Quantum gates by coupled quantum dots and measurement procedure in Si MOSFET

a r X i v :q u a n t -p h /9908021v 1 5 A u g 1999Quantum gates by coupled quantum dots and measurement procedure in Si MOSFETTetsufumi TanamotoCorporate Research and Development Center,Toshiba Corporation,Saiwai-ku,Kawasaki 210-8582,Japan(February 1,2008)We investigated the quantum gates of coupled quantum dots,theoretically,when charging effects can be observed.We have shown that the charged states in the qubits can be ob-served by the channel current of the MOSFET structure.I.INTRODUCTIONSince Shor’s factorization program was proposed,many studies have been carried out in order to real-ize the quantum computer [1–4].Nakamura succeeded in the control of the macroscopic quantum state as the solid devices of Josephson junctions [4].Recently,we have proposed the quantum computer in the asymmet-ric coupled dot in Si nanocrystals [5].The advantage of using the coupled dots attached by the gate electrode is that a quantum state can be controlled by the gate volt-age in the solid circuits.Moreover,when the coupled Si quantum dots are embedded in the gate insulator of the MOSFET [6,7],the quantum state in the two coupled dots (qubits)is expected to be detected by the channel current.In the previous paper [5],we investigated the case where the sizes of Si nanocystals were small such that the energy-levels in the quantum dots were discrete and the two-state system was constituted by the lowest energy-levels of the two quantum dots.Here,in this pa-per,we consider the case where the sizes of the quantum dots are larger (of the order of a few tens of nm)and the energy-levels of the individual quantum dots are almost continuous [8].We also assume that the charging en-ergy of each tunneling junction is large enough that the Coulomb blockade effects can be seen.This was realized in the quantum gate of the Josephson junction by Shnir-man [3]and Averin [9].The superconducting state has the advantage from the viewpoint of the decoherence that the qubits are connected by the coherent circuits.On the other hand,we use the normal state of the capacitively coupled quantum dots,because there exist some predic-tions that the decoherence time in the normal quantum dot array in semiconductors is not expected to be so short [10].First,in this paper,we show that the capacitively coupled dots can be reduced to a two-state system of in-teracting two qubits.Next,we show the detecting mech-anism of the quantum state by the channel current based on the conventional MOSFET model.Asymmetry of the coupled dots is not assumed and we set e =1.II.CAPACITIVELY COUPLED QUANTUMDOTS AS TWO QUBITSThe configuration of the quantum dots and capaci-tances are shown in Fig.1.We assume that the coupling between qubits is weaker than that in a qubit(The full formation of the electrostatic energy is shown in the Ap-pendix),and we takeC 1=C 2=√2C 4≪C 5(=C 8),C 6(=C 9),C 7(=C 10).(1)By this assumption,we can expand the charging energy as a function of n a ≡N A −N B and n b ≡N C −N D (N =N A +N B =N C +N D ),by a small C 3,and obtain the electrostatic energy:U (n a ,n b )=E cn a +ηC 5+C 7N −2C 5C 7C 5+C 7N −2C 5C 72(C 25+C 27−√4(C 5C 6+C 6C 7+C 7C 5)2C 3(3)If we consider the gate voltage region where the electro-static energy of n j =0(j =a,b )state crosses that of n j =1state described by Shnirman [3]and Averin [9],the electrostatic energy and the tunneling amplitude,Ωj ,constitute the Hamiltonian of the two-state system asH = j =a,b(ǫj σzj +Ωj σxj )−(η/4)σza σzb (4)where ǫa =E c (1/2+[(C 7−C 5)N −2C 5C 7V a ]/(C 5+C 7))and ǫb =E c (1/2+[(C 7−C 5)N −2C 5C 7V b ]/(C 5+C 7)).σx and σz are Pauli matrices.When the capacitances are approximated as C i =2πǫox r 2/(d i +(ǫox /ǫSi )r )where ǫox =4,ǫSi =12,d i is the size of the capacitance and r is the radius of each quantum dot,then V a ,V b is of the order of tens of meV.The mechanism of the controlled NOT operation is similar to that of the Josephson junction [9].Whether n b =0or n b =1,the level-crossing gate voltage,V a ,shifts(see Eq.(2)and the controlled NOT operation is realized.The dynamical motion of the excess charge in the two-state system can be easily considered by solvingthe time-dependent Schr¨o dinger equation[11],andtheexcesschargeshowstheoscillating behaviordepend-ingon the energy of the two states.The time period of the oscillation,τδ,can be simply approximated as τδ∼¯h /2α(V 2i −V 2i −1),(5)where β0≡Zµ0C 0/L i (Z is the channel width,µ0is themobility,L i is the channel length of i -th qubit where we set L 1=L 2=···=L N ,and C 0is the capacitance of the SiO 2)and α≡1+1C 0where Q B is the charge within the surface depletion region.V G is the gate voltage,andthe threshold voltage,V (i )th ,is given by V (i )th =V th +∆V (i )th where V th ≡V FB +2ϕB +Q B(V G 1+V G 2)(V G 1V G 2V D −αV G 12(2V g −∆V th )∆V th V 2D .(8)This difference can be observed in the nonlinear I D -V Dregion and,in the pure linear region where the terms which include αdisappear,the changed qubits cannot be distinguished.V.CONCLUSIONSWe have investigated the quantum gates of the capaci-tively coupled quantum dot array in the MOSFET struc-ture,and have derived the two-state Hamiltonian by the capacitances of the quantum dots.We also have shown the detecting mechanism of the MOSFET structure by analyzing the channel current based on the conventional MOSFET model.ACKNOWLEDGMENTSThe author is grateful to N.Gemma,S.Fujita,K.Ichimura,and J.Koga for fruitful discussions.APPENDIX A:THE ELECTROSTATIC ENERGYOF THE TWO QUBITSIn this section we show the electrostatic energy of the two qubits in terms of the capacitances of dots and thegate electrodes (Fig.1)[18,19].We assume C 1=C 2=√2C 4,C 8=C 5,C 9=C 6and C 10=C 7.Thetotal electrostatic energy of the two qubits is given byU =i =1,...,10q 2i16(1C b )n 2a +1C A C B −(C 3+C 6)2N −C 7C 5C C CD −(C 6−C 3)2V −n a+1C a +14C 7−C 5C A C B −(C 3+C 6)2V +−C 6−C 3−C C8(1C b )n a n b +14(C A C B −(C 3+C 6)2)C 3+C 6+C A4(C C C D −(C 6−C 3)2)(C 7V +)2whereC A =C 3+C 5+C 6,C B =C 3+C 6+C 7,C C =̺C 3+C 5+C 6,CD =̺C 3+C 6+C 7,and V ±=V a ±V b ,1/C a =(C 5+C 7)/((C 5+C 7)(C 3+C 6)+C 5C 7),1/C b =[C 5+C 7+2(̺+1)C 3]/[(C 5+C 6+ηC 3)(C 6+C 7+̺C 3)−(C 6−C 3)2]and ̺=2√[1]A.Ekert and R.Jozsa,Rev.Mod.Phys.68,733(1996)and references therein.[2]N.A.Gershenfeld and I.L.Chaung,Science 275,350(1997).[3]A.Shnirman et al.,79,2371(1997).[4]Y.Nakamura et al.,Nature 398,786(1999).[5]T.Tanamoto,preprint,quant-ph/9902031.[6]S.Tiwari et al.,Appl.Phys.Lett.68,1377(1996),[7]L.Guo et al.,Science 275,649(1997).[8]Q.Ye,et al.,Phys.Rev.B 44,1806(1991).[9]D.V.Averin,Solid State Commun.105,659(1998).[10]P.Zanardi and F.Rossi,Phys.Rev.Lett.81,4752(1998).[11]N.Tsukada et al.,Appl.Phys.Lett.56,2527(1990).[12]A.J.Gar´c ia and J.Ferm´a ndez,Phys.Rev.B 55,5546(1997).[13]A.J.Leggett et al.,Rev.Mod.Phys.59,1(1987).[14]A.Shnirman and G.Sch¨o n,Phys.Rev.B 57,15400(1998).[15]S.A.Gurvitz,preprint,quant-phys/980858[16]G.Hackenbroich et al.,Phys.Rev.Lett.81,5896(1998).[17]S.M.Sze,Physics of Semiconductor Devices ,(John Wi-ley &Sons,Inc,New York 1981).[18]K.A.Matveev et al.,Phys.Rev.B 54,5637(1996).[19]C.Livermore et al.,Science 274,1332(1996).FIG.1.Two coupled qubits by the quantum coupled dots.FIG.2.An example of the N coupled dot system of quan-tum computing.Dots are coupled in the longitudinal direc-tion.The electron transfer in the lateral direction is assumed to be negligible.The FET channel structure enables the de-tection of the small signal of the charge distribution in coupled quantum dots.。