore, the energy difference between the antisymmetric orbital wave function (symmetric spin wave function) and the symmetric orbital wave function (antisymmetric spin wave function) to reflect the Pauli repulsion principle^[24], as shown in Fig. 1. In this section, we study the exchange interaction between quantum dots and propose electrical methods to control the exchange interaction.

The Heitler−London (HL) approach was used to describe valence bonding between two atoms and succeeded in explaining the formation of hydrogen molecule^[25]. Consequently, the interaction between double quantum dots with one electron in each dot can be approximated by employing the HL approach^{[34, 35]}.

The HL approach, applied to double quantum dots, associates the Hamiltonian as follows^[34]:

1$\widehat{H}={\displaystyle \sum _{k=1}^2}\widehat{h}(r_k)+C(r_1\mathrm{,}{r}_2)\mathrm{.}$

Here, $\widehat{h}(r_k)=-\frac{{\hslash }^2}{2m}{\nabla }_{r_k}^2+V(r_k)$ is the one-body Hamiltonian of the k-th electron containing kinetic energy and confinement energy in an external potential, $C(r_1\mathrm{,}{r}_2)=\frac{e^2}{4\pi {\epsilon }_0{\epsilon }_{\text{r}}\left|r_1-r_2\right|}$ the Coulomb interaction between two electrons located at $r_{\text{1}}$ and $r_{\text{2}}$ separately, $e=1.602\times {10}^{-19}\text{C}$ the elementary charge, $m=9.1\times {10}^{-31}\text{kg}$ the mass of free electron. In a crystal, we can replace $m$ with the effective mass $m^{*}$ as an approximation^[36].

For gate-defined double quantum dots in two dimensional electron gas (2DEG) with identical confinement profile, we can simulate the $V(r)$ with a quartic potential, in which two harmonic wells possess identical frequency of ${\omega }_0$ centered at $(-a\mathrm{,0})$ and $(+a\mathrm{,0})$, as shown in Fig. 2(a)^[37]. The $V(r)$ reads:

2$V(r)=V(x\mathrm{,}y)=\frac{m{\omega }_0^2}{2}\left[\frac{1}{4a^2}{\left(x^2-a^2\right)}^2+y^2\right]\mathrm{.}$

The (anti)symmetric two-electron orbital state-vector ($|{\psi }_{-}\rangle$) $|{\psi }_{+}\rangle$ can be generated by combining single-dot ground-state orbital wave-functions as follows:

3$\begin{array}{l}|{\psi }_{-}\rangle =\frac{|a\overline{a}\rangle -|\overline{a}a\rangle }{\sqrt{2\left(1-{\Omega }^2\right)}}\mathrm{,}\\ |{\psi }_{+}\rangle =\frac{|a\overline{a}\rangle +|\overline{a}a\rangle }{\sqrt{2\left(1+{\Omega }^2\right)}}\mathrm{.}\end{array}$

Here, $|a\rangle$ and $|\overline{a}\rangle$ are the Dirac notations to indicate the electron ground state in the quantum dot centering at $\left(a\mathrm{,0}\right)$ and $\left(-a\mathrm{,0}\right)$, respectively. $|a\overline{a}\rangle =|a\rangle |\overline{a}\rangle$ and $|\overline{a}a\rangle =|\overline{a}\rangle |a\rangle$ are two-electron product states. $\Omega ={\displaystyle \int }{\phi }_{-a}(r){\phi }_{+a}^{*}(r){\text{d}}^{2}r=\langle a|\overline{a}\rangle$ is the overlap integral of two orbital wave functions. ${\phi }_{-a}(r)=\langle r|\overline{a}\rangle$ and ${\phi }_{+a}(r)=\langle r|a\rangle$ denote the one-electron ground orbital functions centered at $\left(-a\mathrm{,0}\right)$ and $\left(a\mathrm{,0}\right)$, respectively. Analogy to harmonic oscillator eigenfunctions, they can be expressed as follows:

4$\begin{array}{l}{\phi }_{+a}(x\mathrm{,}y)=\sqrt{\frac{m{\omega }_0}{\pi \hslash }}{\text{e}}^{-m{\omega }_0\left[{(x-a)}^2+y^2\right]/2\hslash }\mathrm{,}\\ {\phi }_{-a}(x\mathrm{,}y)=\sqrt{\frac{m{\omega }_0}{\pi \hslash }}{\text{e}}^{-m{\omega }_0\left[{(x+a)}^2+y^2\right]/2\hslash }\mathrm{.}\end{array}$

The energy difference between two states $|{\psi }_{-}\rangle$ and $|{\psi }_{+}\rangle$ in Eq. (3), which differ only in their symmetry, is referred as the exchange energy and is written as follows:

5$J={\epsilon }_{\text{t}}-{\epsilon }_{\text{s}}=\langle {\psi }_{-}\left|\widehat{H}\right|{\psi }_{-}\rangle -\langle {\psi }_{+}\left|\widehat{H}\right|{\psi }_{+}\rangle \mathrm{.}$

From Eqs. (1)−(5) of the Heitler–London approach and making use of the Darwin–Fock solution to the isolated dots, we find^[38] :

6$J=\frac{\hslash {\omega }_0}{\sinh \left(2d^2\right)}\left\{c\left[{\text{e}}^{-d^2}{I}_0(d^2)-1\right]+\frac{3}{4}\left(1+d^2\right)\right\}\mathrm{.}$

Here, unitless $c=\sqrt{2\pi }\left[e^2/\left(4\pi {\epsilon }_0{\epsilon }_{\text{r}}{a}_{\text{B}}\right)\right]/2\hslash {\omega }_0$ is the ratio of Coulomb energy to confinement energy, and $I_0$ the zeroth-order Bessel function. If we set the energy of harmonic wells $\hslash {\omega }_0$ to be $3\text{meV}$, we have $a_{\text{B}}$ at 19 nm and $c$ at 2.4. Numerical calculation to Eq. (6) shows that $J$ decays exponentially with increasing inter-dot spacing $d\equiv a/a_{\text{B}}$, as shown in Fig. 2(b).

The HL approach, however, does not contain any information about the double occupied states^[15]. We have to take into account the hybridization between single-occupied state (1, 1) and double-occupied states (2, 0) as well as (0, 2) by extending the Heitler−London approach to the Hund−Mulliken approach^[26], where $(n_1\mathrm{,}{n}_2)$ means that the number of electrons in quantum dot centered at $(-a\mathrm{,0})$ and $(+a\mathrm{,0})$, respectively. Herein, we limit ourselves in $|S(\mathrm{2,0})\rangle$, $|S(\mathrm{0,2})\rangle$, $|S(\mathrm{1,1})\rangle$, and $|T_0(\mathrm{1,1})\rangle$ four elementary states. $|S\rangle =(|\uparrow \downarrow \rangle -|\downarrow \uparrow \rangle )/\sqrt{2}$ is the singlet state of two electron spins and $T_0=\left(|\uparrow \downarrow \rangle +|\downarrow \uparrow \rangle \right)/\sqrt{2}$ is the triplet state of two electron spins at $S_{\text{z12}}=0$, where $S_{\text{z12}}$ is the spin projection of two electron spins in an arbitrary direction (e.g. the direction of external static magnetic field). The state $|T_0(\mathrm{0,2})\rangle$ is energetically inaccessible due to great intra exchange interaction with the state $|S(\mathrm{0,2})\rangle$^[8]. When an external magnetic field is applied, the triplet states of $|T_{+}\rangle =|\uparrow \uparrow \rangle$ and $|T_{-}\rangle =|\downarrow \downarrow \rangle$ at $S_{\text{z12}}=\pm 1$ are lifted from the state $|T_0(\mathrm{1,1})\rangle$ and are no longer accessible, too.

Two single-occupied states ${\Psi }_{+}^{\text{s}}(r_1\mathrm{,}{r}_2)$ and ${\Psi }_{-}^{\text{s}}(r_1\mathrm{,}{r}_2)$ in either symmetric (+) or antisymmetric (-) forms, and two double-occupied states ${\Psi }_{-a}^{\text{d}}(r_1\mathrm{,}{r}_2)$ and ${\Psi }_{+a}^{\text{d}}(r_1\mathrm{,}{r}_2)$ at $-a$ and $+a$ positions can be constructed in ground states as follows:

7$\{\begin{array}{ll}{\Psi }_{-a}^{\text{d}}(r_1\mathrm{,}{r}_2)={\Phi }_{-a}(r_1){\Phi }_{-a}(r_2)\hfill & S(\mathrm{2,0})\hfill \\ {\Psi }_{+a}^{\text{d}}(r_1\mathrm{,}{r}_2)={\Phi }_{+a}(r_1){\Phi }_{+a}(r_2)\hfill & S(\mathrm{0,2})\hfill \\ {\Psi }_{+}^{\text{s}}(r_1\mathrm{,}{r}_2)=[{\Phi }_{+a}(r_1){\Phi }_{-a}(r_2)+{\Phi }_{-a}(r_1){\Phi }_{+a}(r_2)]/\sqrt{2}\hfill & S(\mathrm{1,1})\hfill \\ {\Psi }_{-}^{\text{s}}(r_1\mathrm{,}{r}_2)=[{\Phi }_{+a}(r_1){\Phi }_{-a}(r_2)-{\Phi }_{-a}(r_1){\Phi }_{+a}(r_2)]/\sqrt{2}\hfill & T_0(\mathrm{1,1})\hfill \end{array}.$

Here, ${\Phi }_{+a}=\left({\phi }_{+a}-g{\phi }_{-a}\right)/\sqrt{1-2\Omega g+g^2}$ and ${\Phi }_{-a}=\left({\phi }_{-a}-g{\phi }_{+a}\right)/\sqrt{1-2\Omega g+g^2}$ are orthogonalized single-electron wave functions centered at $(+a\mathrm{,0})$ and $(-a\mathrm{,0})$, respectively, and $g=\left(1-\sqrt{1-{\Omega }^2}\right)/\Omega$ is a coefficient used to achieve the orthogonalization. Since the symmetricity between orbital and spin wave functions must be opposite, we can write the corresponding spin state respect to the orbital wave function as shown in Eq. (7).

Taking Eq. (7) as the bases, we can express the Hamiltonian of Eq. (1) in a matrix as follows:

8$\widehat{H}=2\epsilon +\left(\begin{array}{cccc}U& V_{\text{x}}& \sqrt{2}{t}_{\text{H}}& 0\\ V_{\text{x}}& U& \sqrt{2}{t}_{\text{H}}& 0\\ \sqrt{2}{t}_{\text{H}}& \sqrt{2}{t}_{\text{H}}& V_{\text{inter}}+V_{\text{x}}& 0\\ 0& 0& 0& V_{\text{inter}}-V_{\text{x}}\end{array}\right)\mathrm{.}$

The elements of matrix in Eq. (8) are as follows:

9$\begin{array}{l}\epsilon =\langle {\Phi }_a\left|\widehat{h}\right|{\Phi }_a\rangle =\langle {\Phi }_{-a}\left|\widehat{h}\right|{\Phi }_{-a}\rangle \mathrm{,}\\ t_{\text{H}}=\langle {\Phi }_a\left|\widehat{h}\right|{\Phi }_{-a}\rangle +W=\langle {\Phi }_{-a}\left|\widehat{h}\right|{\Phi }_a\rangle +W\mathrm{,}\\ W=\langle {\Phi }_a{\Phi }_a\left|C\right|{\Phi }_a{\Phi }_{-a}\rangle =\langle {\Phi }_{-a}{\Phi }_{-a}\left|C\right|{\Phi }_{-a}{\Phi }_a\rangle \mathrm{,}\\ V_{\text{inter}}=\langle {\Phi }_a{\Phi }_{-a}\left|C\right|{\Phi }_a{\Phi }_{-a}\rangle =\langle {\Phi }_{-a}{\Phi }_a\left|C\right|{\Phi }_{-a}{\Phi }_a\rangle \mathrm{,}\\ V_{\text{x}}=\langle {\Phi }_{-a}{\Phi }_a\left|C\right|{\Phi }_a{\Phi }_{-a}\rangle =\langle {\Phi }_a{\Phi }_{-a}\left|C\right|{\Phi }_{-a}{\Phi }_a\rangle \mathrm{,}\\ U=\langle {\Phi }_a{\Phi }_a\left|C\right|{\Phi }_a{\Phi }_a\rangle =\langle {\Phi }_{-a}{\Phi }_{-a}\left|C\right|{\Phi }_{-a}{\Phi }_{-a}\rangle \mathrm{.}\end{array}$

Here, the Coulomb matrix element $U$ is intra-dot Coulomb interaction energy, $V_{\text{inter}}$ inter-dot Coulomb interaction, $V_{\text{x}}$ the exchange energy in the tunnel barrier, and $W$ remaining energy^[39].

The exchange energy between two dots can be determined by the energy gap between the singlet and triplet ground states by solving the eigenvalues of the Hamiltonian in Eq. (8), and reaches at:

10$J=\frac{1}{2}\left(\sqrt{U_{\text{H}}^{\text{2}}+16t_{\text{H}}^{\text{2}}}-U_{\text{H}}\right)-2V_{\text{x}}\mathrm{,}$

where $U_{\text{H}}\equiv U-V_{\text{inter}}$ is on-site repulsion renormalized by long-range Coulomb interactions^[37].

When the two dots are weakly coupled, $V_{\text{x}}$ and $W$ were negligible since they are very small^[40]. Though either $V_{\text{x}}$ or $W$ is not small enough compared with $J$ for relatively strong coupled double quantum dots, they robustly stay constant in most of experiments^[26]. Therefore, we set $V_{\text{x}}$ and $W$ as zero, and Eq. (10) can be simplified as follows^[41]:

11$J=\frac{1}{2}\left(\sqrt{U^2+16t_{\text{c}}^{\text{2}}}-U\right)\mathrm{,}$

where

12$t_{\text{c}}=\langle {\Phi }_a\left|\widehat{h}\right|{\Phi }_{-a}\rangle =\langle {\Phi }_{-a}\left|\widehat{h}\right|{\Phi }_a\rangle \mathrm{.}$

If the double quantum dots are at weak coupling limit, i.e. the Hubbard ratio $t_c/U\ll 1$, the Eq. (11) can be further simplified to:

13$J\approx \frac{4t_{\text{c}}^{\text{2}}}{U}\mathrm{.}$

In order to control the exchange interaction in a realistic device, we correlate the device parameters with the exchange interaction $J$ using the Hubbard model. Gate-defined double quantum dots can be numerically simulated in a GaAs/AlGaAs heterostructure^[42], as shown in Fig. 3(a). By doping the AlGaAs layer with Si atoms, we can introduce free electrons in the quantum well at the interface between AlGaAs and GaAs layers. The electrons accumulate at the GaAs/AlGaAs interface forming a 2DEG, a thin (~10 nm) sheet of electrons that can only move along the interface. A negative potential applied on barrier gates $V_{\text{b}}$, $V_{\text{b1}}$, and $V_{\text{b2}}$ locally depletes the underneath 2DEG to form double quantum dots. By adjusting $V_{\text{b}}$, we can manipulate the barrier height between neighboring quantum dots to control the coupling strength between the two dots. Gate voltages $V_{\text{g1}}$ and $V_{\text{g2}}$ are used to alter the number of electrons residing in left dot 1 and right dot 2, respectively. The total Hamiltonian of the double quantum dots can be expressed by the Hubbard model as follows^{[43, 44]}:

14${\widehat{H}}_{\text{tot}}=\widehat{U}({\widehat{n}}_1\mathrm{,}{\widehat{n}}_2)+{\displaystyle \sum _{\sigma =\uparrow \mathrm{,}\downarrow }}{t}_{\text{c}}\left({\widehat{c}}_{\mathrm{1,}\sigma }^{†}{\widehat{c}}_{\mathrm{2,}\sigma }+h\mathrm{.}c\mathrm{.}\right)\mathrm{,}$

where the operator ${\widehat{c}}_{\mathrm{1,}\sigma }^{†}$(${\widehat{c}}_{\mathrm{2,}\sigma }$) charges (discharges) an electron in QD 1(2) with spin $\sigma =\uparrow \mathrm{,}\downarrow$. ${\widehat{n}}_i={\displaystyle {\sum }_{\sigma =\uparrow \mathrm{,}\downarrow }}{\widehat{c}}_{i\mathrm{,}\sigma }^{†}{\widehat{c}}_{i\mathrm{,}\sigma }={\widehat{n}}_{i\uparrow }+{\widehat{n}}_{i\downarrow }$ is a number operator of QD $i$ ($i$ = 1, 2), $t_{\text{c}}$ a hopping matrix element, known as "tunnel coupling" between two quantum dots^{[45, 46]}.

We limit our discussion on no more than two electrons accommodating in the double quantum dots. The total electrostatic energy of the double quantum dots, $\widehat{U}({\widehat{n}}_1\mathrm{,}{\widehat{n}}_2)$, can be expressed as follows^[45] :

15$\begin{array}{l}\widehat{U}({\widehat{n}}_1\mathrm{,}{\widehat{n}}_2)={\displaystyle \sum _{i=1}^2}\left[\frac{E_{\text{C}i}}{2}{\widehat{n}}_i({\widehat{n}}_i-1)+{\epsilon }_i{\widehat{n}}_i\right]+{\widehat{n}}_1{\widehat{n}}_2{E}_{\text{Cm}}+f(V_{\text{g1}}\mathrm{,}{V}_{\text{g2}})\mathrm{,}\\ {\epsilon }_1\equiv -\left(C_{\text{g1}}{V}_{\text{g1}}{E}_{\text{C1}}+C_{\text{g2}}{V}_{\text{g2}}{E}_{\text{Cm}}\right)/e\mathrm{,}\\ {\epsilon }_2\equiv -\left(C_{\text{g2}}{V}_{\text{g2}}{E}_{\text{C2}}+C_{\text{g1}}{V}_{\text{g1}}{E}_{\text{Cm}}\right)/e\mathrm{,}\\ f(V_{\text{g1}}\mathrm{,}{V}_{\text{g2}})\equiv [\frac{1}{2}{C}_{\text{g1}}^{\text{2}}{V}_{\text{g1}}^{\text{2}}{E}_{\text{C1}}+\frac{1}{2}{C}_{\text{g2}}^{\text{2}}{V}_{\text{g2}}^{\text{2}}{E}_{\text{C2}}+C_{\text{g1}}{V}_{\text{g1}}{C}_{\text{g2}}{V}_{\text{g2}}\times E_{\text{Cm}}]/e^2.\end{array}$

Here, $E_{\text{C}i}$ is the charging energy of individual single dot i, $E_{\text{Cm}}$the electrostatic coupling energy between the two dots.

Since $t_{\text{c}}$($\sim 10\mu \text{eV}$) is significantly smaller than $U(n_1\mathrm{,}{n}_2)$ ($\sim 5\text{meV}$)^[47], the Froehlich−Nakajima transformation (also known as Schrieffer–Wolff transformation) can be used to obtain ${\widehat{H}}_{\text{eff}}$ in (1, 1) charge state subspace^[48−50] from ${\widehat{H}}_{\text{tot}}$ as follows:

16${\widehat{H}}_{\text{eff}}=\widehat{U}(\mathrm{1,1})+J\left({\widehat{S}}_1\cdot {\widehat{S}}_2-\frac{{\hslash }^2}{4}\right)/{\hslash }^2\mathrm{,}$

where ${\widehat{S}}_i=\frac{\hslash }{2}{{\displaystyle \sum }}_{\begin{array}{l}\tau =\uparrow \mathrm{,}\downarrow \hfill \\ {\tau }^{\prime }=\uparrow \mathrm{,}\downarrow \hfill \end{array}}{\widehat{c}}_{i\tau }^{†}{\widehat{\sigma }}_{\tau {\tau }^{\prime }}{\widehat{c}}_{i{\tau }^{\prime }}$ is the spin operator of the electron in dot i^[51], and $\widehat{\sigma }\equiv \left({\widehat{\sigma }}_x\mathrm{,}{\widehat{\sigma }}_y\mathrm{,}{\widehat{\sigma }}_z\right)$ is Pauli operator. $J$ is the exchange interaction between two electron spins and can deduced from Eq. (14):

17$J=\frac{4t_{\text{c}}^{\text{2}}}{\frac{2\left[U(\mathrm{0,2})-U(\mathrm{1,1})\right]\left[U(\mathrm{2,0})-U(\mathrm{1,1})\right]}{\left[U(\mathrm{2,0})-U(\mathrm{1,1})\right]+\left[U(\mathrm{0,2})-U(\mathrm{1,1})\right]}}\mathrm{.}$

Based on Eq. (15), we can rewrite Eq. (17) as follows:

18$J=\frac{4t_{\text{c}}^{\text{2}}}{\frac{2\left[E_{\text{C2}}-\Delta \epsilon -E_{\text{Cm}}\right]\left[E_{\text{C1}}+\Delta \epsilon -E_{\text{Cm}}\right]}{E_{\text{C1}}+E_{\text{C2}}-2E_{\text{Cm}}}}\mathrm{,}$

where $\Delta \epsilon ={\epsilon }_2-{\epsilon }_1$ is a detuning energy and can be manipulated by $V_{\text{g1}}$ and $V_{\text{g2}}$^{[8, 13]} as shown in Eq. (15). The detuning direction is illustrated by a yellow-dashed arrow in the two-dimensional stability diagram of double quantum dots, as shown in Fig. 3(b).

The tunnel coupling $t_{\text{c}}$ in Eq. (18) is defined in Eq. (12). For the single-occupied two quantum dots, the tunneling coupling $t_{\text{c}}$ hybridizes charge states (1, 0) and (0, 1). Taking $|{\Phi }_{+a}\rangle$ and $|{\Phi }_{-a}\rangle$ as bases, we can express the Hamiltonian (Eq. (14)) as follows:

19$\widehat{H}=\left(\begin{array}{cc}{\epsilon }_1& t_{\text{c}}\\ t_{\text{c}}& {\epsilon }_2\end{array}\right)\mathrm{.}$

${\epsilon }_1$ and ${\epsilon }_2$, the energy of charge states (1, 0) and (0, 1), can be controlled by gate voltages $V_{\text{g1}}$ and $V_{\text{g2}}$, respectively, as shown in Eq. (15). The eigenstates and corresponding eigenenergies^[45] of Hamiltonian (Eq. (19)) are shown in Table 1.

The terms in the Table 1 can be expressed as follows:

20$\begin{array}{l}|{\psi }_{\text{A}}\rangle =\cos \frac{\theta }{2}|{\Phi }_{-a}\rangle +\sin \frac{\theta }{2}|{\Phi }_{+a}\rangle \mathrm{,}\\ |{\psi }_{\text{B}}\rangle =-\sin \frac{\theta }{2}|{\Phi }_{-a}\rangle +\cos \frac{\theta }{2}|{\Phi }_{+a}\rangle \mathrm{,}\\ \tan \theta =2t_{\text{c}}/\Delta \mathrm{,}\\ {\epsilon }_{\text{M}}=\left({\epsilon }_1+{\epsilon }_2\right)/2.\end{array}$

The inter-dot tunnel coupling strength $t_{\text{c}}$ hybridizes charge state (0, 1) with (1, 0) around the energy degeneracy point $\Delta \epsilon =0$, resulting in an avoiding cross with an energy splitting of 2$t_{\text{c}}$, as shown in Fig. 4(a). We can adiabatically transfer an electron from the right quantum dot to the left quantum dot by pushing $\Delta \epsilon$ towards the positive side, as shown in Fig. 3(b). The transition process does not contribute a tunnel current through the device but can be monitored by a proximal charge sensor^[52]. When tunnel coupling $t_{\text{c}}$ is smaller than the single-electron level spacing of individual dot, the profile of sensor conductance in the transition from (0, 1) to (1, 0) state, vice versa, can be described by a thermal equilibrium two-level model as follows^{[53, 54]}:

21$g_{\text{s}}=g_0+\delta g\frac{\Delta \epsilon }{\Delta }\tanh \left(\frac{\Delta }{2k_{\text{B}}{T}_{\text{e}}}\right)\mathrm{,}$

where $g_{\text{s}}$ is the conductance of charge sensor, $T_{\text{e}}$ the electron temperature, $\Delta =\sqrt{{\left(\Delta \epsilon \right)}^2+4t_{\text{c}}^{\text{2}}}$ the half of the energy difference between $|{\psi }_{\text{A}}\rangle$ and $|{\psi }_{\text{B}}\rangle$, and $k_{\text{B}}$ the Boltzmann constant. Since $\Delta$ can be extracted experimentally according to Eq. (21), $t_{\text{c}}$ can be obtained subsequently.

Since the confined electrons of quantum dots locate at the interface between GaAs and AlGaAs (Fig. 3(a)) and the transport of electrons is primarily along the X-axis, the potential energy profile along the X-axis at Y ~ 0.08 $\mu \text{m}$ can be used to describe the two quantum dots, as shown in Fig. 4(b). The approximation can help us to simplify the calculation cost of the coupling strength. The barrier height $E_{\text{B}}$ between two quantum dots can be generated and manipulated by the barrier gate voltage $V_{\text{b}}$, as shown in Fig. 4(c). With the help of simplified one-dimensional model, as shown in Fig. 5(a), we can further explore the quantitative relationship between $t_{\text{c}}$ and barrier height $E_{\text{B}}$. The tunnel coupling $t_{\text{c}}$ mixes the wave functions of two quantum dots $|{\Phi }_{+a}\rangle$ and $|{\Phi }_{-a}\rangle$ to form a bonding state $|{\psi }_{\text{B}}\rangle$ and an antibonding state $|{\psi }_{\text{A}}\rangle$, as shown in Table 1 and Eq. (20). By solving single-electron stationary Schrodinger equation in appendix A, we can deduce the tunnel coupling strength $t_{\text{c}}$ from the energy difference between $|{\psi }_{\text{B}}\rangle$ and $|{\psi }_{\text{A}}\rangle$, as follows:

22$t_{\text{c}}=\frac{4{\text{e}}^{-q_{0}L}}{a{q}_0}{E}_{\text{n}}^{\text{(0)}}\mathrm{,}$

where $E_{\text{n}}^{\text{(0)}}\equiv \frac{{\pi }^2{\hslash }^2}{2m{a}^2}$ and $q_0\equiv \sqrt{\frac{2m(E_{\text{B}}-E_{\text{n}}^{\text{(0)}})}{{\hslash }^2}}$.

Eq. (18) provides two independent electrical approaches to manipulate the exchange interaction in experiments as described in the following.

1. The tunnel coupling strength $t_{\text{c}}$ via the barrier gate voltage applied between two neighboring quantum dots. The exponential relationship between tunnel coupling strength $t_{\text{c}}$ and barrier height $E_{\text{B}}$ can be inspired from Eq. (22). Moreover, the simulation shows that $E_{\text{B}}$ is linearly proportional to the barrier gate voltage $V_{\text{b}}$, as shown in Fig. 4(c). Combining with Eq. (18), the facts implies that the exchange interaction $J$ is primarily an exponential function of $V_{\text{b}}$, as shown in Fig. 6(a).

2. The detuning energy $\Delta \epsilon$ via the plunge gate voltage applied on each quantum dot. Two plunge gate voltages can be adjusted to control the detuning energy $\Delta \epsilon$, hence to manipulate the exchange interaction $J$ according to Eq. (18). The exchange interaction $J$ is shown as a function of the detuning energy $\Delta \epsilon$ in Fig. 6(b).

Manipulating the gate voltages $V_{\text{g1}}$ and $V_{\text{g2}}$ also slightly alternates the barrier height $E_{\text{B}}$ by crosstalk effects. Equivalently, $V_{\text{b}}$ makes the crosstalk to the potential of double quantum dots, so that the linear relationship between $E_{\text{B}}$ and $V_{\text{b}}$ is accordingly masked. The crosstalk is routinely and efficiently compensated using virtual gates, which are specific linear combinations of physical gate voltages according to mutual capacitance matrix^{[55, 56]}. The virtual gate can also help to make a sole change of the exchange interaction between target pair quantum dots without affecting the exchange interaction of other pair quantum dots. Several other methods have been formulated to explore the exchange interaction between quantum dots such as Hartree−Fock^[57] and full interaction methods^[58]. Magnetic fields and spin−orbit coupling also influence the exchange interaction, and can be found in details elsewhere^[59−63].

Electrons in multiple quantum dots can be in any available quantum states out of ground states. We routinely choose two lowest energy quantum states as the computational bases to define $|0\rangle$ and $|1\rangle$ states. An effective Hamiltonian of the two-level system can be obtained from the exact Hamiltonian by Schrieffer−Wolff (SW) unitary transformation which decouples the computational basis and the high-energy subspaces^[50]. In 1998, Loss and DiVincenzo proposed a scheme to construct a physical qubit using the single spin state of electron in one single semiconductor quantum dot, known as Loss−Divincenzo (LD) qubit^[64]. A focused local oscillating magnetic fields^[65], however, is required in nanometer-scale devices to manipulate spin states and the power dissipation due to the generation of magnetic field is not compatible with cryogenic environments. One can use a high frequency electric field to control spin states with the assistance of either spin−orbit coupling effect or static slanting field of micromagnet to mitigate the technique challenges^{[66, 67]}. Alternatively, the spin qubit can be realized in a quantum decoherence-free subspace^[15] of spin states in multiple quantum dots, such as S−T₀ qubit^[68] and exchange-only qubit^[69]. Since multiple coupled quantum dots are involved in the schemes, the exchange interaction between neighboring dots is indispensable to achieve the construction and the manipulation of quantum states.

One of the straightforward extensions of LD qubit is the S−T₀ qubit constructed from double quantum dots with charge state (1, 1). The apparent advantage to use S−T₀ qubit is the immunity to the global magnetic fluctuations^[15]. The construction and operation of S−T₀ qubit requires the exchange interaction and a magnetic field gradient across the quantum dots, as shown in Fig. 7(a). The Hamiltonian of the two-spin system is as follows:

23${\widehat{H}}_{\text{S}-{\text{T}}_{\text{0}}}={\widehat{H}}_{\text{S}-{\text{T}}_{\text{0}}}^{\text{1}}+{\widehat{H}}_{\text{S}-{\text{T}}_{\text{0}}}^{\text{2}}=\frac{1}{2}\mu B_1{\sigma }_{\text{z}}^{\text{1}}+\frac{1}{2}\mu B_2{\sigma }_{\text{z}}^{\text{2}}+\frac{1}{4}J\left({\sigma }_1\cdot {\sigma }_2-1\right)\mathrm{,}$

24${\widehat{H}}_{\text{S}-{\text{T}}_{\text{0}}}^{\text{1}}\equiv \frac{1}{2}\mu B_1{\sigma }_{\text{z}}^{\text{1}}+\frac{1}{2}\mu B_2{\sigma }_{\text{z}}^{\text{2}}\mathrm{,}$

25${\widehat{H}}_{\text{S}-{\text{T}}_{\text{0}}}^{\text{2}}\equiv \frac{1}{4}J\left({\sigma }_1\cdot {\sigma }_2-1\right)\mathrm{.}$

Here, $\mu =g^{*}{\mu }_{\text{B}}$ is the magnetic moment of the electron in quantum dot, $g^{*}$ the effective g-factor which is related to spin−orbit interaction^{[70, 71]} and keeps positive as an example, $B_1$($B_2$) the magnetic field strength in the quantum dot 1(2), $J$ the exchange interaction between electron spins in dots 1 and 2, ${\sigma }_1$(${\sigma }_2$) the Pauli operator of quantum dot 1(2), and ${\sigma }_{\text{z}}^{\text{1}}$ (${\sigma }_{\text{z}}^{\text{2}}$) the electron spin Pauli-Z operator of quantum dot 1(2). The eigenstates and corresponding eigenenergies of the Hamiltonians ${\widehat{H}}_{\text{S}-{\text{T}}_{\text{0}}}^{\text{1}}$ and ${\widehat{H}}_{\text{S}-{\text{T}}_{\text{0}}}^{\text{2}}$ are shown in Table 2.

The state subspaces used to define S−T₀ qubit and leakage states are shown in Fig. 7(b), where $S_{\text{z12}}$ is the spin projection of two electron spins in the direction of external static magnetic field and $S_{12}$ is the total spin quantum number. When there is no magnetic field gradient (the Hamiltonian reduced to ${\widehat{H}}_{\text{S}-{\text{T}}_{\text{0}}}^{\text{2}}$), the energy-gap $J$ protected two states $|S\rangle$ and $|T_0\rangle$ form a two-level system and encode $|0\rangle$ and $|1\rangle$, respectively, known as S−T₀ qubit, as shown in Fig. 7(c). The applied magnetic field $B_1$ and $B_2$ separate the leakage states $|\uparrow \uparrow \rangle$ and $|\downarrow \downarrow \rangle$ from the computational state $|T_0\rangle$ to avoid quantum information leakage, as shown in Figs 7(b) and 7(c). We can use energy-selective initialization^[72] to prepare the two electrons in $|S\rangle$ state for further quantum computation.

The $|S\rangle$ and $|T_0\rangle$ states can be set as the north and the south poles of the Bloch sphere, respectively. The quantum states $|\uparrow \downarrow \rangle$ and $|\downarrow \uparrow \rangle$ are thus indicated on the equatorial plane of the Bloch sphere, as shown in Fig. 7(d). For an arbitrary initial state of the system $|\psi \rangle =a|\uparrow \downarrow \rangle +b|\downarrow \uparrow \rangle =c|S\rangle +d|T_0\rangle$ ($a$, $b$, $c$, and $d$ are complex numbers and constrained by ${\left|a\right|}^2+{\left|b\right|}^2=1$ and ${\left|c\right|}^2+{\left|d\right|}^2=1$), the final state reaches at $|\psi \rangle =(a{\text{e}}^{i\mu \Delta Bt/\hslash }|\uparrow \downarrow \rangle +b|\downarrow \uparrow \rangle )/\sqrt{2}$ after time $t$ under the magnetic field gradient $\Delta B$ (the global phase ${\text{e}}^{-i\mu \Delta Bt/2\hslash }$ is ignored). The evolution equals to the state-vector rotation around the Y-axis with precession frequency $\mu \Delta B/h$ on the Bloch sphere. On the other hand, the final state reaches at $|\psi \rangle =(c{\text{e}}^{iJt/\hslash }|S\rangle +d|T_0\rangle )/\sqrt{2}$ after time $t$ under the exchange interaction $J$. The evolution equals to the state-vector rotation around the Z-axis with precession frequency $J/h$ on the Bloch sphere. Consequently, the exchange interaction and the magnetic field gradient provide two independent manipulation axes for the S−T₀ qubit operation. Universal single qubit operations can be achieved by implementing arbitrary rotation of state vector around Y and Z axes^[73] , as shown in Fig. 7(d).

As for a measurement, we can project the charge state (1, 1) to (0, 2). The singlet state $S\left(\mathrm{1,1}\right)$ can transit to $S\left(\mathrm{0,2}\right)$ accompany with one electron spin-conservation tunneling from left to right dot. The triplet state $T_0\left(\mathrm{1,1}\right)$ fails in the transition either to energetically accessible $S\left(\mathrm{0,2}\right)$ state because of spin conservation or to energetically inaccessible $T_0\left(\mathrm{0,2}\right)$ state so that the transfer of electron is forbidden^{[74, 75]}. The event of electron transfer can be monitored by a proximal charge sensor^[54]. Accordingly, the final state can be distinguished in either $|S\rangle$ or $|T_0\rangle$ state.

The LD qubit requires a statistic magnetic field and a high frequency alternating electro-magnetic field as independent manipulation axes. On the other hand, the S−T₀ qubit requires a large magnetic field gradient, which cannot be turned on/off on demand. The implementation of micromagnet and focused alternating electro-magnetic field in quantum chips is also technically challenging^[17]. Exchange-only qubit constructed by the spins of three electrons, alternatively, demonstrates feasibility to achieve universal quantum computation only using the full-electrical exchange interaction between neighboring dots to mitigate the challenges^{[69, 76, 77, 78]}.

Fig. 8(a) shows three-electron spin system, where $J_{12}$ ($J_{23}$) is the exchange interaction between neighboring quantum dots 1(2) and 2(3). The Hamiltonian of the three-spin system is shown as follows:

26${\widehat{H}}_{\text{E}-\text{O}}={\widehat{H}}_{\text{E}-\text{O}}^1+{\widehat{H}}_{\text{E}-\text{O}}^{\text{2}}=\frac{1}{4}{J}_{12}\left({\sigma }_1\cdot {\sigma }_2-1\right)+\frac{1}{4}{J}_{23}\left({\sigma }_2\cdot {\sigma }_3-1\right)\mathrm{,}$

27${\widehat{H}}_{\text{E}-\text{O}}^{\text{1}}\equiv \frac{1}{4}{J}_{12}\left({\sigma }_1\cdot {\sigma }_2-1\right)\mathrm{,}$

28${\widehat{H}}_{\text{E}-\text{O}}^{\text{2}}\equiv \frac{1}{4}{J}_{23}\left({\sigma }_2\cdot {\sigma }_3-1\right)\mathrm{.}$

The eigenstates and corresponding eigenenergies of the Hamiltonians ${\widehat{H}}_{\text{E}-\text{O}}^{\text{1}}$ and ${\widehat{H}}_{\text{E}-\text{O}}^{\text{2}}$ are shown in Table 3.

Where

29$\begin{array}{l}|D_{+1/2}\rangle =(|\uparrow \rangle |T_0\rangle -\sqrt{2}|\downarrow \rangle |T_{+}\rangle )/\sqrt{3}=(|\uparrow \uparrow \downarrow \rangle +|\uparrow \downarrow \uparrow \rangle -2|\downarrow \uparrow \uparrow \rangle )/\sqrt{6}\mathrm{,}\\ |D_{-1/2}\rangle =(|\downarrow \rangle |T_0\rangle -\sqrt{2}|\uparrow \rangle |T_{-}\rangle )/\sqrt{3}=(|\downarrow \downarrow \uparrow \rangle +|\downarrow \uparrow \downarrow \rangle -2|\uparrow \downarrow \downarrow \rangle )/\sqrt{6}\mathrm{,}\\ |D_{+1/2}\text{'}\rangle =|\uparrow \rangle |S\rangle =(|\uparrow \uparrow \downarrow \rangle -|\uparrow \downarrow \uparrow \rangle )/\sqrt{2}\mathrm{,}\\ |D_{-1/2}\text{'}\rangle =|\downarrow \rangle |S\rangle =(|\downarrow \downarrow \uparrow \rangle -|\downarrow \uparrow \downarrow \rangle )/\sqrt{2}\mathrm{.}\end{array}$

30$\begin{array}{l}|{\overline{D}}_{+1/2}\rangle =(|T_0\rangle |\uparrow \rangle -\sqrt{2}|T_{+}\rangle |\downarrow \rangle )/\sqrt{3}=(|\downarrow \uparrow \uparrow \rangle +|\uparrow \downarrow \uparrow \rangle -2|\uparrow \uparrow \downarrow \rangle )/\sqrt{6}\mathrm{,}\\ |{\overline{D}}_{-1/2}\rangle =(|T_0\rangle |\downarrow \rangle -\sqrt{2}|T_{-}\rangle |\uparrow \rangle )/\sqrt{3}=(|\uparrow \downarrow \downarrow \rangle +|\downarrow \uparrow \downarrow \rangle -2|\downarrow \downarrow \uparrow \rangle )/\sqrt{6}\mathrm{,}\\ |{\overline{D}}_{+1/2}\text{'}\rangle =|S\rangle |\uparrow \rangle =(|\uparrow \downarrow \uparrow \rangle -|\downarrow \uparrow \uparrow \rangle )/\sqrt{2}\mathrm{,}\\ |{\overline{D}}_{-1/2}\text{'}\rangle =|S\rangle |\downarrow \rangle =(|\downarrow \uparrow \downarrow \rangle -|\uparrow \downarrow \downarrow \rangle )/\sqrt{2}\mathrm{.}\end{array}$

And

31$\begin{array}{c}|Q_{+3/2}\rangle =|\uparrow \uparrow \uparrow \rangle \mathrm{,}\\ |Q_{+1/2}\rangle =(|\uparrow \uparrow \downarrow \rangle +|\uparrow \downarrow \uparrow \rangle +|\downarrow \uparrow \uparrow \rangle )/\sqrt{3}\mathrm{,}\\ |Q_{-1/2}\rangle =(|\downarrow \downarrow \uparrow \rangle +|\downarrow \uparrow \downarrow \rangle +|\uparrow \downarrow \downarrow \rangle )/\sqrt{3}\mathrm{,}\\ |Q_{-3/2}\rangle =|\downarrow \downarrow \downarrow \rangle \mathrm{.}\end{array}$

The state subspaces of the three electron spins used to define exchange-only qubit and the leakage states are shown in Fig. 8(b), where $S_{\text{z123}}$ is the spin projection of three electron spins in an arbitrary direction, $S_{23}$ the total spin of the electrons confined in the rightmost two quantum dots, $S_{123}$ the total spin quantum number. At the limit of $J_{12}/J_{23}\to 0$ (the Hamiltonian reduced to ${\widehat{H}}_{\text{E}-\text{O}}^{\text{2}}$), the exchange interaction $J_{23}$ provides an energy spacing between $|D\rangle$ and $|D^{\prime }\rangle$ as shown in Fig. 8(c) and Table 3, where $|D\rangle$ is the mixture of $|D_{+1/2}\rangle$ and $|D_{-1/2}\rangle$, and $|D^{\prime }\rangle$ is the mixture of $|D_{+1/2}\text{'}\rangle$ and $|D_{-1/2}\text{'}\rangle$^{[69, 77]}. The two states of $|D\rangle$ and $|D^{\prime }\rangle$ form a two-level system can be defined as $|0\rangle$ and $|1\rangle$, respectively, known as exchange-only qubit. We can first use energy-selective initialization to prepare the last two electrons in a spin singlet state, which lets $S_{23}=0$. As the spin state of the first electron remains random, the three-electron system is initialized in $|D^{\prime }\rangle$ which gives $S_{123}=1/2$. The exchange interactions $J_{12}$ and $J_{23}$ keep the quantum number $S_{123}$ unchanged and allow the computational states within the $S_{123}=1/2$ subspace^[77].

We choose $|D^{\prime }\rangle$ and $|D\rangle$ as the north and the south poles of Bloch sphere. The eigenstates $|\overline{D}\rangle$ and $|{\overline{D}}^{\prime }\rangle$ of ${\widehat{H}}_{\text{E}-\text{O}}^{\text{1}}$ are also indicated on the Bloch sphere, where $|\overline{D}\rangle$ is the mixture of $|{\overline{D}}_{+1/2}\rangle$ and $|{\overline{D}}_{-1/2}\rangle$, and $|{\overline{D}}^{\prime }\rangle$ is the mixture of $|{\overline{D}}_{+1/2}\text{'}\rangle$ and $|{\overline{D}}_{-1/2}\text{'}\rangle$, as shown in Fig. 8(c). According to Eqs. (29) and (30), the angle between the state vector of $|D^{\prime }\rangle$ and the state vector of $|\overline{D}{\text{'}}^{\prime }\rangle$ is 120°, as shown in Fig. 8(d).

Referring to the S−T₀ qubit, we can rotate the state vector separately around the blue-color axis with the precession frequency of $J_{12}/h$, and around the red-color axis with the precession frequency of $J_{23}/h$, as shown in Fig. 8(d). This implies that we can achieve arbitrary single qubit operations by implementing the rotations of Bloch vector around the two axes. Therefore, the two independent exchange interactions between neighboring quantum dots are sufficient to achieve universal single qubit operations^{[13, 15]}.

Like that of S−T₀ qubit, the readout of exchange-only qubit relies on Pauli spin blockade of two neighboring dots. The right most two spins form either the singlet state out of $|D^{\prime }\rangle$ state or the triplet state out of the $|D\rangle$ state, as shown in Fig. 9. We can project the triple-quantum-dot state from charge state (1, 1, 1) to (1, 0, 2) after qubit manipulations. The detection of either singlet or triplet state from a proximal charge sensor reveals the final state of either $|D^{\prime }\rangle$ or $|D\rangle$ state, respectively^[41].

The crucial challenge of exchange-only qubit is to keep the leakage states out of the computational states because the hyperfine interactions from residual nuclear spins fluctuate the total spin and leak the qubit information into $S_{123}=3/2$ states^[77]. An external magnetic field $B$ can be applied additionally to circumvent the leakage to the quantum states $|Q_{\pm 3/2}\rangle$^{[78, 79]}.

The most important application of exchange interaction is the implementation of quantum gate operations between quantum dot qubits^{[80, 81]}. In this section, we take SWAP gate^{[72, 82]} and CNOT gate^{[83, 84]} for examples to explore the role of exchange interaction in achieving two-qubit logical gates.

The SWAP gate is subject to exchange the quantum states of two qubits^[85] and illustrated in a quantum circuit, as shown in Fig. 10(a). $|a\rangle$ and $|b\rangle$ refer to arbitrary quantum states. For LD qubits, we can set spin up as $|1\rangle$ and spin down as $|0\rangle$. The SWAP gate on two LD qubits can be achieved by manipulating the exchange interaction between them^[86]. Taking the schematic diagram of two quantum dots in Fig. 10(b) for an example, we can write the Hamiltonian of the system when the exchange interaction $J$ is on as follows:

32${\widehat{H}}_{\text{SWAP}}=J(t)\left({\widehat{S}}_1\cdot {\widehat{S}}_2-\frac{{\hslash }^2}{4}\right)/{\hslash }^2\mathrm{.}$

The state subspace required for achieving the SWAP gate is shown in Table 4.

The initial state of the system, setting at $|\uparrow \downarrow \rangle =(|S\rangle +|T_0\rangle )/\sqrt{2}$ state, as shown in the upper panel of Fig. 10(b) evolves into $\left(e^{i{\displaystyle {\int }_0^t}J(t^{\prime })\text{d}{t}^{\prime }/\hslash }|S\rangle +|T_0\rangle \right)/\sqrt{2}$ after time $t$. If $t$ satisfies ${\displaystyle {\int }_0^t}J(t^{\prime })\text{d}{t}^{\prime }/\hslash =\pi \mathrm{,3}\pi \mathrm{,5}\pi \mathrm{...}$, the $|\downarrow \uparrow \rangle =(|T_0\rangle -|S\rangle )/\sqrt{2}$ state succeed after the spin SWAP gate, as shown in the bottom panel of Fig. 10(b). $|\uparrow \uparrow \rangle$ and $|\downarrow \downarrow \rangle$ are eigenstates of ${\widehat{H}}_{\text{SWAP}}$, so the swap of these states is trivial.

There is no requirement of magnetic field in the SWAP gate operation. The presence of magnetic field gradient (such as nuclear spin field gradient), on the contrary, compromises the fidelity of the SWAP gate. Therefore, Ⅳ-group material platforms could favor a higher-fidelity SWAP gate operation due to the naturally less nonzero nuclear spins^[82].

The quantum circuit representation of CNOT gate is shown in Fig. 11(a), in which qubit 1 with state $|a\rangle$ acts as control bit and qubit 2 with state $|b\rangle$ acts as target bit. The target qubit sustains if the control qubit is set to $|0\rangle$, whereas the target bit is flipped. The gate action is, thus, expressed as $|a\mathrm{,}b\rangle \to |a\mathrm{,}b\oplus a\rangle$ in formal logic^[85]. Similar to the definition in 3.2.1, we can set spin up as $|1\rangle$ and spin down as $|0\rangle$.

In addition to the exchange interaction, a magnetic field gradient is required to implement the CNOT gate operation in the double quantum dots, as shown in Fig. 11(b). The Hamiltonian of the double dots is close to ${\widehat{H}}_{\text{S}-{\text{T}}_{\text{0}}}$ and can be expressed as follows:

33${\widehat{H}}_{\text{CNOT}}=\mu B_1{S}_{\text{z}}^{\text{1}}/\hslash +\mu B_2{S}_{\text{z}}^{\text{2}}/\hslash +J\left({\widehat{S}}_1\cdot {\widehat{S}}_2-\frac{{\hslash }^2}{4}\right)/{\hslash }^2\mathrm{.}$

${\widehat{S}}_1$(${\widehat{S}}_2$) is the spin operator of electron spin in quantum dot 1(2). $S_{\text{z}}^{\text{1}}$ ($S_{\text{z}}^{\text{2}}$) is the Z-axis component of electron spin operator in quantum dot 1(2).

The state subspace required to achieve CNOT gate is shown in Table 5.

The presence of exchange interaction makes a hybridization between $|\downarrow \uparrow \rangle$ and $|\uparrow \downarrow \rangle$ to generate $|\Downarrow \Uparrow \rangle$ and $|\Uparrow \Downarrow \rangle$ states as the eigenstates. The hybridization states $|\Downarrow \Uparrow \rangle$ and $|\Uparrow \Downarrow \rangle$ collapse to $|\downarrow \uparrow \rangle$ and $|\uparrow \downarrow \rangle$, respectively, at the $\Delta B\gg J$ limit^[83].

The energy diagram of CNOT gate subspace is shown in Fig. 11(c), where $h{f}_{|{\psi }_{\text{R}}\rangle =|\uparrow \rangle }^{\text{L}}$is energy difference between spin states $|\uparrow \rangle$ and $|\downarrow \rangle$ in the left dot with the right spin at state $|\uparrow \rangle$. So as to $h{f}_{|{\psi }_{\text{R}}\rangle =|\downarrow \rangle }^{\text{L}}$, $h{f}_{|{\psi }_{\text{L}}\rangle =|\uparrow \rangle }^{\text{R}}$, and $h{f}_{|{\psi }_{\text{L}}\rangle =|\downarrow \rangle }^{\text{R}}$. The shift of single electron resonance frequency relies on another state and precisely matches the exchange splitting as follows:

34$h{f}_{|{\psi }_{\text{R}}\rangle =|\uparrow \rangle }^{\text{L}}-h{f}_{|{\psi }_{\text{R}}\rangle =|\downarrow \rangle }^{\text{L}}=h{f}_{|{\psi }_{\text{L}}\rangle =|\uparrow \rangle }^{\text{R}}-h{f}_{|{\psi }_{\text{L}}\rangle =|\downarrow \rangle }^{\text{R}}=J\mathrm{.}$