Tetsufumi Tanamoto- One- and two-dimensional N-qubit systems in capacitively coupled quantum dots

8/3/2019 Tetsufumi Tanamoto- One- and two-dimensional N-qubit systems in capacitively coupled quantum dots

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One- and two-dimensional N -qubit systems in capacitively coupled quantum dots

Tetsufumi Tanamoto∗

Corporate R & D Center, Toshiba Corporation, 1, Komukai Toshiba-cho, Saiwai-ku, Kawasaki 212-8582, Japan

(September 5, 2001)

Coulomb blockade effects in capacitively coupled quantumdots can be utilized for constructing an N -qubit system withantiferromagnetic Ising interactions. Starting from the tun-neling Hamiltonian, we theoretically show that the Hamil-tonian for a weakly coupled quantum-dot array is reduced

to that for nuclear magnetic resonance (NMR) spectroscopy.Quantum operations are carried out by applying only electri-cal pulse sequences. Thus various error-correction methodsdeveloped in NMR spectroscopy and NMR quantum comput-ers are applicable without using magnetic fields. A possiblemeasurement scheme in an N -qubit system is quantitativelydiscussed.

03.67.Lx,73.23.-b

Quantum computers have been widely investigatedfrom many perspectives [1–12]. A quantum-dot arrayis a promising candidate for the basic element of a quan-

tum computer from the viewpoint of technological feasi-bility [13,14]. It is noteworthy that controlling of lateraland vertical order in self-organized quantum-dot super-lattices has been realized [15]. Although a spin-basedquantum dot computer has been intensively discussed[5,6], it seems that the quantum dot system using chargedstates is more accessible because the latter will be ableto be constructed of various materials other than III-V group materials such as GaAs. From this perspec-tive, several authors investigated the quantum computerbased on the charged states [10–12]. In Ref. [12], wediscussed the quantum dot computer in the limit of afree-electron approximation. The free-electron approxi-

mation will be valid only when the interdot tunneling isstrong and Coulomb blockade is suppressed, or the size of the quantum dots is as small as that of artificial atoms.Generally speaking, however, given the current state of technology, it seems that it will not be possible to makequantum dots as small as atomic order in the near fu-ture. Moreover, controlled-NOT (CNOT) operation fora quantum computer based on charged states has beendiscussed only in the two neighboring qubits so far; ascheme for constructing more than two qubits on a de-vice remains unclear.

In this paper we advance the analysis of the quantumcomputer based on charged states and show a generalN -qubit scheme. Starting from tunneling Hamiltonian

in the Coulomb blockade regime, we demonstrate thatthe Hamiltonians for one- and two-dimensional arraysof weakly coupled quantum dots are reduced to thosefor standard nuclear magnetic resonance (NMR) spec-troscopy [17]. This enables any quantum computation to

be described by electric pulse sequences. We assume thatinterdot tunneling is not so strong that the capacitancebetween two quantum dots is defined and we can use theHamiltonian which allows the electrostatic energy of thequantum-dot structure to be described in terms of the

capacitances of individual dots and gate voltages. Thistype of quantum-dot system was discussed experimen-tally by Livermore et al [16]. The interaction betweenqubits is derived from the Coulomb interaction betweenthe excess electrons in the quantum dots. We also nu-merically illustrate the measurement process using thefield-effect, which is considered to be a sensitive mea-surement method for electronic states [18]. We set e=1and kB = 1.

The fundamental idea of the quantum-dot qubits basedon charged states is as follows. A qubit is composed of two quantum dots coupled via a thin tunneling barrierand a gate electrode that is attached on a thick insulatingmaterial. This is a bistable well structure [19] where the

electronic quantum state of the coupled dots is controlledby the gate bias. First, we consider the one-dimensionalarrayed qubits(Fig. 1). The quantum dots are, e.g., Sinanocrystals [20–22] or GaAs dots [13,14]. The quantumdots are assumed to be sufficiently small for chargingeffects to be observed. N αi and N βi are the numbersof excess electrons from the neutral states in two quan-tum dots. One excess charge is assumed to be insertedfrom a substrate first and to stay in the two-coupled dots(C Bi > C Ci). When the excess charge exists in the up-per dot and lower dot, we call them |0 = | ↑ state and|1 = |↓ state, respectively. The quantum logic gates areassumed to be operated in the region where the

|0

and

|1 states are near-degenerate, then the system becomesa two-state system [7,8,23]. The qubits are arranged one-dimensionally and are capacitively coupled.

Here we showthe Hamiltonian of the one-dimensionally arrayed cou-pled quantum dots for quantum computation by startingfrom the tunneling Hamiltonian:

H =

N i=1

(ta†i bi+t∗b†i ai+αia†i ai+βi b

†i bi) + H ch, (1)

where ai (bi) describes the annihilation operator whenthe excess electron exists in the upper (lower) dot, and

αi (βi) shows the electronic energy of the upper (lower)dot. There is no restriction on the number of coupledquantum dots, N . H ch is the charging energy that in-cludes the interaction between qubits. Because we con-sider the Coulomb blockade in the weak coupling region,

1



the resistance of the interdot tunneling barrier should belarger than RK(= h/e2 ∼ 25.8kΩ) and the operationaltemperature be less than the charging energies. Thus,the operational speed should be less than the CR con-stant of the capacitance network so that the double wellpotential profile generated by the charging energy is effec-tive (adiabatic region). The criterion of the operation isdiscussed below in detail. As for the interaction betweenqubits, the distribution of the excess charge is consideredto be antiferromagnetic due to the repulsive Coulomb in-teraction. We can show that this interaction betweenqubits is an Ising interaction by minimizing the generalformulation of the charging energy:

H ch=

N i=1

q2Ai

2C Ai+

q2Bi2C Bi

+q2Ci

2C Ci−qAiV gi

+

N −1i=1

q2Di

2C Di+

q2Ei2C Ei

+q2Fi

2C Fi+

q2Gi2C Gi

,

+N i=2

q2Hi

2C Hi−qHiV gi−1

+N −1i=1

q2Ii

2C Ii−qIiV gi+1

, (2)

with constraints −N αi= qAi−qBi+qDi−qDi−1+qEi−qFi−1+qHi−1+qIi and −N βi= qBi−qCi+qGi−qGi−1−qEi−1+qFi

(i = 1, . . ,N ). The last line of Eq.(2) shows the effectsof other gate electrodes on the ith qubit (cross-talk). Byusing Lagrange multiplier constants, the energy of thesystem is obtained as a function of the relative excesscharge ni ≡ N αi −N βi and the total excess charge of thetwo quantum dots N i ≡ N αi + N βi(=1 ):

H ch ∼=N i=1

C bi2Di

ni+

C ciC bi

N i+

1+

C ciC bi

QV

2

+

N i=2

2

DiDi−1(C biC bi−1C ei−1+C ciC ci−1C di−1)nini−1, (3)

with C ai ≡ C Ai+ C Ci + 4C Bi +2(C Di+ C Ei) + C Hi +C Ii , C bi ≡ C Ai+ C Ci+2(C Di+ C Ei) +2(C Di−1+ C Ei−1)+C Hi+ C Ii , C ci ≡ C Ci−C Ai+C Hi+ C Ii , C di ≡ C Di+C Ei ,C ei ≡ C Di − C Ei (C Di = C Gi, C Ei = C Fi) and Di ≡C aiC bi − C 2ci and QV ≡ C AiV gi+C Hi−1V gi−1+C IiV gi+1.It is assumed that the coupling between qubits is smallerthan that within a qubit and we neglect higher orderterms than (C 2di/Di)2( 1). This assumption is validwhen the distance between the quantum dots in differentqubits is larger than that between quantum dots in aqubit. From Eq.(3), we define the characteristic chargingenergy of the system as E C ≡ C b/(2D). We consider thegate voltage region, where the ni=-1 state and ni=1 stateare near-degenerate as in Ref. [7,8,23], and we obtain theHamiltonian of the coupled quantum dot system:

H =

N i=1

[ta†i bi + t∗b†i ai + ΩiI iz]+

N −1i=1

J i,i+1I iz I i+1z , (4)

where |↑i=|ni = 1 and | ↓i= |ni =−1, I iz = (a†i ai−b†i bi)/2, and

Ωi =4C Ci

Di[QV −Qres

V ] , (5)

J i,i+1 =1

2DiDi+1[C biC bi+1C ei+1+C ciC ci+1C di+1]. (6)

QresV includes αi and βi and shows the gate voltage when

the |0 and |1 degenerate (on resonance). We control thetime-dependent quantum states of the qubits in the vicin-ity of on resonant gate bias by applying a gate voltagesuch as V i(τ ) = V res

i+ vi(τ ) where V res

iis the gate bias

of on resonance. In this on resonant region, a transfor-mation of the coordinate:

α+i

α−i

≡ U 0

aibi

, U 0 ≡ 1√

2

1 11 −1

. (7)

is convenient (we neglect the phase of ti for simplicity).Then the Hamiltonian Eq.(4) can be described as

H (τ ) =

N i=1

[2tiI zi−∆i(τ )I xi ]+

N −1i=1

J i,i+1I xiI xi+1 (8)

where

I xi ≡ (α†+iα−i+α†

−iα+i)/2, I yi≡ i(α†+iα−i−α†

−iα+i)/2,

I zi≡(α†+iα+i−α†

−iα−i)/2, (9)

and

∆i(τ ) =4C AiC Ci

Di[vi(τ ) + δvi,cr(τ )]. (10)

δvicr(τ ) ≡ [C Hi−1vi−1(τ )+C Iivi+1(τ )]/C Ai is a cross-talkterm and is an effect of other gate electrodes. Hamilto-nian (8) is an NMR Hamiltonian when we regard ωi = 2tias a Zeeman energy and ∆i(τ ) as a transverse magneticfield, if ω > ∆0

J . Thus, when ∆i(τ ) = ∆0i cos(ωiτ +

δi) and we take a rotating wave approximation by theunitary transformation U rwa(τ ) = exp(−i

N i=1 ωiτI iz),

we have

H rwa=−N i=1

∆i02

[I xi cos δi + I yi sin δi]

+N −1i=1

J i,i+1

2[I xi I xi+1 + I yi I yi+1]. (11)

The pulse process is carried out when the oscillating elec-tric field is applied (∆0 = 0) where the interaction termis considered to be able to be neglected because ∆0

J .

For example, πz-pulse in the (ai, bi) basis corresponds toπx-pulse representation in this (α+i, α−i) basis and is car-ried out when δ = 0 and ∆0τ/2 = π. In general, rotation

Rγ(θ) ≡ e iθI γ (γ = x, y, z) in the (ai, bi) basis is inter-

preted by Ri(θ) = U −1

0 Ri(θ)U 0 in the (α+i, α−i) basis.

2



The evolution process is carried out when the oscillatingvoltage is not applied (∆0 =0). A two-qubit CNOT gate

is described by a pulse sequence of the form U ijCNOT ∝U −10 Rix(π/2)R

jy(π/2)Rjx(π/2)Rij(−π)R

jy (−π/2)U 0

[24,25]. The 2-body interaction Rij(−π) = e−πI iz I i+1z

is obtained by using an average Hamiltonian theory [17].Here we apply the “Carr-Purcell” sequence τ −πy−2τ −πy − τ to the Hamiltonian Eq.(8) in order to average theeffect of Zeeman term to zero. Thus, we can show thatthe Hamiltonian of the coupled quantum dot system canbe reduced to that of the weakly coupled system in NMRand quantum operations can be carried out in a manner

similar to that of operations in NMR quantum comput-ers.

Similarly, we can show that the Hamiltonian of thetwo-dimensionally arrayed coupled dots is reduced tothat of an artificial Ising system controllable by elec-tric fields. Here we consider the case where there areonly nearest-neighbor capacitive couplings (C E and C F are neglected in the above formulation). If we expressthe coupling strength between the sites i ≡ (i, j) andi+x ≡ (i+ 1, j) as J xi and that between the sites i andi+y ≡ (i, j + 1) as J yi (1 ≤ i ≤ N x, 1 ≤ j ≤ N y:N = N xN y), we have a Hamiltonian similar to Eq. (4)where

J xi = 2DiDi+x

(C biC bi+x + C ciC ci+x)C xDi, (12)

J yi =2

DiDi+y

(C biC bi+y + C ciC ci+y)C yDi, (13)

with Di ≡ C aiC bi − C 2ci, C ai ≡ C Ai + C C i + 4C Bi, C bi≡ C Ai+C C i, C ci ≡ C C i−C Ai. C xDi(= C xGi) and C yDi(=C yGi) are capacitances between qubits in x-direction andy-direction, respectively.

Here we summarize the criterion for realizing theabove-mentioned scheme. First of all, the Coulombblockade should be effective at the operational tempera-ture T and we have T

E C . In view of time-dependent

operation, the excess charge is assumed to be affectedby the electronic potential generated by the capacitancenetwork. Therefore, all quantities concerning time evolu-tion should be smaller than the CR constant of the net-work. Here we take CR = C intRint where C int(≡ D/C b)and Rint are capacitance and resistance of the interdottunneling barrier in a qubit, respectively. By includingthe condition for using the pulse sequence as mentionedabove, we have a condition for the operation:

T J ∆0 < t (CR)−1. (14)

We can roughly estimate this criterion by taking typi-cal values, r0 =2.5 nm (radius of a quantum dot), dA=8nm, dB= 1.5 nm, dC = 2.5 nm, and the distance betweenqubits dD is 12 nm (ox =4 (SiO2) and Si=12), reflect-ing several experimental data [20–22]. Using relations,C A,C =2πoxr20/(dA,C +(ox/Si)r0) and C B = 2πoxr20/(dB+2(ox/Si)r0), we obtain E C ∼ 13meV(150K), t ∼

0.4meV and J ∼ 0.1 meV. If Rint is of the order of MΩ,we obtain (C intRint)−

1 ∼ 3.1THz and greater than t ∼100GHz. Thus, the condition (14) is satisfied if the oper-ational temperature should be much less than 1K. If wecould prepare r0 = 0.5nm, dB = 1.2nm and dD = 2nm,we obtain t ∼120 K and J ∼ 90K and quantum cal-culations can be expected to be carried out at aroundliquid nitrogen temperature. The effects of cross-talk areof the order of C Hi/C Ai ∼ dAi/dDi and cannot be ne-glected even when gate electrodes are set closer to thecorresponding quantum dots. However, we can controlthe cross-talk effects by adjusting vi(τ ) to obtain the re-quired ∆i(τ ).

At the near-degeneracy point, the system becomes atwo-state system and the estimation of the decoherence(in order of µsec) discussed in Ref. [12] may be applicable.Until now, we have not known of any corresponding ex-perimental data for the decoherence time [27]. The pointis that various methods developed in NMR spectroscopy,such as the composite pulse method [17], can be utilizedto reduce the imperfections of the pulse and coherencetransfer, that is, errors that are brought about in theoperations. In addition, if the speed of quantum compu-tations can be increased to much more than the shortestdecoherence time, τ c ∼ ω−1

c (∼ 10−14 s), group-theoreticapproaches [28,29] for decreasing the decoherence will be

effective. In this case, we have to reduce the correspond-ing CR of the junction such that Eq.(14) holds and thecycle time of operations should be much less than τ c.Thus, the errors and the effects of decoherence of a cou-pled quantum-dot system will be reduced by developingmany contrivances.

Hereafter we consider the one-dimensionally arrayedqubits for simplicity. Next, we quantitatively illustratethe reading out process based on an FET structure(Fig.1). Measurement is carried out, after quantum cal-culations, by applying a finite bias V D between the sourceand drain [12]. The detection mechanism is such thatthe change of the charge distribution in a qubit induces

a threshold voltage shift ∆V th of the gate voltage abovewhich the channel current flows in the substrate. ∆V this of the order of edq/ox(dq is a distance between thecenters of two quantum dots in a qubit) [21]. The effectof ∆V th differs depending on the position of the qubit,because the width of the depletion region in the sub-strate changes gradually from source to drain. A sim-plified model of a metal-oxide semiconductor field-effect-transistor (MOSFET) including velocity saturation ef-fects (Θ in Eq. (15)) [26] is used for the current thatflows under the ith qubit

I (i)D = Λ

[V gi−V thi](V i−V i−1)−(1/2)ηi(V 2i −V 2i−1)

1+Θ(V i−

V i−1

), (15)

where Λ ≡ Zµ0C 0/L0 (Z is the channel width, µ0 isthe mobility, L0 is the channel length of one qubit, andC 0 is the capacitance of the gate insulator), ηi ≡ 1 + ζ iwhere ζ i is determined by the charge of the surface de-

3



pletion region, and V i is the voltage of ith qubit to be

determined from V N = V D and I (1)D = I (2)D = · · · = I (N )D .The detection should be carried out before the change of the quantum states of the qubits. For this purpose, Λshould be as large as possible [12]. The threshold voltageof ith qubit V thi is given by V thi = V th(ζ i)+∆V thi. Fig-ure 2(a) shows the ratio of the current change |I i-I 0|/I 0as a function of V D in 8 qubits with ζ i = 0, where I 0is the initial current and I i is the current when V gi-V thiof ith qubit changes by 10%. This ratio is largest for aqubit near the drain (i = 8) because it has the narrow-est inversion layer. To show how to distinguish qubits,

we compare the current where only the ith qubit shiftsits threshold voltage with that where only the (i+1)thqubit shifts its threshold voltage. Figure 2(b) shows theresults for (i)i=1, (ii)i=N/2, and (iii)i=N − 1, with thesame voltage shift as in Fig.2(a). The maximum allow-able number of arrayed qubits depends on the sensitivityof the external circuit to the channel current. When thedensity of acceptor in the substrate is of the order of 1017 cm−3, the number of acceptors below one qubit isless than one. If this effect is represented by ζ i as a ran-dom number, the above ratios may increase or decreaseand the overall features are similar to those in Fig.2. Toconstruct a large qubit array, additional dummy qubitsare needed over the source and drain so that separated

qubits on different FETs are connected.In conclusion, we have theoretically shown that the

Hamiltonian for a weakly coupled quantum-dot array inthe Coulomb blockade regime is reduced to that for NMRspectroscopy. The flexible quantum information process-ing developed in the NMR quantum computer and a va-riety of error-correction methods in conventional NMRare applicable to the quantum-dot system that is con-sidered to be the most feasible system in view of thepresent technology. The difficulty of scalability in theNMR computer, namely that the overlap of pulses re-stricts the number of qubits [2], is overcome in the op-erations by individual gate electrodes. The disadvantage

of the short decoherence time is also compensated for byanother advantage, namely that the measurement proce-dure is compatible with classical circuits. It is expectedthat a transmitting loss of signals to classical circuits willdecrease due to this compatibility. Recently, nanofab-rication of two-dimensionally distributed self-aligned Sidoubly-stacked dots has been successfully realized in theform of a non-volatile memory device [22] and the de-tailed analysis of the behavior of electrons, such as anartificial antiferromagnet, is expected to be performed.The N -qubit system composed of arrayed quantum dotsis expected to be developed as a result of further advancesin fabrication technologies.

The author thanks N. Gemma, R. Katoh, K. Ichimura,J. Koga, R. Ohba and M. Ueda for useful discussion.

∗ Email address: [email protected][1] D. P. DiVincenzo, Fortschr. Phys. 48, 771 (2000).[2] C. H. Bennett and D. P. DiVincenzo, Nature 404, 247

(2000).[3] S. Lloyd, Science261, 1569 (1993); ibid 273, 1073 (1996).[4] B. E. Kane, Nature 393, 133 (1998).[5] D. Loss and D. P. DiVincenzo, Phys. Rev. A 57, 120

(1998).[6] D. P. DiVincenzo et al., Nature 408, 339 (2000).[7] D. V. Averin, Solid State Comm. 105, 659 (1998).[8] Y. Makhlin, G. Schon and A. Shnirman, Nature 398, 305

(1999).[9] J. E. Mooij et al ., Science 285, 1036 (1999).

[10] A. Barenco, D. Deutsch, A. Ekert and R. Jozsa, Phys.Rev. Lett. 74, 4083 (1995).

[11] L. A. Openov, Phys. Rev. B 60, 8798 (1999).[12] T. Tanamoto, Phys. Rev. A 61, 022305 (2000).[13] N. B. Zhitenev et al ., Science 285, 715 (1999).[14] T. H. Oosterkamp et al ., Nature 395, 873 (1998).[15] G. Springholz et al ., Phys. Rev. Lett. 84, 4669 (2000).[16] C. Livermore, et al . Science 274, 1332 (1996).[17] R. R. Ernst, G. Bodenhausen and A. Wokaun, Principles

of Nuclear Magnetic Resonance in One and Two Dimen-

sions (Clarendon Press, Oxford, 1987).[18] R. Landauer, J. Phys: Condens. Matter 1 8099 (1989).[19] R. Landauer, Science 272, 1914 (1996).[20] S. Tiwari et al ., Appl. Phys. Lett. 68,1377 (1996).

[21] L. Guo, E. Leobandung and S. Y. Chou, Science 275,649 (1997).

[22] R. Ohba et al., 2000 IEEE International Electron Devices

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Conference on Solid State Devices and Materials (Sendai,2000), p.122.

[23] Y. Nakamura, C. D. Chen and J. S. Tsai, Phys. Rev.Lett. 79, 2328 (1997).

[24] N. Linden, B. Herve, R. J. Carbajo and R. Freeman,Chem. Phys. Lett. 305, 28 (1999).

[25] L. M. K. Vandersypen et al., Appl. Phys. Lett. 76, 646(2000).

[26] B. Hoefflinger, IEEE. Trans. Electron. Devices ED-28,971 (1981).

[27] Experiments by T. Fujisawa, Y. Tokura and Y. Hi-rayama(Phys. Rev. B 63 081304 (2001)) show that thedecoherence time of a quantum dot is long in order of µsec in some situations.

[28] L. Viola, E. Knill and S. Lloyd, Phys. Rev. Lett. 82, 2417(1999).

[29] P. Zanardi and F. Rossi, Phys. Rev. Lett. 81, 4752(1998).

FIG. 1. Schematic of the capacitance network of aone-dimensionally coupled quantum-dot array. The capaci-tances C Ai, C Bi, and C Ci, the gate V gi and the two quantumdots constitute the ith qubit. The excess electron moves be-

tween the two quantum dots in a qubit via the tunnelingbarrier, and electron transfer between qubits is prohibited.

4



FIG. 2. I i represents the current where only the ith qubitshifts its threshold voltage V th(0)+∆V thi. I 0 represents thecurrent where all qubits have the same threshold voltageV th(0). (a) The ratio of current change |I i − I 0|/I 0 as a func-tion of V D in an 8-qubit quantum computer (i= 1, i=4, andi= 8). (b) The ratio of change |I i − I i+1|/I 0 as a function of the number of qubits, in the case i = 1(near source), i = N/2(middle) and i = N − 1(near drain) at V D = 1.5V. The ef-ficiency of detecting the quantum state of qubits is highestwhen the qubits near the drain change their states. In (a)and (b), V g − V th=2V and Θ=0.3 V−1 in Eq.(7) in text. Thethreshold shift is 10% of V g − V th.

5



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Tetsufumi Tanamoto- One- and two-dimensional N-qubit systems in capacitively coupled quantum dots