Long-Range Interaction of Spin-Qubits via Ferromagnets

We propose a mechanism of coherent coupling between distant spin qubits interacting dipolarly with a ferromagnet. We derive an effective two-spin interaction Hamiltonian and estimate the coupling strength. We discuss the mechanisms of decoherence induced solely by the coupling to the ferromagnet and show that there is a regime where it is negligible. Finally, we present a sequence for the implementation of the entangling CNOT gate and estimate the corresponding operation time to be a few tens of nanoseconds. A particularly promising application of our proposal is to atomistic spin-qubits such as silicon-based qubits and NV-centers in diamond to which existing coupling schemes do not apply.


I. INTRODUCTION
Quantum coherence and entanglement lie at the heart of quantum information processing. One of the basic requirements for implementing quantum computing is to generate, control, and, measure entanglement in a given quantum system. This is a rather challenging task, as it requires to overcome several obstacles, the most important one being decoherence processes. These negative effects have their origin in the unavoidable coupling of the quantum systems to the environment they are residing in.
A guiding principle in the search for a good system to encode qubits is the smaller the system the more coherence, or, more precisely, the fewer degrees of freedom the weaker the coupling to the environment. Simultaneously, one needs to be able to coherently manipulate the individual quantum objects, which is more efficient for larger systems. This immediately forces us to compromise between manipulation and decoherence requirements.
Following this principle, among the most promising candidates for encoding a qubit we find atomistic twolevel systems, such as NV-centers and silicon-based spin qubits. [1][2][3][4][5][6][7][8][9][10][11][12] The latter are composed of nuclear (electron) spins of phosphorus atoms in a silicon nanostructure. They have very long T 2 times of 60 ms 13 for nuclei and of 200µs for electrons. 14 Recently, high fidelity single qubit gates and readout have been demonstrated experimentally. 14 Nitrogen-vacancy centers 15 in diamond have also been demonstrated experimentally to be very stable with long decoherence times of T * 2 ≈ 20 µs and T 2 ≈ 1.8 ms. 16 Both types of spin qubits have the additional advantage that noise due to surrounding nuclear spins can be avoided by isotopically purifying the material. Unfortunately, it is hardly possible to make these spin qubits interact with each other in a controlled and scalable fashion. They are very localized and their position in the host material is given and cannot be adjusted easily. Therefore, if during their production two qubits turn out to lie close to each other they will always be coupled, while if they are well-isolated from each other they will never interact. It is thus of high interest to propose a scheme to couple such atomistic qubits in a way that allows a high degree of control.
We fill this gap in the present work by proposing a setup to couple two spin qubits separated by a relatively large distance on the order of micrometers, see Fig. 1. The coupling is mediated via a ferromagnet with gapped excitations to which the spin qubits are coupled by magnetic dipole-dipole interaction. Since the ferromagnet is gapped only virtual magnons are excited but in order to obtain the sizable coupling one needs to tune the splitting of the qubit close to resonance with the gap of the ferromagnet. The on and off switching of the qubitqubit interaction is therefore achieved by tuning qubits off resonance (see below). The resulting system is thus realizable with present state-of-the-art technologies. We point out that our analysis is not restricted to a precise type of spin qubit but is in principle applicable to any system that dipolarly interact with the spins of a ferromagnet. In particular, our proposal is also applicable to an electron spin localized in a semiconductor quantum dot, gate-defined or self-assembled. 17,18 While other schemes exist to couple such qubits over large distances, [19][20][21][22][23] none of them is applicable to atomistic qubits. The main novelty of our proposal is thus the possibility to also couple atomistic qubits that are of high technological relevance.
Before we proceed with the quantitative analysis, let us first give an intuitive picture of the qubit-qubit coupling. The coupling between two distant qubits is mediated via a coupler system. The relevant quantity of this coupler is its spin-spin susceptibility-in order to have a long-range coupling, a slowly spatially decaying susceptibility is required. The dimensionality of the coupler plays an important role since, in general, it strongly influences the spatial decay of the susceptibility, which can be anticipated from purely geometric considerations. Furthermore, since the coupler interacts with the qubits via magnetic dipolar forces, we require that a large part of the coupler lies close to the qubits. To this end we immediately see that a dog-bone shape depicted in Fig. 1 satisfies these two requirements-strong dipolar coupling to the qubits and slow spatial, practically 1D, susceptibility decay between the qubits. The orange dog-bone shape denotes the ferromagnet that is coupled via magnetic dipole interaction to spins of nearby quantum dots (red sphere with green arrow). The ferromagnet is assumed to be a monodomain and its magnetization is denoted by blue arrows (M ) that can take arbitrary orientation. L is the length of the quasi-1D ferromagnetic channel that is approximately equal to the distance between the qubits. The shape of the ferromagnetic coupler is chosen such that it enables strong coupling to the spin-qubits while maintaining the spatially slowly decaying 1D susceptibility between the two discs.

II. MODEL
The system we consider consists of two spin-1 2 qubits coupled dipolarly to the ferromagnet where H F is for the moment unspecified Hamiltonian of the dog-bone shaped ferromagnet that is assumed to be polarized along the x-axis. We first assume that the qubits are also polarized along the x-axis, H σ = i=1,2 ∆i 2 σ x i , while the ferromagnet disc axes are along z, see Fig. 1. The magnetic dipole coupling between the ferromagnet and the spin-qubits can be written as where A r , B r , C r are given by with S ± r = S y r ± iS z r and lattice constant a. Here we denote the real part of a complex number with prime and the imaginary part with double prime. The operator S r describes the spin of the ferromagnet at the position r.
Next, we release the assumptions about the mutual orientation of the disc axes, the axes of polarization of the ferromagnet, and the direction of the qubits splitting and assume that these can take arbitrary directions. Now the interaction Hamiltonian reads where S r and σ r have, in general, different quantization axes. The expressions of the coefficients in Eq. (6) are now more complicated, nevertheless it is important to note that the integrals of these coefficients are experimentally accessible. The qubits can be used to measure the stray field of the ferromagnet which is given by . . , e i } r . In order to measure the remaining coefficients, one needs to apply the magnetic field externally in order to polarize sequentially the ferromagnet along the two perpendicular directions to the ferromagnet easy axis. The coefficients are obtained then by measuring again the stray fields (with the aid of the qubits) which now are given by (d +e , d −e , c ) and (d +e , d −e , c ). Furthermore, all the results that we are going to obtain for the qubitqubit coupling as well as the estimates of the decoherence will depend only on the integrals of the coefficients, i.e., on {a i , . . . , e i } rather than {a i , . . . , e i } r .

A. Coherent coupling
We proceed to derive the effective qubit-qubit coupling by performing a Schrieffer-Wolff (SW) transformation. 24 We assume that the excitations in the ferromagnet are gapped due to some magnetic anisotropy (e.g. shape-anisotropy), with the gap being denoted by ∆ F . This is important because when the qubit splitting ∆ is smaller than ∆ F , flipping the qubit spin cannot excite magnons in the ferromagnet, thus there are only virtual magnons excited via coupling to the qubitsotherwise such a coupling would lead to strong decoherence in the qubits. Due to the presence of the gap in the ferromagnet, its transversal susceptibility χ ⊥ (ω, r) decays exponentially for ω < ∆ F with the characteristic length l F ∝ 1/ √ ∆ F − ω, thus we take into account only terms with ω ∼ ∆ F , see Appendix. Straightforward application of lowest order SW transformation accompanied by tracing out the degrees of freedom of the ferromagnet yields the effective qubit-qubit coupling Hamiltonian where χ 1D ⊥ is the transverse susceptibility (i.e. transverse to the z direction) of a quasi-1D ferromagnet, since we assumed a dog-bone shaped ferromagnet. We have neglected the longitudinal susceptibility χ since it is smaller by factor of 1/S compared to the transverse one and it is suppressed by temperature. It is readily seen from the above expression that in order to obtain a sizable coupling between the qubits we have to tune at least one of the qubits close to resonance, ∆ i ∼ ∆ F . This can be achieved by conveniently positioning the qubit such that the Zeeman splitting produced by the stray field of the ferromagnet is close to the excitation gap of the ferromagnet. The fine tuning can be then achieved by applying locally a small external magnetic field from a coil. The on resonance requirement offers an elegant way to switch on/off the coupling between the qubits. The idea is to tune the qubit splitting close to resonance to switch on the mediated interaction and to tune it off resonance to switch off the mediated interaction. 25 For the sake of completeness, in the Appendix we present a detailed discussion of the effective coupling mediated by the dog-bone when the qubits are exchange coupled to the ferromagnet which requires a tunnel coupling between spin qubit and ferromagnet.

B. Implementation of two-qubit gates
Two qubits interacting via the ferromagnet evolve according to the Hamiltonian H eff , see Eq. (7). The Hamiltonian is therefore the sum of Zeeman terms and qubitqubit interaction. These terms, by and large, do not commute, making it difficult to use the evolution to implement standard entangling gates. Nevertheless, if we assume that ∆ 1 = ∆ 2 , H σ acts only in the subspace spanned by {|↑, ↑ , |↓, ↓ } and the Zeeman splitting of the qubits is much larger than the effective qubit-qubit coupling, we can neglect the effect of H eff in this part of the subspace and approximate it by its projection in the space spanned by the vectors {|↑, ↓ , |↓, ↑ } where α = −8Re(e 1 e * 2 ) and β = −4Re(d 1 e * 2 + d 2 e * 1 ). Within this approximation, the coupling in H eff and the Zeeman terms now commute. From here we readily see that the stray field components, a i , b i , as well as the coefficient c i do not determine the operation time of the two qubit gates-the operation time depends only on d i and e i . To proceed we perform a rotation on the second qubit around the z-axis by an angle tan θ = β/α and arrive at the Hamiltonian We consider the implementation of the iSWAP gate x 2 +σ y 1σ y 2 )π/4 , which can be used to implement the CNOT gate. 26 The Hamiltonian H can be transformed to the desired form by changing the sign of σ x 1σ x 2 term. This is achieved with the following sequence 27 where t = π/(4 α 2 + β 2 ). When iSWAP is available, the CNOT gate can be constructed in the standard way 28 Since H eff is an approximation of H eff , the above sequence will yield approximate CNOT, U CNOT , when used with the full the Hamiltonian. The success of the sequences therefore depends on the fidelity of the gates, F (U CNOT ). Ideally this would be defined using a minimization over all possible states of two qubits. However, to characterize the fidelity of an imperfect CNOT it is sufficient to consider the following four logical states of two qubits: 19 |+, ↑ , |+, ↓ , |−, ↑ , and |−, ↓ . These are product states which, when acted upon by a perfect CNOT, become the four maximally entangled Bell states |Φ + , |Ψ + , |Φ − , and |Ψ − , respectively. As such, the fidelity of an imperfect CNOT may be defined, The choice of basis used here ensures that F (U CNOT ) gives a good characterization of the properties of U CNOT in comparison to a perfect CNOT, especially for the required task of generating entanglement. For realistic parameters, with the Zeeman terms two order of magnitude stronger than the qubit-qubit coupling, the above sequence yields fidelity for the CNOT gate of 99.976%.
To compare these values to the thresholds found in schemes for quantum computation, we must first note that imperfect CNOTs in these cases are usually modeled by the perfect implementation of the gate followed by depolarizing noise at a certain probability. It is known that such noisy CNOTs can be used for quantum computation in the surface code if the depolarizing probability is less than 1.1%. 29 This corresponds to a fidelity, according to the definition above, of 99.17%. The fidelities that may be achieved in the schemes proposed here are well above this value and hence, though they do not correspond to the same noise model, we can expect these gates to be equally suitable for fault-tolerant quantum computation.

III. DECOHERENCE
In this section we study the dynamics of a single qubit coupled to the ferromagnet. In particular we want to answer the question whether the effective coupling derived in the previous section is coherent, i.e., whether the decoherence time solely due to the dipolar coupling to the ferromagnet is larger than the qubit operation time.
A ferromagnet has two types of fluctuationslongitudinal and transverse ones. The longitudinal noise stems from fluctuations of the longitudinal Sz component (we recall that the ferromagnet is polarized along z), while the transverse one is related to fluctuations of S±. In what follows we study these two noise sources separately. The general noise model that describes both types of noise is then given by where the ferromagnet operators X (Y ) with zero expectation value couple longitudinally (transversally) to the qubit. The noise model given in Eq. (13) leads to the following relaxation and decoherence times within Born-Markov approximation 30 where we defined the fluctuation power spectrum of an operator A in the following way, In order to obtain estimates for the decoherence times we need a specific model for the ferromagnet Hamiltonian, herein taken to be a gapped Heisenberg model H F = −J r,r S r · S r + ∆ F r S z r , J being the exchange coupling and ∆ F the excitation gap induced by some magnetic anisotropy.

A. Longitudinal noise
The power spectrum of longitudinal fluctuations is given by the following expression (see Appendix) where D = 2JS. We readily observe that the power spectrum is sub-ohmic, i.e., it diverges at low frequencies S 3D (ω) ∝ 1/ √ ω-this is a direct consequence of the fact that longitudinal fluctuations are gapless. Due to this divergence, the perturbation theory (Born approximation) cannot be used when there is longitudinal coupling to the longitudinal noise. In order to deal with this singularity, we study transverse (Y ) and longitudinal (X) coupling separately. The transverse coupling can be treated perturbatively, while for the longitudinal coupling we solve the problem exactly.

Transverse coupling to longitudinal noise
The part of the Hamiltonian that describes transverse coupling to the longitudinal noise reads Using Eq. (15) and the inequality we obtain the relaxation time The above expression readily shows that relaxation time can be tailored arbitrarily small by choosing the ratio T /∆ F sufficiently small.

Longitudinal coupling to longitudinal noise
Here we consider only longitudinal coupling to longitudinal noise thus the Hamiltonian reads with V = dra r Sz r . To simplify the problem further, 31 we substitute Sz r → Sx r since the latter is linear in magnon operators while the former is quadratic. When the final formula for the decoherence time is obtained we substitute back the power spectrum of Sz r instead of Sx r .
In order to study decoherence we have to calculate the following quantity 31 with (T ) T the (anti-) time ordering operator. The average in the above expression can be evaluated using a cluster expansion 32 and since the perturbation V is linear in the bosonic operators, only the second order cluster contributes. Therefore, the final exact result for the time-evolution of σ − (t) reads where S(t) = [V (t), V (0)] + . After performing the Fourier transformation we obtain Note that this expression is of exactly the same form as the one for a classical Gaussian noise. 33 Now we substitute back Sx r → Sz r For long times t /T the dynamics is of the form where we have used the inequality S 3D (ω, r) ≤ S 3D (ω, r = 0). Thus, this type of decoherence can be suppressed by choosing the ratio T /∆ F sufficiently small.

B. Transverse noise
The power spectrum of transverse fluctuations of the ferromagnet is gapped and thus vanishes for ω < ∆ F (see Appendix), Since the transverse fluctuations are gapped and the precession frequency of the qubits is below the gap, this noise source does not contribute in the second order (Born approximation) because only virtual magnons can be excited. In this section we choose the quantization axes such that qubit splitting is along the z-axis, while the ferromagnet is polarized along the x-axis (see Fig. 1), this is done solely for simplicity and all the conclusions are also valid for the most general case. The Hamiltonian of the coupled system is of the form Eq. (13) with operators X (Y ) with S ± r = S y r ± iS z r and the definitions where A r , B r , C r are given by Eqs.
(3)-(5). To proceed further we perform the SW transformation on the Hamiltonian given by Eq. (13). We ignore the Lamb and Stark shifts and obtain the effective Hamiltonian with the following notation The model given by Eq. (13) yields the following expressions for the relaxation and decoherence times After a lengthy calculation we obtain the following expressions for T 1 and T 2 (see Appendix for a detailed derivation) with the function f (x, y) defined as follows It is important to note that f (x, y) ∝ e −y , i.e., we obtain, as before for the longitudinal noise, that the effect of transverse fluctuations can be suppressed by choosing the temperature much smaller than the excitation gap of the ferromagnet. As anticipated, Eq. (42) shows that the transverse noise becomes more important as the resonance is approached (∆ ∼ ∆ F ).

IV. ESTIMATES
In this section we give numerical estimates for the coherent coupling mediated by the ferromagnet and the associated decoherence times. These estimates are valid for both silicon-based and NV-center qubits.
Assume that the qubits lie close to the disc axis at a distance h = 25 nm below the disc and that the ferromagnet has in-plane polarization (along x-axis). Assume the thickness of the disk to be 20 nm, its radius to be 50 nm, and a lattice constant of 4Å. In this case the stray field at the plane x = 0 is along x and has a magnitude that can reach values up to 1 T depending on the precise position of the qubit. Similarly, when the ferromagnet is polarized out-of-plane (along the z-axis), then the stray field at position x = y = 0 is along z and can take values up to 1 T . For these cases and when the qubit splitting is brought close to resonance, ∆ F − ∆ ≈ 10 −2 µeV , we obtain operation times on the order of tens of nanoseconds when the qubits are separated by a distance of about 1 µm. The decoherence times T 2 depend strongly on the ratio k B T /∆ F and the additional decoherence source can be made negligible if this ratio is sufficiently small. For a magnon gap ∆ F = 100 µeV (corresponding to a magnetic field of about 1 T ) and a temperature T = 0.1 K, we obtain decoherence times solely due to the coupling to the ferromagnet that are much bigger than the operation times and the typical decoherence times of the qubits.

V. CONCLUSIONS
We propose a scheme to coherently couple two atomistic qubits separated over distances on the order of a micron. We present a sequence for the implementation of the entangling CNOT gate and obtain operation times on the order of a few tens of nanoseconds. We show that there is a regime where all fluctuations of the ferromagnet are under control and the induced decoherence is non-detrimental: this is achieved when the temperature is smaller than the excitation gap of the ferromagnet. The main novel aspect of our proposal is its applicability to the technologically very important silicon qubits and NV-centers to which previous coupling methods do not apply.

VI. ACKNOWLEDGEMENTS
We would like to thank A. Yacoby, A. Morello, and R. Warburton for useful discussions. This work was supported by SNF, NCCR QSIT, and IARPA.

Appendix A: Holstein-Primakoff transformation
For the sake of completeness we derive in this Appendix explicit expressions for the different spin-spin correlators used in this work For this purpose, we make use of a Holstein-Primakoff transformation in the limit n i 2S, with a i satisfying bosonic commutation relations and n i = a † i a i . 34 The creation operators a † i and annihilation operators a i satisfy bosonic commutation relations and the associated particles are called magnons. The corresponding Fourier transforms are straightforwardly defined as a † q = 1 √ N i e −iq·Ri a i . In harmonic approximation, the Heisenberg Hamiltonian H F reads where q = ω q + ∆ F = 4JS[3 − (cos(q x ) + cos(q y ) + cos(q z ))] + ∆ F is the spectrum for a cubic lattice with lattice constant a = 1 and the gap ∆ F is induced by the external magnetic field or anisotropy of the ferromagnet.
Let us now define the Fourier transforms in the harmonic approximation From this it directly follows that with q ≈ Dq 2 + ∆ F in the harmonic approximation. The Fourier transform is then simply given by The corresponding correlator in real space is then simply given by (q := |q|) Let us now perform the following substitution which gives for ω > ∆ F We remark that We note the diverging behavior of the above correlation function for ∆ F = 0 and ω → 0, namely Similarly, it is now easy to calculate the corresponding commutators and anticommutators. Let us define It is then straightforward to show that and therefore Following essentially the same steps as the one performed above, we obtain the 3D real space anticommutator for Let us now finally calculate the transverse susceptibility defined as As before, in the harmonic approximation, one finds In the frequency domain, we then have and thus in the small q expansion In real space, for the three-dimensional case, we obtain Making use of the Plemelj formula we obtain for ω > ∆ F It is worth pointing out that the imaginary part of the susceptibility vanishes, and therefore the susceptibility is purely real and takes the form of a Yukawa potential where l F = D ∆ F −ω . Note also that the imaginary part of the transverse susceptibility satisfies the well-know fluctuation-dissipation theorem In three dimensions the susceptibility decay as 1/r, where r is measured in lattice constants. For distances of order of 1µm this leads to four orders of magnitude reduction.
For quasi one-dimensional ferromagnets such a reduction is absent and the transverse susceptibility reads where l F is defined as above and the imaginary part vanishes as above, i.e., Similarly for ω > ∆ F we have and The longitudinal susceptibility reads Applying Wick's theorem and performing a Fourier transform, we obtain the susceptibility in frequency domain where n k is the magnon occupation number given by the Bose-Einstein distribution where k is again the magnon spectrum ( k = ω k +∆ F ≈ Dk 2 + ∆ F for small k). Note that the longitudinal susceptibility is proportional to 1/S, due to the fact that Since we are interested in the decoherence processes caused by the longitudinal fluctuations, we calculate the imaginary part of χ (ω, q) which is related to the fluctuations via the fluctuation-dissipation theorem. Performing a small q expansion and assuming without loss of generality ω > 0, we obtain for the imaginary part Next, since we are interested in the regime where ω T (and thus βω 1), we have n k n k+q . Further-more, we approximate the distribution function n k = e −β(∆ F +ω k ) 1−e −β∆ F +βω k (this is valid when βω k 1) and arrive at the following expression where Ei(z) is the exponential integral function. We also need the the real space representation obtained after inverse Fourier transformation, In order to perform the above integral we note that the imaginary part of the longitudinal susceptibility, given by Eq. (C5), is peaked around q = ω/D with the width of the peak (1/ √ βD) much smaller than its position in the regime we are working in (ω T ). For r = 0, the integration over q can be then performed approximately and yields the following expression where Erfc(z) denotes the complementary error function. It is readily observed from the above expression that the longitudinal fluctuations are exponentially suppressed by the gap. Assuming that ∆ F T , we obtain the following simplified expression We observe that, since J(ω) = χ (ω, r) , the longitudinal noise of the ferromagnet is-as the transverse one-subohmic. 30 It is interesting to obtain the behavior of the longitudinal susceptibility in the opposite limit, when βω 1. In this limit, the difference of the two Boltzmann factors in Eq. (C4) can be expanded to the lowest order in the small quantity βω, In order to calculate the Fourier transform to real space, we note that for βω 1 the denominator of the above expression depends only weakly on ω, thus we ignore this dependence and obtain the Fourier transform for r = 0 The above formula shows that the longitudinal noise of a ferromagnet at high temperatures (βω 1) behaves as ohmic rather than sub-ohmic bath.
Next we calculate the longitudinal fluctuations for the case of a quasi-one-dimensional ferromagnet (∆ F T ) and obtain where γ is a numerical factor of order unity. Note that S (ω, r) is defined through the fluctuation dissipation theorem as S (ω, r) = coth(βω/2)χ (ω, r) .
(C12) D ∆ F −ω and D = 2JS. In what follows, we assume that the external gap is always larger than the qubit splitting, ∆ < ∆ F , as this ensures that the transverse noise is not contributing to decoherence in second order since transverse noise is related to the vanishing imaginary part of the transverse susceptibility, χ ⊥ (ω) = 0 (ω < ∆ F ). The spatial dependence of the effective two spin coupling given by Eq. (D4) is of Yukawa type due to presence of the external gap. If we assume a realistic tunnel coupling to the ferromagnet of 100µeV, 36,37 the Curie temperature of 550K [as for example for yttrium iron garnet (YIG)] and a gap of ∆ F = 100µeV, and the qubit splitting close to the resonance ∆ F − ∆ = 3 × 10 −3 µeV (corresponding to a magnetic field of about B = 60µT) we obtain for the qubit-qubit coupling strength a value on the order of 4 × 10 −11 eV for a lattice constant of about 4Å. This coupling strength gives rise to the operation times of 5µs-significantly below the relaxation and decoherence times of the spin qubit, T 1 = 1s 38 and T 2 > 200µs 39 respectively. Furthermore, the error threshold-defined as the ratio between the two-qubit gate operation time to the decoherence time-we obtain with such an operation time is about 10 −2 , which is good enough for implementing the surface code error correction. 40 Here we used T 2 instead of T * 2 since spin-echo can be performed together with two-qubit gates. 41 Alternatively, the decoherence time of GaAs qubits can be increased without spin-echo by narrowing the state of the nuclear spins. 42,43 The dimensionality of the ferromagnet plays an important role-if we assume 10nm width of the trench where the ferromagnet is placed, then, for energies below 0.1meV, the ferromagnet behaves as quasi onedimensional (1D). In this case we obtain wherefrom it is evident that at distances r l F the susceptibility of a quasi-1D ferromagnet is practically constant in contrast to the 3D case, where a 1/r decay is obtained, see Eq. (D4). Additionally, we require l F D/(AS) = 2J/A for the perturbation theory to be valid. Thus, for the same parameters as above, but without the need to tune very close to the resonance (we set herein ∆ F − ∆ = 0.5µeV, corresponding to about B = 10mT) a coupling strength of 10 −8 eV is obtained.
For 1D case there is yet another rather promising possibility-to use magnetic semiconductors. 44 These materials are characterized by a particularly low Curie temperature of 30K or below, 44 and the distance between the ions that are magnetically ordered via RKKY interaction is about 10 − 100nm. Such a large lattice constant is very beneficial for the long range coupling-if we take the lattice constant to be 10nm, the coupling to the ferromagnet A = 15µeV and the qubit splitting close to resonance (∆ F − ∆ = 0.5µeV, corresponding to about B = 10mT), the qubit-qubit coupling becomes of the order of 1µeV. Such a coupling strength in turn leads to an error threshold on the order of 10 −8 . Therefore, even the standard error correction protocol can be used in this case.
c. Derivation of the effective Hamiltonian (exchange coupling) Here we give a detailed derivation of the qubit-qubit effective Hamiltonian. As stated above, the total Hamiltonian of the system reads (D6) where we identify the main part as H 0 = H F + H σ and the small perturbation as the exchange coupling V = A i σ i · S ri . The Hamiltonian of the ferromagnet reads H F = −J r,r S r · S r , while the Hamiltonian for the two distant qubits is H σ = ∆ 2 i=1,2 σ x i . The second order effective Hamiltonian 24 is given by H where V (t) = e iH0t V e −iH0t . We have Recalling that the zz susceptibility can be neglected and that only the transverse susceptibility contributes, we obtain the following result from Eq. (D7), U = lim Finally, by rewriting cos(∆t) = e i∆t +e −i∆t with the following notation The model given by Eq. (E10) yields the following expressions for the relaxation and decoherence times where, again, S A (ω) = dte −iωt {A † (t), A(0)}.
In order to obtain the estimates for relaxation and decoherence time, we consider the ferromagnet to be in shape of infinite plane. Furthermore, we are not aiming at performing an exact evaluation of the integrals in Eqs. (E20)-(E21), but rather at finding the lower bound for the relaxation and decoherence times. To this end we note that |C +− (ω, r − r )| ≤ |C +− (ω, r = 0)| and arrive at the following inequalities where we used notation B = drB r . Finally we arrive at the estimates for the relaxation and decoherence times with the function f (x, y) defined as follows Assuming the same parameters as in the main text, we obtain decoherence times of about 0.5 hours, while