20 25 30 35 40 45 50 $_{0}$ (meV) 0 0.5 1 1.5 2 N i A (meV) 0 10 20 30 B (T) 0.5 1 N i A  2e 2MI 2MI 2e 15]. Here µ$_{l}$ and R$_{l}$ are magnetic moment, and position of the l th nucleus and sum goes over all nucleus in the lattice. In second quantization it can be written as where A$_{αβ}$ ( R$_{l}$ ) = ˜ A$_{l}$ϕ ∗ $_{α}$( R$_{l}$ ) ϕ$_{β}$ ( R$_{l}$ ). Finally where ⃗ Q$_{αα}$ ′ = ∑ I J$_{αα}$ ′ ( ⃗ R$_{j}$ ) ⃗ M$_{j}$ − ∑ n A$_{αα}$ ′ ( ⃗ R$_{l}$ ) ⃗ I$_{l}$ . From the total Hamiltonian H an effective Hamilto- nian is obtained by tracing over the degrees of freedom of the electron wavefunction and by taking the interaction term H$_{in}$ into account to second order of perturbation theory, which yields H$_{en}$ = − ∑ αβ ∑ l A$_{αβ}$ ( R$_{l}$ ) 2 [ I$_{zl}$ ( c † α $_{↑}$c$_{β}$$_{↑}$ − c † α $_{↓}$c$_{β}$$_{↓}$ ) + I + l c † α $_{↓}$c$_{β}$$_{↑}$ + I − l c † α $_{↑}$c$_{β}$$_{↓}$ ] , (8) H$_{int}$ = − 1 2 ∑ αα ′ ∑ σσ ′ ⃗ Q$_{αα}$ ′ · τ$_{σσ}$ ′ c † $_{ασ}$c$_{α}$ $^{′}$σ ′ , (9) H$_{eff}$ = H$_{m}$ + H$_{n}$ + ∑ x |〈 Ψ$_{x}$ | H$_{in}$ | Ψ$_{g}$ 〉| 2 E$_{g}$ − E$_{x}$ . (10) 3 FIG. 2: N$_{i}$ , the maximum number of elementary gate op- erations within the coherence-time normalized by factor 2 πA ( A ≡ A$_{e}$ ≈ A$_{m}$ ) in meV in a Fe:ZnSe DQD (2MI), com- pared with ZnSe DQD filled with two-electrons (2e) versus parabolic strength of confining potential of each dot, ω$_{0}$ , for various gate voltages, V$_{g}$ = 110 (circles), V$_{g}$ = 154 (squares), V$_{g}$ = 198 (triangles), and V$_{g}$ = 242 (diamonds) in meV. N$_{i}$ as a function of external magnetic field for ω$_{0}$ = 27 . 5 meV and V$_{g}$ = 110 meV is shown in the inset. Here we limit our calculation to a two electron and two MI system in a DQD. The two-electron wavefunction is confined to the Hilbert sub-space constructed from the bonding and anti-bonding (HOMO, LUMO) one-electron orbitals of the DQD, ϕ$_{±}$ . Below the magnetic field cor- responding to spin singlet-triplet transition, this results in six basis functions of two-electron, a spin singlet ( S$_{0}$ ) ground state Ψ$_{g}$ that can be expressed as superposition of ϕ$_{+}$ ( ⃗ r$_{1}$ ) ϕ$_{+}$ ( ⃗ r$_{2}$ ) | S$_{0}$ 〉 and ϕ$_{−}$ ( ⃗ r$_{1}$ ) ϕ$_{−}$ ( ⃗ r$_{2}$ ) | S$_{0}$ 〉 with binding energy E$_{g}$ , and five excited states Ψ$_{x}$ with energy E$_{x}$ , consisting of three degenerate first excited triplet states, and two higher excited singlet states [11]. Calculating the matrix elements in Eq. 10 yields the effective Hamiltonian and two electrons, where each MI represents a spin qubit. The result is compared to a two qubit system realized by two electrons in a DQD. The calculation is performed by using the quasi-static bath approximation [15–17], where the host nuclear spins are approximated by a random magnetic field ⃗ B$_{n}$ with a Gaussian distribution. In this limit the two-electron and two-MI nuclear bath Hamilto- nian are given by where H$_{mm}$ = ∑ j,j ′ Δ$_{jj}$ ′ ⃗ M$_{j}$ · ⃗ M$_{j}$ ′ is the electron medi- ated (RKKY-type [14]) interaction between the MIs, and H$_{mn}$ = $^{∑}$$_{j,l}$Δ$_{jl}$ ⃗ I$_{l}$ · ⃗ M$_{j}$ is the electron mediated interac- tion between MIs and nuclear spin bath; the electron mediated interaction between host nuclear spins H$_{nn}$ is neglected. Here, Δ$_{jj}$ ′ = − γ $^{2}$J2 $_{em}$U ( ⃗ R$_{j}$, ⃗ R$_{j}$ ′ ) / (2Δ$_{e}$) and Δ$_{j,l}$ = γ 2 ˜ AJ$_{em}$U ( ⃗ R$_{j}$, ⃗ R$_{l}$ ) / Δ$_{e}$. Further, γ = α$_{+}$ − α$_{−}$ , where α$_{+}$ , and α$_{−}$ are the coefficients of the two-electrons ground state that is expressed as linear combination of bonding-antibonding in two level model Ψ$_{g}$( ⃗ r$_{1}$, ⃗ r$_{2}$ ) = [ α$_{+}$ϕ$_{+}$ ( ⃗ r$_{1}$ ) ϕ$_{+}$ ( ⃗ r$_{2}$ ) + α$_{−}$ϕ$_{−}$ ( ⃗ r$_{1}$ ) ϕ$_{−}$ ( ⃗ r$_{2}$ )] | S$_{0}$ 〉 , Δ$_{e}$ is the two electron singlet-triplet splitting, ˜ A = 1 /L ∑ L l =1 ˜ A$_{l}$ , and U ( ⃗ R$_{j}$, ⃗ R$_{λ}$ ) = ϕ$_{+}$ ( ⃗ R$_{j}$ ) ϕ$_{−}$ ( ⃗ R$_{j}$ ) ϕ$_{+}$ ( ⃗ R$_{λ}$ ) ϕ$_{−}$ ( ⃗ R$_{λ}$ ) with λ = j $^{′}$, l . where k = e , m and ⃗ K = ⃗ S, ⃗ M for electrons and MI, respectively. The coherence time is obtained by solving the equation of motion for ⃗ K$_{1}$ and ⃗ K$_{2}$ with initial state |↑↓〉 and by averaging over the Gaussian magnetic field distribution. From that we obtain 〈 B$_{e}$ 〉 = 〈 B$_{m}$ 〉 = 0, 〈 B 2 $_{e}$〉 = 1 / ( g$_{e}$µ$_{b}$ ) 2 ∑ l I$_{l}$ ( I$_{l}$ + 1) ˜ A 2 l | ϕ$_{+}$ ( R$_{l}$ ) | $^{4}$, and 〈 B 2 $_{m}$〉 = 1 / ( g$_{m}$µ$_{b}$ ) 2 ∑ l I$_{l}$ ( I$_{l}$ + 1) ˜ A 2 l | ϕ$_{+}$ ( R$_{l}$ ) ϕ$_{−}$ ( R$_{l}$ ) | $^{2}$. From there an effective Zeeman splitting ˜ Δ$_{k}$ = (2 〈 B 2 $_{k}$〉 / 3) 1 / 2 is cal- culated, hence τ$_{k}$ = ¯ h/ ( g$_{k}$µ$_{b}$ ˜ Δ$_{k}$). Assuming I$_{l}$ = 1 / 2 we find the spin relaxation time τ$_{e}$ = ¯ h/ (2 A$_{e}$ ) and τ$_{m}$ = C ¯ h/ (2 A$_{m}$ ). Here A$_{e}$ = ( ∑ l ˜ A 2 l | ϕ$_{+}$ ( R$_{l}$ ) | $^{4}$)1 / $^{2}$, A$_{m}$ = ( ∑ l ˜ A 2 l | ϕ$_{+}$ ( R$_{l}$ ) ϕ$_{−}$ ( R$_{l}$ ) | $^{2}$)1 / $^{2}$, and C = Δ$_{e}$ / [ γ $^{2}$J$_{em}$ Λ( ⃗ R$_{1}$, ⃗ R$_{2}$ )] is the RKKY correction to the MI coherence time, stems from the MI-nuclear-spin in- In the following we use the effective Hamiltonian 11 to calculate the decoherence time of the DQD with two MIs H$_{kn}$ = 2 ∑ i =1 g$_{k}$µ$_{b}$ ⃗ B$_{n}$ · ⃗ K$_{i}$, (12) H$_{eff}$ = H$_{m}$ + H$_{n}$ + H$_{mm}$ + H$_{mn}$, (11)