MI V$_{g}$ em J e A n ( ϕ$_{ασ}$ ) are calculated by discretizing T + Z in real space, and diagonalizing the resulting matrix. By using the cre- ation (annihilation) operators c † ασ ( c$_{ασ}$ ) for an electron in a non-interacting single-particle (SP) state | α, σ 〉 , the Hamiltonian of an interacting system in second quanti- zation can be written as 2 where the first term represents the single particle Hamiltonian and V$_{ασ,βσ}$ $^{′}$,µσ $^{′}$,νσ = ∫ d⃗ r ∫ d ⃗ r $^{′}$ϕ ∗ $_{ασ}$( ⃗ r ) ϕ ∗ βσ ′ ( ⃗ r $^{′}$) e 2 ε | ⃗ r − ⃗ r ′ $_{|}$ϕ$_{µσ}$ ′ ( ⃗ r $^{′}$) ϕ$_{νσ}$ ( ⃗ r ), is the two-body Coulomb matrix element. The Hamiltonian for the MIs accounts for MI-MI direct exchange interaction and MI-Zeeman coupling, FIG. 1: Schematics of two spin qubit unit in a DQD consid- ered here. Each spin qubit is represented by a MI. Interaction between qubits is mediated by the two electrons; the electron MI coupling coefficient is denoted by J$_{em}$ . As interaction of MIs with nuclear spin bath is negligible, the dominant noise source is electron mediated coupling between MIs and nuclear spin bath; the electron nuclear spin bath coupling coefficient is given by A . The interaction strength between the MIs and between MI and nuclear spin bath depends on the singlet- triplet electron energy gap Δ$_{e}$, which can be controlled by external electric gate voltage ( V$_{g}$ ), magnetic field ( B ), and shape/size of the DQD. This allows active noise engineering and optimization of the two qubit system. where J$_{jj}$ ′ is the direct MI-MI antiferromagnetic (AFM) coupling, g$_{m}$ is the MI g-factor, and M$_{zj}$ is the z- component of the MI spin operator. The nuclear-nuclear direct dipole interaction in the host semiconductor is neglected. The nuclear Hamilto- nian is given by the Zeeman coupling term, H$_{e}$ = ∑ α ∑ σ ϵ$_{ασ}$c † $_{ασ}$c$_{ασ}$ + 1 2 ∑ αβµν ∑ σσ ′ V$_{ασ,βσ}$ $^{′}$,µσ ′ $_{,νσ}$c † $_{ασ}$c † βσ $_{′}$c$_{µσ}$ ′ c$_{νσ}$, (2) H$_{m}$ = M ∑ j,j $^{′}$=1 J$_{jj}$ ′ ⃗ M$_{j}$ · ⃗ M$_{j}$ ′ + ∑ j g$_{m}$µ$_{b}$M$_{zj}$B, (3) H$_{n}$ = L ∑ l =1 g$_{n}$µ$_{b}$I$_{zl}$B, (4) H$_{em}$ = − J$_{em}$ ∑ i,j ⃗ S$_{i}$ · ⃗ M$_{j}$δ ( r$_{i}$ − R$_{j}$ ) , (5) H$_{e}$ = N ∑ i =1 ( T$_{i}$ + Z$_{i}$ ) + e 2 2 ε ∑ i = j 1 | ⃗ r$_{i}$ − ⃗ r$_{j}$ | , (1) H$_{em}$ = − ∑ αβ ∑ I J$_{αβ}$ ( R$_{j}$ ) 2 [ M$_{zj}$ ( c † α $_{↑}$c$_{β}$$_{↑}$ − c † α $_{↓}$c$_{β}$$_{↓}$ ) + M + j c † α $_{↓}$c$_{β}$$_{↑}$ + M − j c † α $_{↑}$c$_{β}$$_{↓}$ ] , (6) H$_{en}$ = ∑ i,l ˜ A$_{l}$ ⃗ S$_{i}$ · ⃗ I$_{l}$δ ( r$_{i}$ − R$_{l}$ ) (7) with ˜ A$_{l}$ = (16 π/ 3) µ$_{b}$µ$_{l}$/I$_{l}$ the isotropic (Fermi contact) part of the electron-nucleus hyperfine interaction [6, 13, where J$_{αβ}$ ( R$_{j}$ ) = J$_{em}$ϕ ∗ $_{α}$( R$_{j}$ ) ϕ$_{β}$ ( R$_{j}$ ). Similarly, we de- scribe the electron - nuclear spin bath hyperfine interac- tion by with J$_{em}$ the exchange coupling between electron spin ⃗ S$_{i}$ at r$_{i}$ and impurity spin ⃗ M$_{j}$ located at the position R$_{j}$ [10]. In second quantization it can be written as The e-MI exchange interaction is modeled by where g$_{n}$ is the nuclear g-factor and I$_{zl}$ is the z- component of nuclear spin operator. between the nuclear spins of MI and host nuclei has no influence on our system. Electrons confined in quasi-two-dimensional quantum dots in a uniform perpendicular magnetic field can be described by the effective mass Hamiltonian where T = 1 / (2 m $^{∗}$) $^{(}$(¯ h/i ) ⃗ ∇ + ( e/c ) A ( ⃗ r ) 2 + V ( ⃗ r ) is the single electron Hamiltonian in an external magnetic field ⃗ B = B ˆ z , perpendicular to the plane of the confining po- tential. Here ( ⃗ r ) = ( x, y, z ) describes the electron posi- tion, V ( ⃗ r ) denotes the quantum dots confining potential, and A ( ⃗ r ) = (1 / 2) ⃗ B × ⃗ r is the vector potential. Further, m ∗ is the conduction-electron effective mass, − e is the electron charge, and ε is the host semiconductor dielec- tric constant. Finally, Z$_{i}$ = (1 / 2) g$_{e}$µ$_{b}$S$_{zi}$B determines the Zeeman spin splitting, g$_{e}$ is the electron g-factor in host semiconductor, µ$_{b}$ refers to the Bohr magneton, and S$_{zi}$ represents the z -Pauli matrix of electron i . The single particle eigenvalues ( ϵ$_{ασ}$ ) and eigenvectors