In this section, we first summarize the derivation of an effective Hamiltonian allows us to identify how the noises appear during gate operation in the (1, 1) charge regime. More details are provided in the Supplementary Information S1. Then, we present the decoherence formula, the spectral density of the effective noise, and the spin decoherence results.

Effective Hamiltonian

When \(t_0 \ll U - \epsilon _0\), which is generally satisfied in experiments, an effective two-qubit Hamiltonian can be obtained for the system. The way to obtain the effective Hamiltonian is through an Schrieffer–Wolff transformation that decouples the higher energy states. Since we are considering only one of double occupation states, the procedure can be simplified. Consider the Hamiltonian without noise. The Hamiltonian H C without noise can be diagonalized and the corresponding eigenstates can be denoted as \(\left| {(1,1)S^{\prime} } \right\rangle\) and \(\left| {(2,0)S^{\prime} } \right\rangle\). Here the prime denotes that the new state is close to the original state. The higher eigenstate \(\left| {(2,0)S^{\prime} } \right\rangle\) is decoupled from the rest of the Hamiltonian up to the first order of \({\textstyle{{t_0} \over {U - \epsilon _0}}}{\textstyle{{\delta E_Z} \over {U - \epsilon _0}}}\) (Supplementary Information S1), which is in general small. The lower eigenstate \(\left| {(1,1)S^{\prime} } \right\rangle\) is approximately \(\left| {(1,1)S} \right\rangle\) with certain admixture from \(\left| {(2,0)S} \right\rangle\),

$$\left| {(1,1)S^{\prime} } \right\rangle \approx \left| {(1,1)S} \right\rangle + \frac{\theta }{2}\left| {(2,0)S} \right\rangle ,$$ (4)

where the admixture factor θ = \(- 2\sqrt 2 t_0{\mathrm{/}}\left( {U - \epsilon _0} \right)\). The charge admixture lowers the energy of \(\left| {(1,1)S^{\prime} } \right\rangle\), which results in an effective exchange interaction J. Due to this charge admixture, the charge noise also couples to the qubit subspace. \(\left| {(1,1)T_ + } \right\rangle\) and \(\left| {(1,1)T_ - } \right\rangle\) are decoupled from \(\left| {(1,1)T_0} \right\rangle\) and \(\left| {(1,1)S^{\prime} } \right\rangle\), so, the effective Hamiltonian for the states \(\left( {\left| {(1,1)T_0} \right\rangle } \right.\), \(\left. {\left| {(1,1)S^{\prime} } \right\rangle } \right)\) that are affected by noise is

$${H^{\prime}} = \left[ {\begin{array}{*{20}{c}} 0 & {\frac{{\delta E_Z}}{2}} \\ {\frac{{\delta E_Z}}{2}} & {J + \hat n^{\prime} } \end{array}} \right],$$ (5)

$$\hat n^{\prime} = \sqrt 2 \theta \hat n_t - \left( {\theta ^2{\mathrm{/}}4} \right)\hat n_\epsilon ,$$ (6)

where \(J\) = \(\left\langle {S^{\prime} } \right|H_C\left| {S^{\prime} } \right\rangle\) ≈ \(- {\textstyle{{2t_0^2} \over {U - \epsilon _0}}}\) is the exchange interaction, and the noise \(\hat n^\prime\) is formally \(\hat n^{\prime}\) = \(\left( {\partial J{\mathrm{/}}\partial t_0} \right)\hat n_t\) + \(\left( {\partial J{\mathrm{/}}\partial \epsilon _0} \right)\hat n_\epsilon\). Both tunneling noise \(\hat n_t\) and detuning noise \(\hat n_\epsilon\) act on the spin qubit subspace but in different orders of the admixture factor θ. The effective noise due to the tunneling fluctuation is first order in the small charge admixture, ∂J/∂t 0 ∝ θ; while the noise due to the detuning fluctuation is a second order effect, \(\partial J{\mathrm{/}}\partial \epsilon _0 \propto \theta ^2\). Tunneling noise is a first order effect because tunneling noise \(\sqrt 2 \hat n_t\left| {(1,1)S} \right\rangle \left\langle {(2,0)S} \right|\) is coupled to \(\left| {(1,1)S} \right\rangle\) in the qubit subspace even without charge admixture, while \(- \hat n_\epsilon \left| {(2,0)S} \right\rangle \left\langle {(2,0)S} \right|\) is decoupled from the qubit subspace and requires admixture to couple to \(\left| {(1,1)S} \right\rangle\). Even if the tunneling noise is smaller than the detuning noise, which is generally true as shown below, the effect of tunneling noise may still be dominant.

The eigenstates of H′′ without noise can be obtained and denoted as \(\left| { \uparrow \downarrow ^{\prime\prime} } \right\rangle\) and \(\left| { \downarrow \uparrow ^{\prime\prime} } \right\rangle\) (see Fig. 1b). In the basis of \(\left| { \uparrow \downarrow ^{\prime\prime} } \right\rangle\) and \(\left| { \downarrow \uparrow ^{\prime\prime} } \right\rangle\), the effective Hamiltonian including noise is

$$H^{\prime\prime} = \left[ {\begin{array}{*{20}{c}} {\frac{J}{2} + \frac{{{\mathrm{\Omega }}_J}}{2} + \hat n_{ \uparrow \downarrow ^{\prime\prime} , \uparrow \downarrow ^{\prime\prime} }} & {\hat n_{ \uparrow \downarrow ^{\prime\prime} , \downarrow \uparrow ^{\prime\prime} }} \\ {\hat n_{ \downarrow \uparrow ^{\prime\prime} , \uparrow \downarrow ^{\prime\prime} }} & {\frac{J}{2} - \frac{{{\mathrm{\Omega }}_J}}{2} + \hat n_{ \downarrow \uparrow ^{\prime\prime} , \downarrow \uparrow ^{\prime\prime} }} \end{array}} \right],$$ (7)

where \({\mathrm{\Omega }}_J\) = \(\sqrt {J^2 + \delta E_Z^2}\) is the energy splitting of \(\left| { \uparrow \downarrow ^{\prime\prime} } \right\rangle\) and \(\left| { \downarrow \uparrow ^{\prime\prime} } \right\rangle\), \(\hat n_{ \uparrow \downarrow ^{\prime\prime} , \uparrow \downarrow ^{\prime\prime} }\) = \(\left( {1 + J{\mathrm{/\Omega }}_J} \right)\hat n^{\prime} {\mathrm{/}}2\), \(\hat n_{ \uparrow \downarrow ^{\prime\prime} , \downarrow \uparrow ^{\prime\prime} }\) = \(\hat n_{ \downarrow \uparrow ^{\prime\prime} , \uparrow \downarrow ^{\prime\prime} }\) = \(\left( {\delta E_Z{\mathrm{/\Omega }}_J} \right)\hat n^{\prime} {\mathrm{/}}2\), \(\hat n_{ \downarrow \uparrow ^{\prime\prime} , \downarrow \uparrow ^{\prime\prime} }\) = \(\left( {1 - J{\mathrm{/\Omega }}_J} \right)\hat n^{\prime} {\mathrm{/}}2\).

When \(J \ll \delta E_Z\), which is satisfied in a two-qubit gate experiment,15 states \(\left| { \uparrow \downarrow ^{\prime\prime} } \right\rangle\) and \(\left| { \downarrow \uparrow ^{\prime\prime} } \right\rangle\) are approximately spin product states, where \(\left| { \uparrow \downarrow ^{\prime\prime} } \right\rangle\) ≈ \(\left( {\left| {(1,1)T_0} \right\rangle + \left| {(1,1)S^{\prime} } \right\rangle } \right){\mathrm{/}}\sqrt 2\), \(\left| { \downarrow \uparrow ^{\prime\prime} } \right\rangle\) ≈ \(\left( {\left| {(1,1)T_0} \right\rangle - \left| {(1,1)S^{\prime} } \right\rangle } \right){\mathrm{/}}\sqrt 2\), and the effective noise on \(\left| { \uparrow \downarrow ^{\prime\prime} } \right\rangle\) and \(\left| { \downarrow \uparrow ^{\prime\prime} } \right\rangle\) are \(\hat n_{ \uparrow \downarrow ^{\prime\prime} , \uparrow \downarrow ^{\prime\prime} } \approx \hat n_{ \downarrow \uparrow ^{\prime\prime} , \downarrow \uparrow ^{\prime\prime} } \approx \hat n^{\prime} {\mathrm{/}}2\). In this limit, the control-qubit and target-qubit are well defined. The control-qubit state effectively selects the subspace of the system (See Fig. 1b). If control-qubit is spin-down (spin-up), then, the system is in the subspace of \(\left| { \downarrow \uparrow ^{\prime\prime} } \right\rangle\) and \(\left| { \downarrow \downarrow } \right\rangle\) \(\left( {\left| { \uparrow \uparrow } \right\rangle } \right.\) and \(\left. {\left| { \uparrow \downarrow ^{\prime\prime} } \right\rangle } \right)\). Thus, for a given state of the control-qubit, we can reduce the decoherence of a two-qubit system to the decoherence of a two-level system. We focus on pure spin dephasing, which is generally much faster than spin relaxation.

Decoherence formula

In this subsection, we develop the expression that determines the effect of noise. For two states \(\left| \alpha \right\rangle\) and \(\left| \beta \right\rangle\) of interest, the system dephases as exp[−ϕ(τ)],49,50

$$\phi (\tau ) = {\int}_{\omega _0}^\infty {\kern 1pt} d\omega J_{\alpha \beta }^{(zz)}(\omega )[2{\kern 1pt} {\mathrm{sin}}(\omega \tau {\mathrm{/}}2){\mathrm{/}}\omega ]^2,$$ (8)

$$J_{\alpha \beta }^{(zz)}(\omega ) = \frac{2}{{\hbar ^2}}{\int}_{ - \infty }^\infty \left\langle {\hat h_{\alpha \beta }^{(z)}(0)\hat h_{\alpha \beta }^{(z)}(\tau )} \right\rangle {\mathrm{cos}}(\omega \tau )d\tau ,$$ (9)

where \(\hat h_{\alpha \beta }^{(z)}\) = \(\left( {\hat n_{\alpha ,\alpha } - \hat n_{\beta ,\beta }} \right){\mathrm{/}}2\) is the relative noise of the two states \(\left| \alpha \right\rangle\) and \(\left| \beta \right\rangle\) of interest, \(J_{\alpha \beta }^{(zz)}(\omega )\) is the spectral density for the noise, and the cutoff frequency ω 0 represents the inverse of the measurement time of coherence dynamics.

Here, we emphasize the difference between a two-qubit gate system and a S − T 0 qubit before our detailed discussion of decoherence in the system. A two-qubit gate system with two electrons in a DQD shares many similarities with a S − T 0 qubit in a DQD; however, there is an important difference. The difference is due to the fact that spin dephasing depends on the relative noise of two states rather than the noise of each individual states. For a S − T 0 qubit, the qubit is encoded in states \(\left| { \uparrow \downarrow ^{\prime\prime} } \right\rangle\) and \(\left| { \downarrow \uparrow ^{\prime\prime} } \right\rangle\), the effective noise is \(\hat h_{ \uparrow \downarrow ^{\prime\prime} , \downarrow \uparrow ^{\prime\prime} }^{(z)}\) = \(\left( {\hat n_{ \uparrow \downarrow ^{\prime\prime} , \uparrow \downarrow ^{\prime\prime} } - \hat n_{ \downarrow \uparrow ^{\prime\prime} , \downarrow \uparrow ^{\prime\prime} }} \right){\mathrm{/}}2\) = \(\left( {J{\mathrm{/\Omega }}_J} \right)\hat n^{\prime} {\mathrm{/}}2\). Increasing δE Z , which reduces the ratio J/Ω J , can reduce the effective noise \(\hat h_{ \uparrow \downarrow ^{\prime\prime} , \downarrow \uparrow ^{\prime\prime} }^{(z)}\) and suppresses the decoherence of S − T 0 qubit, as shown in a recent experiment.22 However, in a two-qubit gate system, for a given state of control-qubit, only one of \(\left| { \uparrow \downarrow ^{\prime\prime} } \right\rangle\) and \(\left| { \downarrow \uparrow ^{\prime\prime} } \right\rangle\) is involved. The relative noise will be either \(\hat n_{ \uparrow \downarrow ^{\prime\prime} , \uparrow \downarrow ^{\prime\prime} }\) or \(\hat n_{ \downarrow \uparrow ^{\prime\prime} , \downarrow \uparrow ^{\prime\prime} }\), which is not suppressed with increasing δE Z . Therefore, in contrast to a S − T 0 qubit, spin decoherence in a two-qubit logic gate is not suppressed by increasing δE Z .

Spectral densities

In order to study spin dephasing in the system, we need the corresponding spectral density. In a DQD, charge noise can induce detuning noise that arises from the non-identical noise on the two QDs, and tunneling noise that arise from fluctuations in barrier height. When the control-qubit is initialized to be spin-down, the relevant two states are \(\left| { \downarrow \uparrow ^{\prime\prime} } \right\rangle\) and \(\left| { \downarrow \downarrow } \right\rangle\). The effective noise is \(\hat h_{ \uparrow \downarrow ^{\prime\prime} , \downarrow \downarrow }^{(z)} = \hat n^{\prime} {\mathrm{/}}4\). (Dephasing for the spin-up control-qubit is the same, since \(\hat n_{ \uparrow \downarrow ^{\prime\prime} , \uparrow \downarrow ^{\prime\prime} } \approx \hat n_{ \downarrow \uparrow ^{\prime\prime} , \downarrow \uparrow ^{\prime\prime} }\) when \(J \ll \delta E_Z\)). Since the charge noise is believed to be from noise-producing defects, homogeneously distributed in the plane of the device, fluctuation of the tunnel barrier height due to charge noise is of the same order as detuning fluctuations. For non-correlated noises, the effective noise spectral density is given by (Supplementary Information S2)

$$J_{ \uparrow \downarrow ^{\prime\prime} , \downarrow \downarrow }^{(zz)}(\omega ) = A_{eff}{\mathrm{/}}\omega ,$$ (10)

where A eff = \({\textstyle{A \over {8\hbar ^2}}}\left[ {2\theta ^2\left( {\partial t_0{\mathrm{/}}\partial E_b} \right)^2 + {\textstyle{{\theta ^4} \over {16}}}} \right]\). The first term accounts for tunneling noise; the second for detuning noise. ∂t 0 /∂E b converts the barrier fluctuation to fluctuations of the tunneling rate. In the WKB approximation,

$$\partial t_0{\mathrm{/}}\partial E_b \approx t_0{\mathrm{/}}\left( {2{\mathrm{\Delta }}_b} \right),$$ (11)

where \({\mathrm{\Delta }}_b\) ≡ \(\sqrt {\left( {E_b - E_0} \right)\hbar ^2{\mathrm{/}}\left( {2m^ \ast l_b^2} \right)}\), E 0 is the orbital energy of a single QD, E b , and l b are the effective barrier height and width. With knowledge of \(J_{ \uparrow \downarrow ^{\prime\prime} , \downarrow \downarrow }^{(zz)}(\omega )\), the spin dephasing dynamics in exp[−ϕ(τ)] can be calculated from Eq. (8).

Equations (10) and (11) indicate that the relative strength of \(\sqrt 2 \partial t_0{\mathrm{/}}\partial E_b \approx \sqrt 2 t_0{\mathrm{/}}\left( {2{\mathrm{\Delta }}_b} \right)\) and \(\theta {\mathrm{/}}4 = \sqrt 2 t_0{\mathrm{/}}\left( {2\left( {U - \epsilon _0} \right)} \right)\) determines whether tunneling noise or detuning noise dominates. If \({\mathrm{\Delta }}_b > U - \epsilon _0\), detuning noise dominates; if \(\Delta _b < U - \epsilon _0\) tunneling noise dominates.

Spin decoherence results

The spin dephasing has exp[−A eff τ2 ln(1/(ω 0 τ))] dependence (Supplementary Information S3), approximately exp[−(τ/T φ )2] dependence, where T φ is the spin dephasing time. We have evaluated Eq. (8) and fitted exp[−ϕ(τ)] to exp[−(τ/T φ )β], and found that β ≲ 2 (Supplementary Information S4). Therefore, the spin dephasing shows \(1{\mathrm{/}}T_\varphi \propto A_{eff}^{1/\beta } \approx \sqrt {A_{eff}}\) scaling, i.e. the dephasing rate 1/T φ has approximately \(1{\mathrm{/}}\left( {U - \epsilon _0} \right)^2\) dependence for detuning noise and \(1{\mathrm{/}}\left( {U - \epsilon _0} \right)\) dependence for tunneling noise.

Figure 2a shows the detuning dependence of the dephasing rate of the target-qubit including only detuning noise and only tunneling noise (the control-qubit is spin-down). The dephasing due to detuning and tunneling noise show different detuning dependence, which enables the identification of different noise sources. For tunneling noise, results have been shown for different values of ∂t 0 /∂E b , which modifies the crossover between tunneling noise and detuning noise. The dephasing due to tunneling noise can dominate over detuning noise in a wide range of detuning (note that U = 25 meV).

Fig. 2 Spin dephasing 1/T φ as a function of detuning. a 1/T φ as a function of detuning \(\epsilon _0\) due to tunneling noise (TN) only or detuning noise (DN) only. For tunneling noise, we choose representative values ∂t 0 /∂E b = 10−2 (Dashed), 10−3 (Dotted), and 10−4 (Dash-dotted). b J/(2ħ) and 1/T φ as a function of detuning \(\epsilon _0\), where A = (2 μeV)2 and ∂t 0 /∂E b = 3.2 × 10−3. The dots are the experimental data for J/(2ħ) (circle) and 1/T φ (square).15 Full size image

Figure 2b shows a log-log plot of J/(2ħ) and dephasing rate 1/T φ for only tunneling noise and only detuning noise. The experimental data shown as dots is extracted from ref. 15. The calculated 1/T φ due to detuning noise shows approximately \(1{\mathrm{/}}\left( {U - \epsilon _0} \right)^2\) dependence, which is different from the experimental data, while 1/T φ due to tunneling noise shows approximately \(1{\mathrm{/}}\left( {U - \epsilon _0} \right)\) dependence. To match the experimental spin dephasing, we find ∂t 0 /∂E b = (3.2 ± 0.2) × 10−3, which for WKB estimate ∂t 0 /∂E b ≈ t 0 /(2Δ b ) ≈ 4 × 10−4. We attribute the discrepancy to the simplicity of WKB method, the simplicity of the model barrier used, and the exponential dependence of tunneling on the parameters. (Note that the value of ∂t 0 /∂E b also depends on the choice of the amplitude A of charge noise.) J/(2ħ) and 1/T φ are almost parallel, indicating that they show the same \(1{\mathrm{/}}\left( {U - \epsilon _0} \right)\) dependence, and that the dephasing is dominated by the tunneling noise. This parallel dependence doesn’t change with variation of α cz or V cz0 .

The dominance of tunneling noise is counter to what is usually assumed. To understand the qualitative behavior, we consider the WKB approximation Eq. (11). Tunneling noise is dominant because Δ b is small (lower tunnel barrier E b and bigger distance between QDs) compared to \(U - \epsilon _0\) (U is big in small dots). This tends to be satisfied in small silicon QDs using accumulation mode.