Compact spin qubits using the common gate structure of fin field-effect transistors

Scalability and affinity with conventional computers are the most important features of semiconductor spin qubits¹ when building a quantum circuit. Recently, a number of significant developments have been achieved, greatly improving coherence time and fidelity.^2–5 The benefits of semiconductor qubits enable us to use the accumulated knowledge and technologies of the miniaturization of semiconductor devices, the gate lengths of which are already less than 20 nm in commercial use. In this respect, the qubits based on current complementary metal–oxide–semiconductor (CMOS) field-effect transistor structures^6,7 have become more important in handling the trend of miniaturization of transistors. However, it is questionable whether the qubit structures considered so far can be translated smoothly to mass production. First, most of the previous qubit structures require approximately ten electrodes to define, control, and read out a qubit. This is because the direct qubit–qubit interaction requires a small distance between the two qubits, and the measurement structures are separated from the qubit–qubit interaction parts. Although these setups have succeeded in a few qubit systems, if these qubits are to be integrated in a chip, the number of complicated wires will become a significant problem (referred to as the “jungle of wires” problem; the wiring problem was first raised in Ref. 8). Moreover, when qubit structures are far from the commercial base transistor architectures, a huge cost incurred in building the chips is unavoidable. Advanced nano-size transistors require several lithography masks via numerous complicated manufacturing processes.⁹ The high cost can only be made affordable if a large number of chips are expected to be sold in a large market, such as smartphones, which would be far-future for quantum computers because they currently only work at very low temperatures. From this perspective, qubit structures should be as similar to those of conventional transistors as possible.

Herein, we theoretically investigate a compact spin-qubit system embedded in common multi-gate fin field-effect transistor (FinFET) devices,^10,11 with all gates electrically tied together as the common gate. The quantum dots (QDs) as qubits are coupled with their nearest fin conducting channels. We have combined several concepts of spin qubits and embedded them into the conventional FinFET aimed at the smooth extension of the present CMOS technologies into the fabrication of the spin-qubit system. We have also investigated the possibility of quantum annealing application of spin qubits other than quantum computations. The manipulations and measurements of the qubits are carried out by the common gate via the fin channels, in addition to the local magnetic fields across the qubits. The measurement is described as a resonant behavior between the FinFET channel and QDs,¹² and it is shown that the energy difference between the nearest qubits is enhanced by the resonant structure. Note that, because two channels couple with a QD and each channel other than the edge channel is shared with two QDs, the resonance is enhanced, resulting in amplifying the detection of the qubit state. Although the present setup does not include the single-shot readout mechanism¹³ by using the fin channels as the couplers between the qubits as well as the measurement current lines, the number of wires is reduced, and we pave the way for solving the “jungle of wires” problem.

We start with the conventional FinFET structure. FinFET types of transistors are widely used and can be extended to one-dimensional (1D) nanowires with gate lengths of less than 5 nm.¹⁴ FinFET devices are developed to address the problems of orthodox planar CMOS transistors.^10,11 Their ultra-thin bodies of less than 30 nm thickness enables them to solve the planar CMOS problem of leakage current between the source and the drain. The FinFET devices also solve the problem of random doping in the channel. In addition, note that the thickness of the FinFET body (<30 nm) is less than that of the devices in previous spin-qubit experiments.^2–5 

There are two choices of methods to embed the spin qubits into the FinFET device. Lansbergen et al. located single-donor spin qubits in the channel of the FinFET device.¹⁵ The other choice is to embed the spin qubits outside the channels. The simplest structure is to array the spin qubits between the fin structure, as depicted in Fig. 1. The common gate structure is used in the same manner as the conventional FinFET. A qubit is defined by an electron or a hole in a QD (QDs can be replaced by trap sites¹⁶). The source and drain electrodes are separated to detect the channel current independently. The fabrication of this structure is within the scope of existing technologies.^17–20 Excess charges are added to the QDs by biasing the two different channels surrounding the QDs. The spin qubits are controlled by two orthogonal magnetic fields, B_x and B_z. The uniform magnetic field B_z is applied to the sample, and the dynamic magnetic field B_x is generated by the wires (LCL in Fig. 1) over the common gate. The gate length L and width W are assumed to be less than 28 nm. Both 2D and 1D channel electrons are assumed. The 2D electron gas is mainly formed on the surface of the channel structures of the conventional FinFET. The 1D channel case corresponds to the nanowire FinFETs.

Because the qubits are spatially separated, the direct exchange interactions between the qubits cannot be used. Instead, the interaction must be mediated by the channel electrons, and the RKKY interaction^21–23 using the channel electrons is the origin of the qubit–qubit interaction. The RKKY interaction between two spin operators S₁ and S₂ is expressed by

where J^RKKY is the coupling constant between the qubits. The tunneling of charges between the channel and the QDs forms the s–d interaction between the spins in the QDs and those in the fin channel, and the RKKY interaction consists of the second-order perturbation of the s–d interaction. Thus, the RKKY interaction is weaker than the direct-exchange interaction. As demonstrated later, the magnitudes of the RKKY interactions are estimated²⁴ as 0.01 meV and 0.2 μeV for 1D and 2D FinFET devices at the tunneling coupling energy Γ = 0.15 meV for FinFETs of 28 and 14 nm gate length, respectively. The corresponding coherence times are 10⁻⁹ ∼ 10⁻⁸ s. The magnitude of the RKKY interaction depends on the Fermi level of the channel (see Table I), and the RKKY interaction is controlled by the applied gate voltage V_G with source voltage $VS(i)$ and drain voltage $VD(i)$ (i ∈ N). The RKKY interaction is formed when the channel electrons transfer the spin state across the channel. This happens when there is no voltage difference between the source and the drain.

The two main operation modes (measurements and qubit manipulation) are implemented by changing V_G, $VS(i)$⁠, and $VD(i)$ [Figs. 1(c) and 1(d)]. The measurement mode and the qubit-manipulation mode are changed by the Fermi energy of the channel (Fig. 2). The qubit states are measured by the channel current of the FinFET devices. The channel current reflects the spin up(↑) and down(↓) states of two QDs when the Fermi energy lies between the two upper energy states (Fig. 3). For example, when the upward magnetic field is applied to the device, the current for the ↓-spin state is larger than that of the ↑-spin state (spin-filter effect). The shot noise and thermal noise are analyzed, and the signal-to-noise ratio is found to be larger than 100 if the applied magnetic field is sufficiently large. In the quantum computing case, the idling mode is optional and discussed in  Appendix C.

The spin states are controlled by the local field B_x and the global field B_z, in which the two qubit states (↑-spin and ↓-spin) are distinguished by the Zeeman-energy splitting gμ_BB_z (hereafter, we take g = 2). B_x is generated by the currents of the LCLs over the gate electrodes [Fig. 1(a)]. Assuming that the distance r between the QD and the LCL is 20 nm, a magnetic field of B_x = 1 mT is obtained by the current I = 2πrB_x/μ_Si ≈ 10 μ A for μ_Si = 10μ₀ from Ampére’s law (μ₀ = 1.26 × 10⁻⁶ kg m⁻² s⁻² A⁻²).

Figures 1(e) and 1(f) show an example of the 2D qubit system and the corresponding qubit network. In Fig. 1(e), only the contacts to the first wiring layer (generally called “M1”) are shown because, in general, the patterning pitch of the contacts is tightest in chips. The higher layers (M1, M2, M3, …) and the corresponding vias (V1, V2, …) are not shown. These contacts are connected to the controlling conventional transistors (FinFETs) through the higher wiring layers, which are finally connected to the IOs of the system. Each FinFET can connect the qubits that belong to different FinFETs. The magnitude of the RKKY interaction decreases with increasing distance between the qubits because of the Bessel function, as shown below. Thus, the diagonal interactions between different FinFETs are weaker than the interactions between neighboring qubits in the same FinFET. Thus, the distances between different FinFETs should be minimized. Note that there are always strict design rules in the process technologies of each factory; hence, the distances cannot be shorter than fixed values. Here, we focus on a single FinFET device, and FinFET networks will be discussed in the near future.

In the case of general quantum computing, the global magnetic field is chosen as the quantized axis. In this case, the spin direction is changed through the magnetic resonance, and the frequency of the local field ω must satisfy ℏω ∼ 2μ_BB_z.²⁵ Van Dijk et al.²⁶ investigated the low temperature CMOS technologies in the range of 2–20 GHz operations. The 2 and 20 GHz local field approximately correspond to 143 mT (∼8.3 μ eV) and 1.43 T (∼83 μ eV), respectively. Detailed CMOS circuits are the future problems.

In the case of the quantum annealing machine (QAM),^27–30 the quantized axis is generated by the LCL, and the uniformly applied external field is chosen as B_x. The Hamiltonian is given by $H=∑i<jJijRKKYσ⃗iσ⃗j/4+∑i[Bizσiz+Δi(t)σix]$ [$σiα$ (α = x, z) are the Pauli matrices]. The various data of the combinatorial problems are inputted into the RKKY interactions $JijRKKY$ and the local magnetic field $Biz$⁠. $JijRKKY$ is adjusted by the magnitudes of the Fermi energies of the fin channel, and $Biz$ is adjusted by the LCLs. In the present case, the Ising term is replaced by the Heisenberg coupling term of Eq. (1), and the tunneling term Δ(t) is produced by the global magnetic field. Δ(t) is gradually switched off when the annealing process is complete. In this case, high-frequency operation of the magnetic field is not necessary, but a large magnetic field should be produced by the LCL. Note that there is a maximum current density to prevent electromigration of thin wires.³¹ Hu et al.³² investigated Cu wires with different cap materials for 7 and 14 nm transistors and demonstrated a reliable current density of 1.5 MA cm⁻². The wire with an area 28 nm (width) × 56 nm (height) allows ∼2.35 × 10⁻⁴ A, which produces a $Bz(i)$ of 23.5 mT (∼2.722 μeV ∼31.5 mK), assuming a distance of 20 nm between the LCL and the qubits. This is small relative to the current possible operating temperature of 100 mK. If we can use 3 × 10⁸ A cm⁻² NiSi nanowires,³³ the wire can generate a magnetic field of 470.4 mT (∼54.5 μeV ∼632.5 mK). Thus, the use of the QAM is feasible if we can prepare reliable wires with the current density greater than 100 MA cm⁻². Thus, if the QDs are embedded between the FinFET devices, we can pave the way for solving the “jungle of wires” problem, although the single-shot readout is a future challenge toward the quantum error-correction. In the following, we describe the detailed analysis of our model.

Hereafter, we describe the theoretical detail of the common gate spin qubits. In the FinFET device,^10,11 the carrier density of the fin channel can be changed within a range between 10¹⁵ and 10¹⁹ cm⁻³ by controlling the gate voltage within 0.3 ≲ V_G ≲ 1.2 V. Here, we consider the carrier density from the 10¹⁵ to 10²⁰ cm⁻³ region. The corresponding Fermi energy E_Fd of the 1D (d = 1) and 2D (d = 2) electron gas (hole gas) is estimated as 0.188 meV (75.2 μeV) ≲ E_F1 ≲ 0.405 eV (0.162 eV) and 0.196 meV (47.8 μeV) ≲ E_F2 ≲ 0.484 eV (0.103 eV), respectively (see  Appendix A). The advantage of using the FinFET channel is that the adjustment of the gate bias V_G enables us to control the Fermi energy of the channel, which leads to control of the measurement process and the qubit–qubit interaction. We assume a Coulomb-blockade region of QDs where the charging energy is estimated as U ≈ 46.4 meV for L = W = 10 nm, assuming a cubic QD of size L_QD = L/2 (see  Appendix A). The discrete energy levels of the cubic QD ϵ_n are simply estimated by $ϵn=∑l=x,y,zπ2ℏ2(nl+1)2/(2m*LQ2)$ (where m^* is an effective mass and n = {n_x, n_y, n_z} is an integer set where n_l = 0, 1, …), and we obtain ϵ₀ ≈ 3.76 meV for electrons and 1.50 meV for holes. The corresponding energy of the first excited state ϵ₁ is given as ∼0.675 eV (0.270 eV), and we can consider single energy levels of the QDs (assuming that there is no offset to ϵ₀ in the QDs). Hereafter, we consider the case of electrons. The two energy levels of the qubit state are defined by the ↑-spin state and the ↓-spin state under an external magnetic field B_z in the resonant tunneling region,³⁴ such as (Fig. 2)

The channel current reflects the QD states when the Fermi energy of the channel is close to the energy levels of the QDs, as shown in Fig. 2(b). The positions of the upper energy levels are determined such that the upper and lower energy levels form the singlet states $|S〉=[|↑↓〉−|↓↑〉]/2$ (the triplet states are not considered because of their higher energy levels³⁵). The singlet energy state E_S↓ for the ↓-state qubit is lower than that of the ↑-state qubit, given by E_S↓ = E_S↑ − Δ_z. Thus, as shown in Fig. 2(b), if we set the Fermi energy between E_S↑ and E_S↓, the ↑-spin electrons can tunnel between the QDs and the channel, but the ↓-spin electron tunneling is blocked (Fig. 3), which is a spin-filter effect similar to that in Ref. 12. The RKKY interaction is ineffective in this measurement mode because the channel electrons that mediate the interaction between flow of two QDs from the source to the drain, and the RKKY interaction only works when both the ↑-spin and ↓-spin states are below the Fermi level. Determination of the ↑-spin or ↓-spin state is performed by comparing the corresponding channel current with that of the reference channel current in which both neighboring QDs have the same spin direction.

Here, we analyze the conductance of the multi-channel FinFET device. As a typical example, we consider the two QDs surrounded by three channels, as shown in Fig. 2(a). The basic setup is similar to that of the two-channel Kondo problem, except that we have to consider three current lines. As Newns and Read^36,37 demonstrated, the standard approach to this problem is to apply the mean-field slave-boson approximation, in which the number of electrons in the localized state is less than 1 and the spin-flop process is included in the tunneling between the localized state and the conducting channel. When B_z is applied (Fig. 2), the flip between ↑-spin and ↓-spin in the tunneling process is suppressed,³⁴ and we can investigate this setup in the range of the resonant-level model.³⁸ We assume that the scattering in the conducting channel is mainly caused by localized spins in the QDs. All tunneling processes between the QDs and the fin channel are included. In conventional FinFET circuit simulations, the drift–diffusion model is used as the core model to analyze the current characteristics.^10,11 However, even in a conventional FinFET, more than 50% of the current flows without scattering (ballistic transport).³⁹ Thus, to examine the basic transport properties, we assume the scattering is caused only by the QDs.

We derive the conductance using the Kubo formula⁴⁰ based on the tunneling Hamiltonian (see  Appendix B and the supplementary material). Figure 4(a) shows the conductance g_yy of the summation of the three current lines $gyy(i)$ (i ∈ 1, 3, 5) as a function of the energy levels of the two QDs, where E_SL and E_SR are either E_S↓ or E_S↑. We can observe a double-peak structure around the Fermi energy where E_SL is close to E_SR but E_SL ≠ E_SR. Because the double-peak structure can be observed even for a single fin channel (⁠$gyy(1)=gyy(5)=0$⁠, not shown), we analyze the peak structure of the single channel $gyy(3)$⁠. The expression for $gyy(3)$ is

where k₁ = 1, k₂ = πn_e2W², $Δ=(2EkF−ESL−ESR−s11−s55)/2$⁠, and δ = E_SL − E_SR. n_e2 is the number of the carriers per nm², and $sij≡∫|Vtun(ki)|2/(Eki−Ekj)$ is the self-energy. Γ_i ≈ 2π|V_tun(k_i)|²ρ_F (ρ_F is the density of state at Fermi energy E_F, and V_tun is the overlap of wave functions between the channel and the QDs in the tunneling Hamiltonian).

The symmetric case δ = 0 gives the conventional resonant tunneling form $g=4/[(Δ−2s33)2+4Γ32]2$⁠. In contrast, for the asymmetric case where δ ≠ 0 and Δ ≪ δ, we have

Thus, the conductance increases as the asymmetry δ of the two QDs decreases for the region very close to the Fermi energy. This is the origin of the sharp double peaks shown in Fig. 4(a). In general, realistic applications will require robustness to variations in device parameters, and the double peaks might not be suitable for practical qubit detection because they are sensitive to changes in {E_SL, E_SR}. Instead, we consider the region where {E_SL, E_SR} are more distant from E_F and the conductance changes gently. Figure 4(b) shows the conductance changes as a function of E_S↑ − E_S↓, the scale of which is converted to B_z. We can see that the conductance is approximately ten times larger than R_K/2 = 2 × 10²/h [R_K ≈ 25.8 kΩ (von Klitzing constant)], which corresponds to the conductance of mS, because 2 × 10²/h ≈ 7.75 × 10⁻⁵ S. Note that the transconductance of the FinFET is in the order of mS.^10,11 Thus, our results show that the FinFET devices can detect the energy difference E_S↑ − E_S↓ of different qubits. Moreover, because there are many fin channels, we can identify the spin direction of each qubit. For example, in the case of three fin channels, by setting $gyy(1)≠0$ and $gyy(3)=gyy(5)=0$⁠, it is possible to identify whether the left qubit is in the ↑-state or ↓-state. In general, for N QDs and N + 1 fin channels, the ith channel current is measured while (i − 1)th and (i + 1)th channel currents are switched off (i < N). The i − 1th and i + 1th channel current can then be measured while the ith channel current is switched off. By comparing the two cases, we can determine the spin directions of the i − 1th and i + 1th qubits. We can perform these processes in parallel to reduce the total measurement time.

We now consider the effect of noise. There are unexpected trap sites in the FinFET devices. If the traps are stable, the local electric field could be adjusted by changing the channel voltage. Regarding the dynamic traps, random telegraph noise (RTN) caused by capture and release of electrons at trap sites occurs in the order of μs.⁴¹ The RTN becomes a major problem when we consider a sequence of quantum algorithms because the voltage shift caused by the RTN is in the order of mV.⁴² Thus, we need to repeat the quantum operations to extract the desirable results. Here, we focus on the shorter time region of two gate operations. In this region, the shot noise and thermal noise are the main obstacles. These types of FinFET device noise are in the order of 10⁻²³ A² Hz⁻¹.¹⁰ The shot noise is higher than the thermal noise (see  Appendix D), and its effect is described using our conductance formula. The shot noise is given by S_q = 2qI = 2qg_yyV_D ∼ 6.21 × 10⁻²⁴ R_Kg_yy for V_D = 0.5 V. The conductance fluctuation originating from this shot noise is then given by $Δgyy=SqΔf/VD=2qgyyΔf/VD$⁠. The condition g_yy > Δg leads to g_yy > 2qΔf/V_D. Figure 4(c) shows the comparison of Δg_yy with the conductance difference. As can be seen from this figure, the effect of the shot noise is small at qubit energy levels close to the Fermi energy. Figure 4(d) estimates the fidelity caused by the shot noise (see  Appendix D). The required B_z decreases as the energy levels approach E_F.

The physics regarding the coupling between localized-state and conduction electrons has a long history as the Kondo effect,⁴³ other than the RKKY interaction.^44,45 The Kondo effect is observed below the Kondo temperature T^K. In the Kondo regime T < T^K, the localized electrons in the QDs and channel electrons are coherently coupled, and the initial qubit state is lost. Therefore, the Kondo effect is undesirable in our system. For the RKKY interaction to be used effectively, the energy scale of the RKKY interaction should be larger than T^K,^46,47 and the target parameter region is given by J^RKKY > T^K. The present setup is similar to the two-channel Kondo case. Experimentally, it appears to be more difficult to observe the two-channel Kondo effect than the single-channel Kondo effect.^48,49 Here, we numerically compare the Kondo effect with the RKKY interaction.

The RKKY interaction is caused by the s–d interaction between the QDs and the channel. The magnitude of the s–d interaction J_sd is derived from the tunneling Hamiltonian such that $Jsd≈Vtun2U/(U−Em)/Em,$ where E_m = E_F − ϵ₀.⁵⁰ Thus, we can change J_sd by controlling E_F through V_G. It is convenient to use $zd≡ΓU(U−Em)Em$ to express J_sd given by

In Eq. (5), there are restrictions of V_tun ≪ E_m ≪ U − V_tun and V_tun ≪ U/2,⁵⁰ which lead to Γ ≪ Γ_max ≡ πρ(E_F)U²/4. As E_m decreases (E_F is close to ϵ₀), J_sd increases, and we take E_m = 2V_tun as an example. One of the advantages of using the transistors is that the carrier density can be changed by the gate electrodes V_G. Hereafter, we describe parameters by using the carrier densities n_ed intended to represent V_G (Table I).

The 1D and 2D RKKY interactions $JdRKKY$ and the decoherence rate $γdRKKY$ (d = 1, 2) are estimated using the formulas of Ref. 24. They are given by $JdRKKY=αdηdEFFd′(kFW)$ and $γdRKKY=4αdkBT$⁠, where $α1=m*2Jsd2/(2πℏ4kF2)$⁠, $α2=m*2Jsd2/32π2ℏ4$⁠, η₁ = 2, η₂ = 8/π, and F_d′(x) is a Bessel function (Table I). The coherence time is given by $τcoh=ℏ/γdRKKY$⁠. Although $γdRKKY$ originally includes Bessel functions, we use the constant part of $γdRKKY$ to estimate the shortest coherence time²⁴ (see also  Appendix C). Using the z_d defined in Eq. (5), we obtain J_sd = ℏk_Fz₁/m^* for 1D and J_sd = z₂ℏ²/m^* for 2D, and

where ξ₁ = 1 and ξ₂ = 1/(4π²) (Table I). The Kondo temperature T^K estimated by $TK≈ΓU/2⁡exp(πϵ0(ϵ0+U)/[ΓU])$⁵¹ is rewritten as

Figures 5(a) and 5(b) show $JdRKKY$ and $TdK$ as a function of Γ. We can see $J1RKKY>T1K$ for all Ls, but the region of $J2RKKY>T2K$ becomes narrower as L increases. The magnitude of $J1RKKY$ is much larger than that of $J2RKKY$⁠, reflecting the corresponding magnitudes of $Jdsd$ in Fig. 5(c). For example, for Γ = 0.2 meV, the magnitude of $J1RKKY$ of L = 28 nm is ∼0.01 meV (∼116 mK), and that of $J2RKKY$ of L = 14 nm is ∼0.2 μeV (∼2.32 mK). Thus, the 1D case is better than the 2D case. It is also seen that larger L enables larger $JdRKKY$ because $JdRKKY$ is proportionate to E_F. However, as shown in Fig. 5(d), larger L induces shorter coherence time. Because $JdRKKY$ is a function of k_FW, the relative magnitude of L(= W) dependence changes depending on L [see Figs. 5(e) and 5(f)].

In the Heisenberg coupling, $SWAP$ is the basic element of the operation, which requires a time τ^op determined by J^RKKYτ^op = ℏπ/2. The number of possible operations is estimated using the number of possible operations during the coherence time, given by

Because $EF=ℏ2kF2/(2m*)$ and k_F are expressed by the density n_ed (Table I), this equation indicates that the ratio is determined by T, m^*, n_ed, and W. Figure 5(e) (1D) and Fig. 5(f) (2D) show the ratios $JdRKKY/γdRKKY$ as functions of the density n_ed and the distance W(=L) between the two qubits. The oscillations in the figure originate from the Bessel functions. As the device size W decreases, the number of possible operations increases. Figure 5(g) shows the time of $SWAP$⁠. In addition, for smaller W(=L), the 1D cases appear preferable because it can be seen that a number in the order of 10² operations are possible. The s–d interaction is affected by the magnetic fields (see the supplementary material); therefore, the RKKY interaction is also affected by B_z. However, we assume B_z < 1 T (∼0.11 meV), which means B_z ≪ E_F (∼200 meV), and we can neglect the effect of B_z in the form of the RKKY interaction.

As shown in Fig. 1, each LCL affects the neighboring qubits (referred to as the crosstalk problem). This problem can be mitigated by changing the direction of the neighboring current lines.⁵² The detailed analysis and condition are presented in  Appendix F.

Because the size of the QDs is less than 28 nm, the variation in the size of the QDs is unavoidable. In this section, we consider the effects of the variations in the QDs on the device operations. When the energy levels of the QDs are given by $ϵn=∑l=x,y,zπ2ℏ2(nl+1)2/(2m*LQ2)$⁠, where n_x, n_y, n_z = 0, 1, 2, …, the effect of the variation L_Q → L_Q + ΔL induces the variation in the energy levels, given by

for L_Q = 10 nm [we use $ℏ2/(2m)=a02Ry$ and m^*/m₀ = 0.5]. For example, when ΔL/L_Q = 0.1 that corresponds to 1 nm variations, it is possible that the ground state varies around Δϵ₀ ∼ 4.65 meV. As shown in  Appendix A, the on-site Coulomb energy is given by U ∼ 46.4 meV, and it is expected that the operations are carried out by adjusting the Fermi energies between the nearest two QDs in the range of U. That is, the variations in the size of the QDs are mitigated by controlling the appropriate Fermi levels of the channels. According to the variations in the sizes of the QDs, the magnitude of J^RKKY also changes depending on the tunneling coupling between the QDs and channel. The time of the two qubit operations should be adjusted depending on the individual couplings. In this process, the appropriate Fermi energies are registered in some digital memory circuits.

The insertion of excess electrons into each QD is carried out by applying voltages between two channels. Because each channel is connected to different electrodes, the transport properties of each QD between the neighboring channels can be detected in the same way as the conventional measurement of single-electron devices.

The channel currents reflect the spin state of neighboring two QDs. The width of the Zeeman splitting (≲1 meV) is smaller than Δϵ₀ for L_Q = 10 nm, and the energy potentials of the electrodes, which enable the spin-filter effect, are different depending on the channels. However, because the source–drain current can be measured independently by the channel, we can detect the spin-filtered channel current by changing the potentials of the source and drains in the measurement phase. Figure 6 shows an example where there are three different energy levels. The energy-level of QD₁ is lowest, and that of the QD₃ is highest. The initial voltages of the source and drain at which current changes are registered in a conventional memory such as static random-access memory (SRAM). The source and drain voltages are raised with respect to ground. In the measurement phase, we equally raise the Fermi level of the channels such as E_F0 = E_F1 = E_F2 = E_F3 from below level A. Then, we can detect the spin-dependent current for QD₁ first. Next, the spin-dependent currents of QD₂ and the QD₃ are measured in order. As seen from Eq. (B6), the enhancement of the resonant tunneling comes from the energy terms to the fourth power and is expected to appear despite the existence of the variations. The detail analysis of the robustness to the variation is a future problem.

Regarding the corner effect of the FinFETs,¹⁵ because we are targeting the 2D electron gas state, which is realized under a relatively larger gate voltage region, the effect of the localized state at the corner is considered to be low. The detail analysis requires technology computer-aided design (TCAD) simulations, which are beyond the scope of this paper and a future problem.

If there is a finite difference between g-factors, the swap gate is disturbed.⁵³ It is desirable that the difference in the g-factors is small in our scheme. The local magnetic field is controlled by the local electric current, as in the study by Li et al.,⁵² and the difference in the g-factor can be adjusted by changing the magnitude of the current. The information of the different g-factors will be also registered in the SRAM, which enlarges the overhead of the system. On the other hand, it is shown that different values of g help the qubit operations.^54–56 Whether g-factors should be uniform or not depends on the future experiments.

In this paper, we have discussed the conductance of the FinFET devices. However, the current–voltage characteristics of the wide ranges of V_D and V_G are required to design a large circuit. The nonlinear current–voltage characteristic is also the origin of the amplifying mechanism of the transistors. This is a future problem.

In Secs. II and III, the quantum computations were described such that the qubit–qubit operations are carried out by changing the magnitudes of the RKKY interactions. Instead, the always-on method⁵⁷ might be suitable for our system because in this method, J^RKKY is constant and the Zeeman energies Δ_z are adjusted with pulses. This method also requires high-frequency control of B_x. Thus, to realize general quantum operations, higher-frequency circuits are required.

As mentioned above, the quantum annealer is also a candidate device because the high-frequency switching on/off of the local fields is not always necessary. The changing values of RKKY interactions depending on gate bias are also suitable for the QAM because the interaction between qubits corresponds to the input data of the various combinatorial problems. It is noted that the present interaction between the qubits has a Heisenberg form, whereas the conventional QAMs exhibit Ising interactions. The practical application of the Heisenberg type will be studied in the future.^58,59

The dilution refrigerator restricts the power consumption of the chip to the mW order at most. Assuming that NiSi nanowires have a current density of 3 × 10⁸ A cm⁻²³³ and a resistivity of 10 μΩ cm, the power consumption of a wire with an area of 28 × 56 nm² and a length of 300 nm is given by 1.72 × 10⁻¹⁰ W (the thinnest wire is usually assigned only at the lowest layer, referred to as the “M1” layer). We can implement ∼5.8 × 10⁶ wires in the chip. If the thinnest wires are used as connections between the qubits, the length of the wire is L and we can afford to use 5.8 × 10⁶ qubits in a single chip.

The present setup does not include the single-shot readout mechanism¹³ because the readout is carried out by using the average current. Thus, the feedback for the quantum error correction is not included. The single-shot readout is a future issue.

See the supplementary material for the complete derivation process of the equations.

We acknowledge useful discussions with Takahiro Mori, Shiro Kawabata, Tomosuke Aono, and Hiroshi Fuketa. This work was partly supported by the MEXT Quantum Leap Flagship Program (MEXT Q-LEAP), Japan (Grant No. JPMXS0118069228).

The data that support the findings of this study are available within the article.

Physical parameters are calculated based on basic equations as follows. The Fermi energy $EF=ℏ2kF2/(2m*)$ for 1D and 2D is given by

where Ry = 13.606 eV (Rydberg constant), a₀ = 0.0529 nm (Bohr radius), and m₀ = 9.109 × 10⁻³¹ kg is the electron mass. The Si effective mass m^* is given by m^*/m₀ = 0.2 for the electrons and m^*/m₀ = 0.5 for the holes. For a density of 10¹⁵ and 10¹⁸ cm⁻³, we have n_e1 = 0.01 and n_e1 = 0.1 nm⁻¹, respectively.

The charging energy is estimated by U ≈ e²/(2C), where both sides of the capacitance of the QD to the two channels are considered. With $C=2ϵsiwd/LQD2$⁠, assuming a cube QD of the size L_QD = L/2, and the thickness of the tunneling barrier w_d (ϵ_Si is the dielectric constant of silicon), we have U ∼ 46.4 meV for L = W = 10 nm and w_d = 1 nm devices. Note that the gate capacitance changes depending on the V_G and around 1 aF at V = 1V in Ref. 11, which corresponds to 80 meV.

As a typical example, we calculate two QDs with three current lines. The Hamiltonian of the QDs and the channel is given by the tunneling Hamiltonian,

where the channels are numbered 1, 3, and 5 and the two QDs are numbered 2 and 4. d_is and c_k,s are the annihilation operators of QD i and the conducting electrons in the channel, respectively. The qubit states are detected by the channel currents. The conductance of the channel is calculated using the Kubo formula.⁴⁰ From Ohm’s law, under the electric field E_y, the current density in the y-direction is given by

where the conductance g_yy(ω) is calculated from the Kubo formula⁴⁰ given by

The current operator $Jyi$ of the ith channel is given by

where L is the channel length and the summation of k_i is carried out over the channel. From the current density j_y in Eq. (B2), the conventional conductance is given by G = Vg_yy (V is a volume), where

where k₁ = 1 and k₂ = πn_e2W²,

where

The detailed derivation is given in the supplementary material.

When V_G = 0, the Fermi energy is below the energy level of the QDs, and the excess electrons leave the QDs. Thus, to preserve the qubits, finite V_G is necessary. This means that this system is a volatile memory. Because at present it is difficult to maintain the spin-qubit state for more than an hour, this volatile mechanism is sufficient. When V_G ≠ 0 and V_S = V_D, neighboring qubits exhibit RKKY interactions. Thus, this system shows an always-on interaction qubit system. The independent qubit state requires two extra QDs between them, as shown in Fig. 7.

The shot noise is given by $Sq=〈ΔIq2〉/Δf=2qI=2qgyyVD$⁠. For Δg_q = ΔI_q/V_D and g′ = g_yyR_K, we have

where R_K = h/e² = 25.812 kΩ (von Klitzing constant). The thermal noise is given by $ST=〈ΔIT2〉/Δf=4kTgyy$⁠. For Δg_T = ΔI_T/V_D, we have

When Δf is in the order of 10¹² s⁻¹, T = 100 mK, and V_D = 1 V, we have $Δgq′=0.0909g′Δf/VD$ and $ΔgT′=3.78×10−4g′Δf/VD2$⁠. Thus, we mainly consider the effect of the shot noise.

In Ref. 24, the coherence time is estimated by $τcoh=ℏ/γdRKKY$ using the definition of $γdRKKY$ in Table II. The second terms in G₁′(x) and G₂′(x) suppress the relaxation between the singlet–triplet transitions and extend the coherence time. To estimate the decoherence strictly, we take G₁′(x) = G₂′(x) = 1, similar to that mentioned in Ref. 24.

Suppose that there are N + 1 current lines in parallel. The magnetic fields h_i(i = 0, …, N) estimated by Ampére’s law are given by

where r is the distance between the qubits and the current lines, L is the distance between the current lines, and $p≡r/r2+L2$⁠. When only the magnetic field of the nth qubit is switched on while those of the other qubits are switched off, the corresponding condition h₁ = ⋯ = h_n−1 = h_n+1 = ⋯ = h_N = 0 leads to

Let us consider a case of switching on the n = 3 qubit out of the six qubits (N = 5); we have

The magnetic field to control the third qubit is given by

From this simple analysis, we obtain the condition of the crosstalk problem given by np² ≠ 1 (n = 1, 2, …), which equals $L≠n−1r$⁠.

We assume a Gaussian distribution of conductance. The conductance g_yy is a function of E_SR and E_SL, with variation Δg_q caused by the noise discussed above. Thus, when we consider the probabilistic distribution regarding g_yy, the conductance g_yy is considered to have maximum probability at g = g_yy and a distribution at around g_yy proportionate to