XOR logic gate on electron spin qubits in quadruple coupled quantum dots
XXOR logic gate on electron spin qubits in quadruple coupled quantum dots
A. Kwa´sniowski and J. Adamowski
Faculty of Physics and Applied Computer Science,AGH University of Science and Technology, Krak´ow, Poland (Dated: November 2, 2018)The spin-dependent localization of electrons in quadruple quantum dots (QD’s) has been studiedby the configuration interaction method. We have investigated two nanodevices that consist oflaterally coupled quadruple QD’s. We have shown that – in both the nanodevices with suitablychosen parameters – the exclusive OR (XOR) logic gate can be realized by all-electrical controlwith the readout of output via the spin-to-charge conversion. We have determined the nanodeviceparameters that are optimal for the performance of the XOR logic gate.
PACS numbers: 03.67.Lx,03.67.Mn,73.21.La
Recent studies of quantum dot (QD) nanodevices to-wards a possible application in quantum computationfocus on a physical realization of qubits and logic op-erations on them [1, 2, 3, 4, 5, 6, 7, 8]. In the QD’s,the qubits can be encoded as two-level confined electronstates being either the orbital or spin states. The orbital(charge) qubits are characterized by a short coherencetime [9], which is disadvantageous for a quantum infor-mation processing. The spin qubits possess the coherencetime of the order of microseconds [10], which makes themgood candidates for a storage and processing of quantuminformation. An electrical manipulation of the electronqubits can been realized in electrostatically gated QD’s[11]. The results of the recent investigations of the lat-erally coupled multiple QD’s [2, 3, 5, 6, 7, 8, 12, 13, 14]are very promising. In the double QD’s, a coherent ma-nipulation of electron spin qubits has been demonstratedby Hayashi et al. [2] and Petta et al. [5]. In the quadru-ple QD’s, a controlled tunnelling and correlated chargeoscillations have been found by Shinkai et al. [14].Quantum computations can be performed using the el-ementary logic gates. Barrenco et al. [15] showed that aset of one-qubit quantum gates and a two-qubit exclusiveOR (XOR) gate is universal, i.e., an arbitrary many-qubitgate can be decomposed into these gates. The possiblerealization of arbitrary one-qubit gate in quantum-wirenanodevices has been proposed in Ref. [16] using thespin-orbit coupling and induced charge effects. The sim-ulations of the controlled NOT (CNOT) two-qubit gatehave been performed [17] for electron charge qubits incoupled QD’s.In this Letter, we report on the results for the spin-dependent localization of electrons in the nanodevicesthat consist of four laterally coupled gated QD’s. Inthese nanodevices, the potential confining the electronscan be tuned by changing the voltages applied to thegates and/or source and drain electrodes. We show thatthe quadruple QD nanodevices with suitably chosen pa-rameters can perform the XOR logic gate on electron spinqubits with the readout of output exploiting the spin-to-charge conversion. -60-40-20 0U [meV]-200 -100 0 100 200x [nm] -100 0 100y [nm]U [meV] e l e r l1l2 r1r2 b) Device B -40-20 0U [meV]-200 -100 0 100 200x [nm] -100 0 100y [nm]U [meV] l2 l1 r1r2 a) Device A FIG. 1: (Color online) Potential energy U of the electron inquadruple coupled QD’s as a function of x and y coordinatesfor device A (a) and B (b). Ovals schematically show the sizesof the QD’s labelled by l l r
1, and r
2. In device B, electricfield F = ( F, ,
0) with F = 1.01 kV/cm is applied. In the quasi-two-dimensional quadruple QD [14], theconfinement potential energy of the electron can be de-scribed by the following formula: U c ( r ) = − (cid:88) µν U µν exp {− [( r − r µν ) /R µν ] p/ } , (1)where r = ( x, y ), the QD’s are labelled by µ = l, r and ν = 1 , U µν is the depth of the potential well( U µν > R µν is the range of the confinement potentialthat determines the size of QD( µν ), r µν is the position ofthe center of QD( µν ), and parameter p ≥ U µν , which can be tuned by changing the voltagesapplied to the nearby gates. In nanodevice B [Fig. 1 (b)],the confinement potential can be modified by switchingon/off the bias voltage V between electrodes e l and e r .Then, the electron gains the additional potential energy∆ U . Taking into account the finite range of electric field F = ( F, ,
0) [18] we put ∆ U = − eV and 0 at the rightand left electrode, respectively, and ∆ U = − eF ( x + L/ L is the dis-tance between electrodes e l and e r and F = V /L . The a r X i v : . [ c ond - m a t . m e s - h a ll ] J u l -60-40-20 0-200 -100 0 100 200 U [ m e V ] x [nm]b) Device B l1, l2 r1 r2if r1 r2 -60-40-20 0-200 -100 0 100 200 U [ m e V ] x [nm]a) Device A l1, l2 r1 r2if
FIG. 2: (Color online) Potential energy U of the electron asa function of x for y fixed at the values corresponding tothe straight lines joining the centers of QD( l r
1) andQD( l r
2) for device A (a) and B (b). Dashed (blue)[solid (red)] curves show the initial ( i ) [final ( f )] potentialenergy profiles. total potential energy U = U c + ∆ U of the electron asa function of x and y is displayed in Figs. 1 and 2. Inthe initial state of the nanodevice, the electron potentialenergy of the left QD’s is set to be lower than that of theright QD’s (cf. Fig. 2).We have solved the Schr¨odinger equation for one andtwo electrons in the quadruple QD with the confinementpotential energy U (Fig. 1). For this purpose we haveextended the method applied previously to the doubleQD [18]. The two-electron problem has been solved bythe configuration interaction method [18] on a space gridwith the one-electron wave functions obtained by thenumerical variational method, described in Appendix ofRef. [18]. The computational method [18] provides ac-curate solutions to few-electron eigenproblems with anarbitrary confinement potential and electric field of finiterange. The present calculations have been performedfor the GaAs material parameters with donor rydberg R = 5 .
93 meV and donor Bohr radius a = 9 .
79 meV. Theinitial potential energy is characterized by U l = U l = 38meV, R l = R l = 58 . U r = 33 . R r = 25 . U r = 34 . R r = 49 . U l = U l = 26 . F = 1 .
01 kV/cm. The centers of QD’s arelocated at the vertices of the square that are lying at dis-tance 127.3 nm from each other. The softness parameteris taken as p = 4 [cf. Eq. (1)]. For each set of nan-odevice parameters we have calculated the lowest singlet( S ) and triplet ( T ) energy levels for the initial and finalpotential energy profiles (cf. Figs. 1 and 2). In the ab-sence of magnetic field, the three triplet states ( T , T ± )are degenerate, therefore, we are dealing with the triplydegenerate level T . We have also determined the local-ization of electrons in the QD’s for the lowest-energy S and T states by calculating the one-electron probabilitydensity [18]. The localization of two electrons obtained for the ini-tial and final potential profiles is schematically shown inFig. 3 (b). Fig. 4 displays contours of the one-electronprobability density calculated for the final potential pro-file (cf. Fig. 2). Fig. 4 shows that the electron distribu-tion in the single QD possesses the rotational symmetryin nanodevice A and is slightly shifted towards the leftcontact in nanodevice B, which results from the effectof the homogeneous electric field. The results of Fig. 4can be analyzed with the help of the truth table for theXOR logic gate [Fig. 3(a)]. The initial (input) state hasbeen prepared so that the left QD’s are singly occupiedby the electrons both in the singlet and triplet states.This electron localization is ensured by setting the po-tential wells of the left QD’s to be deeper than those ofthe right QD’s (cf. Fig. 2). We ascribe the logical val-ues 0 and 1 to the initial spin states associated with the z spin component +¯ h/ − ¯ h/
2, respectively (Fig. 3). Ifwe perform the controlled evolution of the electron statesby slowly changing the potential energy profile (Fig. 2),the electrons become localized in the right QD’s (Fig.4), since – in the final state – the potential wells of theright QD’s are considerably deeper than those of the leftQD’s. The heights and thicknesses of energy barriersseparating the QD’s have been chosen sufficiently largeto prevent the tunnelling of electrons between QD( l l
2) in the initial state. During the changes ofthe potential energy profile, the energy levels of the elec-
FIG. 3: (Color online) (a) Truth table for the XOR logic gate.In columns X and Y ( Z ) the input (output) logical values arelisted, the numbers (1-4) in the left column identify the XORgate operations. (b) Schematic of the nanodevice with thelocalization of electrons in QD’s (circles) for operations (1-4). The input logical value 0 (1) is encoded as the spin up(down) state of the electron localized in the left QD. Theoutput logical values are defined as the charge states q ( r r
1) as follows: q ( r
1) = 0 corresponds to logical value1 and q ( r
1) = − e corresponds to logical value 0. Charge q ( r
1) is measured by the quantum point contact (QPC). Thespin states of the electrons are depicted by arrows: up (down)arrow corresponds to the spin up (down). The double (blue)[solid (red)] arrows schematically show the localization of elec-trons in the input [output] states. trons confined in the left and right QD’s become equalto each other, which means that the resonant tunnellingconditions are satisfied, i.e., the electrons tunnel fromthe left to the right QD’s with the probability ∼
1. Theoccupancy of the right QD’s is determined by the initialspin states of the electrons in the left QD’s. If we con-sider the QD( r r q ( r
1) ofthe QD( r
1) by a sensitive charge detector, e.g., the quan-tum point contact schematically shown as QPC1 on Fig.4. Alternatively, we can use the quantum point contactQPC2, placed near the QD( r q ( r
2) of QD( r q ( r
1) = 0 or q ( r
1) = − e and either q ( r
2) = − e or q ( r
2) = − e for twodifferent output charge states (Fig. 4). This means thatboth the nanodevices A and B operate as spin-chargeconverters. Moreover, these nanodevices can act as theXOR logic gates (Fig. 3). When measuring the charge ofthe single right QD, we can distinguish XOR logic opera-tions (1) and (4) from operations (2) and (3) [Fig. 3(a)].If we additionally know the initial electron spin states,we can uniquely determine each of the four XOR logic y [ n m ] -100-50 0 50 100 QPC1QPC2 l1l2 r1 r2
Device A x [nm] y [ n m ] QPC1QPC2 l1l2 r1 r2
QPC1QPC2 l1l2 r1 r2
Device B s i ng l e t x [nm] 2001000-100 QPC1QPC2 l1l2 r1 r2 t r i p l e t FIG. 4: (Color online) Contours of one-electron probabilitydensity on the x − y plane in the final state with the electronslocalized in the right QD’s. The nanodevice parameters arechosen so that the XOR gate operations are realized in boththe nanodevices: A (left panels) and B (right panels). Circlescorrespond to the sizes of the QD’s, QPC1 and QPC2 denotethe quantum point contact in two alternative positions. Thetwo upper panels correspond to operations (2) and (3) (cf.Fig. 3) with the singlet-state electrons localized in QD( r r
1) andQD( r gate operations [Fig. 3(a)]. During operations (1-4) theinput spin states of the left QD’s are transformed intothe output charge states of the right QD’s.The nanodevices A and B can perform the XOR logicgate only for the suitably chosen nanodevice parame-ters. In order to determine the corresponding parameterregimes, we have performed many computational runsfor different sets of parameters of QD( r
1) keeping fixedthe parameters of other QD’s. The localization of elec-trons in the right QD’s obtained for the final potentialprofile is schematically depicted in the right inset of Fig.5. Fig. 5 shows that – based on the criterion of elec-tron localization in the final state – we can distinguishfive regimes, labelled by I through V, on the R r − U r plane. In regime I, both the electrons are localized in theQD( r
2) in either spin state. In regime II, we obtain thespin-dependent localization of the electrons in QD( r r FIG. 5: (Color online) Boundaries of five regimes I-V (solidcurves) of parameters R r and U r of QD( r r regime II, the role of QD( r
1) and QD( r
2) is interchanged,which means that – in regime IV – the QPC2 will providethe same measurement outcomes as the QPC1 in regimeII and vice versa . In regime V, both the electrons occupythe QD( r
1) in each final spin state.The left upper inset in Fig. 5 displays the zoom of therectangular region on the main panel with the boundariesof regimes I-V obtained for nanodevices A and B. Weobserve that the boundaries of these parameter regimeschange only slightly (these changes are of the order of thethickness of curves on the main panel). The smallness ofthese changes means that both the nanodevices A and Bare essentially equivalent in the proposed realization ofthe XOR logic gate.If the initial spin state is prepared as schematicallyshown in Fig. 3(b), i.e., the left QD’s are singly occupiedby the electrons with either parallel or antiparallel spins,then the controlled quantum-state evolution performedby changing the electrostatic potential profile leads to theuniquely determined final quantum states of electrons.This occurs in all parameter regimes I-V (Fig. 5). Inthe final state, the electrons are localized in either oneor two right QD’s in the well-defined spin states (cf. theright inset of Fig. 5). However, only in regimes II andIV, the localization of electrons in QD( r
1) and QD( r r r
2) allow us to distinguish the singlet and tripletfinal spin states (Fig. 4). Therefore, the nanodevices Aand B can be used to the electrical readout of spin viathe spin-to-charge conversion.The present mechanism of the XOR logic gate is basedon the results obtained for the stationary quantum states.In order to perform the controlled evolution betweenthese states we have to apply the external electric fieldthat appropriately modifies the electron potential energy.The electric field has to be switched on/off sufficientlyslowly in order to cause the adiabatic transition fromthe initial to final state. The non-adiabatic transitionscan lead to several spin flips during the quantum-stateevolution, which makes the operation time longer [19].Nevertheless, the present mechanism leads to the real-ization of the XOR gate, if the spins of electrons are thesame in the initial and final states [cf. Fig. 3(b)]. Theelectrons only change their localization as a result of tun-nelling from the left to right QD’s. Therefore, the XORgate operation time may be short enough in order to per-form the sufficiently large number of operations duringthe spin coherence time [10].We note that the two-qubit XOR logic gate is com-monly defined as follows: | X (cid:105)| Y (cid:105) −→ | X (cid:105)| Z = X ⊕ Y (cid:105) ,where ⊕ is the addition modulo two. This two-qubitXOR gate is equivalent to the CNOT gate [20]. How-ever, according the present mechanism, the nanodevicesA and B perform the following operation: | X (cid:105) l | Y (cid:105) l −→ | Z = X ⊕ Y (cid:105) r , where X, Y , and Z are defined in Fig.3(a) and the output is detected by QPC1 as the charge ofQD( r
1) (cf. Fig. 4). If we alternatively apply the QPC2to measure the charge of QD( r | Z = X ⊕ Y (cid:105) r is also uniquely determined by theinput spin states of the left QD’s. Therefore, the logicgate studied in this Letter possesses all the properties ofthe XOR gate defined in Fig. 3(a). The logic gate op-erations realized in nanodevices A and B result from thequantum transitions between the well-defined quantumstates of the electrons. During these operations, the in-put spin qubits are transformed into the output chargequbits by changing the external voltages, i.e., by the all-electrical control. The nanodevices A and B can realizethe entire cycle of the XOR logic gate, i.e., the inputstate preparation, electrically driven evolution of quan-tum states, and readout of the output via the spin-to-charge conversion. The XOR logic gate, proposed in thisLetter, can be realized in the quadruple QD’s recentlystudied by Shinkai et al. [14].In summary, we have proposed the all-electrical im-plementation of the XOR logic gate in two nanodevicesbased on the laterally coupled quadruple QD’s.This paper has been partly supported by the PolishScientific Network LFPPI ”Laboratory of Physical Fun-damentals of Information Processing”. [1] G. Burkard et al. , Phys. Rev. B , 2070 (1999).[2] T. Hayashi el al. , Phys. Rev. Lett. , 226804 (2003).[3] J.M. Elzerman et al. , Phys. Rev. B , 161308(R) (2003);Nature , 431 (2004).[4] S. Sangu et al. , Phys. Rev. B , 115334 (2004).[5] J.R. Petta et al. , Science , 2180 (2005).[6] R. Hanson et al. , Phys. Rev. Lett. , 196802 (2005).[7] P. Hawrylak and M. Korkusinski, Solid State Commun. , 508 (2005); M. Korkusinski et al., Phys. Rev. B ,115301 (2007).[8] T. Meunier et al. , Phys. Rev. B , 195303 (2006).[9] S. Sauvage et al. , Phys. Rev. Lett. , 177402 (2002).[10] R. Hanson et al. , Phys. Rev. Lett. , 196802 (2003).[11] J. Adamowski et al. , Handbook of Semiconductor Nanos-tructures and Nanodevices , Vol. 1, ed. A.A. Balandin andK.L. Wang (CA: American Scientific Publishers), p. 389(2006).[12] M. Stopa et al., Physica E , 616 (2006).[13] Z. Jiang et al. , Phys. Rev. B , 035307 (2008).[14] G. Shinkai et al. , Appl. Phys. Lett. , 103116 (2007);cond-mat arXiv: 0905.0805v2.[15] A. Barenco et al. , Phys. Rev. A , 3457 (1995).[16] S. Bednarek and B. Szafran, Phys. Rev. Lett. ,216805 (2008); Nanotechnology , 065402 (2009).[17] S. Moskal et al. , Phys. Rev A , 062327 (2005).[18] A. Kwa´sniowski and J. Adamowski, J.Phys.: Condens.Matter , 235601 (2009).[19] S. Moskal et al. , Phys. Rev A , 032302 (2007).[20] D.P. DiVincenzo, Proc. R. Soc. London A454