The sizes of commercial transistors are of nanometer order, and there have already been many proposals of spin qubits using conventional complementary metal–oxide–semiconductor transistors. However, most of the previously proposed spin qubits require many wires to control a small number of qubits. This causes a significant “jungle of wires” problem when the qubits are integrated into a chip. Herein, to reduce the complicated wiring, we theoretically consider spin qubits embedded into fin field-effect transistor (FinFET) devices such that the spin qubits share the common gate electrode of the FinFET. The interactions between qubits occur via the Ruderman–Kittel–Kasuya–Yosida interaction via the channel of the FinFET. The possibility of a quantum annealing machine is discussed in addition to the quantum computers of the current proposals.

## I. INTRODUCTION

Scalability and affinity with conventional computers are the most important features of semiconductor spin qubits^{1} 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 mechanism^{13} 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.

## II. RESULTS

### A. Implementing qubits between the fin channels

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 sites^{16}). 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 **S**_{1} and **S**_{2} 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 estimated^{24} as 0.01 meV and 0.2 *μ*eV 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.

d . | $JdRKKY$ . | $\gamma dRKKY$ . | k_{F}
. | F_{d}′(x)
. |
---|---|---|---|---|

1 | $z12EF\pi F1\u2032(kFW)$ | $2z12kBT\pi $ | πn_{e1} | si(2x) |

2 | $z22EF4\pi 3F2\u2032(kFW)$ | $z22kBT8\pi 2$ | $2\pi ne2$ | J_{0}(x)N_{0}(x) + J_{1}(x)N_{1}(x) |

d . | $JdRKKY$ . | $\gamma dRKKY$ . | k_{F}
. | F_{d}′(x)
. |
---|---|---|---|---|

1 | $z12EF\pi F1\u2032(kFW)$ | $2z12kBT\pi $ | πn_{e1} | si(2x) |

2 | $z22EF4\pi 3F2\u2032(kFW)$ | $z22kBT8\pi 2$ | $2\pi ne2$ | J_{0}(x)N_{0}(x) + J_{1}(x)N_{1}(x) |

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*_{B}*B*_{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* = 2*πrB*_{x}/*μ*_{Si} ≈ 10 *μ* A for *μ*_{Si} = 10*μ*_{0} from Ampére’s law (*μ*_{0} = 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* ∼ 2*μ*_{B}*B*_{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 $H=\u2211i<jJijRKKY\sigma \u20d7i\sigma \u20d7j/4+\u2211i[Biz\sigma iz+\Delta i(t)\sigma ix]$ [$\sigma i\alpha $ (*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 *μ*eV ∼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 × 10^{8} A cm^{−2} NiSi nanowires,^{33} the wire can generate a magnetic field of 470.4 mT (∼54.5 *μ*eV ∼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.

### B. Detailed analysis of the common-gate spin qubits

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 10^{15} and 10^{19} cm^{−3} by controlling the gate voltage within 0.3 ≲ *V*_{G} ≲ 1.2 V. Here, we consider the carrier density from the 10^{15} to 10^{20} 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 *μ*eV) ≲ *E*_{F1} ≲ 0.405 eV (0.162 eV) and 0.196 meV (47.8 *μ*eV) ≲ *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 $\u03f5n=\u2211l=x,y,z\pi 2\u210f2(nl+1)2/(2m*LQ2)$ (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 ϵ_{0} ≈ 3.76 meV for electrons and 1.50 meV for holes. The corresponding energy of the first excited state ϵ_{1} 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 ϵ_{0} 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)

### C. Measurement process

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 $|S\u3009=[|\u2191\u2193\u3009\u2212|\u2193\u2191\u3009]/2$ (the triplet states are not considered because of their higher energy levels^{35}). 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 formula^{40} 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 *k*_{1} = 1, *k*_{2} = *πn*_{e2}*W*^{2}, $\Delta =(2EkF\u2212ESL\u2212ESR\u2212s11\u2212s55)/2$, and *δ* = *E*_{SL} − *E*_{SR}. *n*_{e2} is the number of the carriers per nm^{2}, and $sij\u2261\u222b|Vtun(ki)|2/(Eki\u2212Ekj)$ is the self-energy. Γ_{i} ≈ 2*π*|*V*_{tun}(*k*_{i})|^{2}*ρ*_{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 $g=4/[(\Delta \u22122s33)2+4\Gamma 32]2$. 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 × 10^{2}/*h* [*R*_{K} ≈ 25.8 kΩ (von Klitzing constant)], which corresponds to the conductance of mS, because 2 × 10^{2}/*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 $gyy(1)\u22600$ 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 *i*th 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 *i*th 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 *μ*s.^{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} A^{2} 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} = 2*qI* = 2*qg*_{yy}*V*_{D} ∼ 6.21 × 10^{−24} *R*_{K}*g*_{yy} for *V*_{D} = 0.5 V. The conductance fluctuation originating from this shot noise is then given by $\Delta gyy=Sq\Delta f/VD=2qgyy\Delta f/VD$. The condition *g*_{yy} > Δ*g* leads to *g*_{yy} > 2*q*Δ*f*/*V*_{D}. Figure 4(c) shows the comparison of Δ*g*_{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}.

### D. RKKY interaction and coherence time

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 $Jsd\u2248Vtun2U/(U\u2212Em)/Em,$ where *E*_{m} = *E*_{F} − ϵ_{0}.^{50} Thus, we can change *J*_{sd} by controlling *E*_{F} through *V*_{G}. It is convenient to use $zd\u2261\Gamma U(U\u2212Em)Em$ 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})*U*^{2}/4. As *E*_{m} decreases (*E*_{F} is close to ϵ_{0}), *J*_{sd} increases, and we take *E*_{m} = 2*V*_{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 $\gamma dRKKY$ (*d* = 1, 2) are estimated using the formulas of Ref. 24. They are given by $JdRKKY=\alpha d\eta dEFFd\u2032(kFW)$ and $\gamma dRKKY=4\alpha dkBT$, where $\alpha 1=m*2Jsd2/(2\pi \u210f4kF2)$, $\alpha 2=m*2Jsd2/32\pi 2\u210f4$, *η*_{1} = 2, *η*_{2} = 8/*π*, and *F*_{d}′(*x*) is a Bessel function (Table I). The coherence time is given by $\tau coh=\u210f/\gamma dRKKY$. Although $\gamma dRKKY$ originally includes Bessel functions, we use the constant part of $\gamma dRKKY$ to estimate the shortest coherence time^{24} (see also Appendix C). Using the *z*_{d} defined in Eq. (5), we obtain *J*_{sd} = *ℏk*_{F}*z*_{1}/*m*^{*} for 1D and *J*_{sd} = *z*_{2}*ℏ*^{2}/*m*^{*} for 2D, and

where *ξ*_{1} = 1 and *ξ*_{2} = 1/(4*π*^{2}) (Table I). The Kondo temperature *T*^{K} estimated by $TK\u2248\Gamma U/2\u2061exp(\pi \u03f50(\u03f50+U)/[\Gamma U])$^{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 *L*s, 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 *μ*eV (∼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*_{F}*W*, 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*^{RKKY}*τ*^{op} = *ℏπ*/2. The number of possible operations is estimated using the number of possible operations during the coherence time, given by

Because $EF=\u210f2kF2/(2m*)$ 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 $JdRKKY/\gamma dRKKY$ 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 10^{2} 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.

### E. Crosstalk

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.

## III. THE VARIATION IN THE SIZE OF QDs

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 $\u03f5n=\u2211l=x,y,z\pi 2\u210f2(nl+1)2/(2m*LQ2)$, where *n*_{x}, *n*_{y}, *n*_{z} = 0, 1, 2, …, the effect of the variation *L*_{Q} → *L*_{Q} + Δ*L* induces the variation in the energy levels, given by

for *L*_{Q} = 10 nm [we use $\u210f2/(2m)=a02Ry$ and *m*^{*}/*m*_{0} = 0.5]. For example, when Δ*L*/*L*_{Q} = 0.1 that corresponds to 1 nm variations, it is possible that the ground state varies around Δϵ_{0} ∼ 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 Δϵ_{0} 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 QD_{1} is lowest, and that of the QD_{3} 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 QD_{1} first. Next, the spin-dependent currents of QD_{2} and the QD_{3} 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.

## IV. DISCUSSION

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 method^{57} 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 × 10^{8} A cm^{−2}^{33} and a resistivity of 10 *μ*Ω cm, the power consumption of a wire with an area of 28 × 56 nm^{2} 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 × 10^{6} 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 × 10^{6} qubits in a single chip.

The present setup does not include the single-shot readout mechanism^{13} 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.

## SUPPLEMENTARY MATERIAL

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

## ACKNOWLEDGMENTS

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).

## DATA AVAILABILITY

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

### APPENDIX A: EQUATIONS FOR ESTIMATING PARAMETERS

Physical parameters are calculated based on basic equations as follows. The Fermi energy $EF=\u210f2kF2/(2m*)$ for 1D and 2D is given by

where *Ry* = 13.606 eV (Rydberg constant), *a*_{0} = 0.0529 nm (Bohr radius), and *m*_{0} = 9.109 × 10^{−31} kg is the electron mass. The Si effective mass *m*^{*} is given by *m*^{*}/*m*_{0} = 0.2 for the electrons and *m*^{*}/*m*_{0} = 0.5 for the holes. For a density of 10^{15} and 10^{18} cm^{−3}, we have *n*_{e1} = 0.01 and *n*_{e1} = 0.1 nm^{−1}, respectively.

The charging energy is estimated by *U* ≈ *e*^{2}/(2*C*), where both sides of the capacitance of the QD to the two channels are considered. With $C=2\u03f5siwd/LQD2$, 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* = 1*V* in Ref. 11, which corresponds to 80 meV.

### APPENDIX B: LINEAR RESPONSE THEORY

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 formula^{40} given by

The current operator $Jyi$ of the *i*th 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 *k*_{1} = 1 and *k*_{2} = *πn*_{e2}*W*^{2},

where

The detailed derivation is given in the supplementary material.

### APPENDIX C: IDLING MODE

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.

### APPENDIX D: NOISE

The shot noise is given by $Sq=\u3008\Delta Iq2\u3009/\Delta f=2qI=2qgyyVD$. For Δ*g*_{q} = Δ*I*_{q}/*V*_{D} and *g*′ = *g*_{yy}*R*_{K}, we have

where *R*_{K} = *h*/*e*^{2} = 25.812 kΩ (von Klitzing constant). The thermal noise is given by $ST=\u3008\Delta IT2\u3009/\Delta f=4kTgyy$. For Δ*g*_{T} = Δ*I*_{T}/*V*_{D}, we have

When Δ*f* is in the order of 10^{12} s^{−1}, *T* = 100 mK, and *V*_{D} = 1 V, we have $\Delta gq\u2032=0.0909g\u2032\Delta f/VD$ and $\Delta gT\u2032=3.78\xd710\u22124g\u2032\Delta f/VD2$. Thus, we mainly consider the effect of the shot noise.

### APPENDIX E: COHERENCE TIME

In Ref. 24, the coherence time is estimated by $\tau coh=\u210f/\gamma dRKKY$ using the definition of $\gamma dRKKY$ in Table II. The second terms in *G*_{1}′(*x*) and *G*_{2}′(*x*) suppress the relaxation between the singlet–triplet transitions and extend the coherence time. To estimate the decoherence strictly, we take *G*_{1}′(*x*) = *G*_{2}′(*x*) = 1, similar to that mentioned in Ref. 24.

d . | $\gamma dRKKY$ . | G_{d}′(x)
. |
---|---|---|

1 | $2z12kBT\pi G1\u2032(kFW)$ | G_{1}′(x) = [1 − cos(2x)]/2 |

2 | $z22kBT8\pi 2G2\u2032(kFW)$ | $G2\u2032(x)=1\u2212J02(x)$ |

d . | $\gamma dRKKY$ . | G_{d}′(x)
. |
---|---|---|

1 | $2z12kBT\pi G1\u2032(kFW)$ | G_{1}′(x) = [1 − cos(2x)]/2 |

2 | $z22kBT8\pi 2G2\u2032(kFW)$ | $G2\u2032(x)=1\u2212J02(x)$ |

### APPENDIX F: ANALYSIS OF CROSSTALK

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 $p\u2261r/r2+L2$. When only the magnetic field of the *n*th qubit is switched on while those of the other qubits are switched off, the corresponding condition *h*_{1} = ⋯ = *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 *np*^{2} ≠ 1 (n = 1, 2, …), which equals $L\u2260n\u22121r$.

### APPENDIX G: FIDELITY

We assume a Gaussian distribution of conductance. The conductance *g*_{yy} is a function of *E*_{SR} and *E*_{SL}, with variation Δ*g*_{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

The spin direction is determined by comparing the conductance with the reference conductance *g*_{↑=↓}. As |*E*_{S↑} − *E*_{S↓}| decreases, the overlap between $Pg\u2191\u2260\u2193$ and $Pg\u2191=\u2193$ increases. Thus, we define the fidelity of the measurement by

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^{2}high density 6-T SRAM cell for mobile SoC applications