Universal quantum computing with semiconductor double-dot molecules on a chip Peng Xue Department of Physics, Southeast University, Nanjing 211189, P. R. China and Institute for Quantum Information Science, University of Calgary, Alberta T2N 1N4, Canada (Dated: January 18, 2010) We develop a scalable architecture for quantum computation using controllable electrons of double-dotmoleculescoupledtoamicrowavestriplineresonatoronachip,whichsatisfiesallDivin- cenzo criteria. We analyze the performance and stability of all required operations and emphasize that all techniquesare feasible with current experimental technologies. PACSnumbers: 03.67.Lx,42.50.Pq,73.21.La 0 1 0 I. INTRODUCTION (a) |T (b) 2 |1 (cid:1) (cid:3398)(cid:1732) n Quantum computing enables some computational |T |(1,1)S (cid:1) (cid:1732) a 0 problems to be solvedfaster than would everbe possible T J with a classical computer [1] and exponentially speeds (cid:1) (cid:1732) (cid:1) (cid:1732) |0(cid:600) (cid:168) 8 |(0,2)S 1 upsolutions to otherproblems overthe best knownclas- | T (cid:1) (cid:1732) sical algorithms [2]. Of the promising technologies for + (cid:1) (cid:1732) ] quantumcomputing,solid-stateimplementationssuchas (cid:1) (cid:1732) h spin qubits in quantum dots [3] and bulk silicon [4], and FIG.1: (a)Energyleveldiagramshowingthe(0,2)and(1,1) p - charge qubits in bulk silicon [5] and in superconduct- singlets in solid lines and the three (1,1) triplets in dashed t ing Josephson junctions [6], are especially attractive be- lines, and thelogical qubitstates |0i and |1i in red lines. (b) n cause of stability and expected scalability of solid-state Schematic of the double-well potential with an energy offset a u systems; ofthese competing technologies,semiconductor ∆ provided by theexternal electric field along x axis. q double-dotmolecules(DDMs)areparticularlyimportant [ becauseofthecombinationspinandchargemanipulation to take advantage of long memory times associated with ent life time of qubits. Thus we address all Divincenzo 4 v spinstates andatthe sametime to enable efficient read- criteria [14] and show all play important roles in the dy- 9 out and coherent manipulation of charge states. namics of the two-electronsystem but none represents a 1 fundamental limit for quantum computing. Here we develop a scalable architecture for semi- 7 3 conductor quantum computation based on two-electron . states in DDMs [7–11] coupled to a microwave stripline 8 resonator [12, 13]. The quantum information is encoded II. QUBITS 0 9 in the superpositions of double-dot singlet states. The 0 initialization of qubit states can be implemented by an We consider the system with two electrons located in : adiabatic passage. A universal set of gates including adjacent quantum dots coupling via tunneling. Imag- v single- and two-qubit gates can be implemented via the ine one of the dots is capacitively coupled to a stripline i X resonator-assistedinteraction with a microwavestripline resonator [10, 12, 13]. With an external magnetic field r resonator and the requirement for electrically driving B =100mTalongz axis,the spinalignedstates T = z + a | i DDMs directly is released, which avoids moving the sys- and T = , and the spin-anti-aligned states − |↑↑i | i |↓↓i tem away from the optimal point (where the coupling is T =( + )/√2 and (1,1)S =( )/√2 0 achieved to be strongest) because of the potential differ- |havie en|e↓r↑giy g|a↓p↑si due to th|e Zeemian s|↓p↑liitt−in|g↓↑sihown ence caused by the electric drive and increasing the ex- in Fig. 1(a). The notation (n ,n ) labels the number L R tended dephasing due to the fluctuations of the electric of electrons in the left and right quantum dots. The field. Comparedtothepreviousprotocol[13],thesystem doubly occupied state (0,2)S is coupled via tunneling inourschemealwaysworksinthestrongcouplingregime T to the singlet state| (1,1)Si . The double-dot sys- and the second order dephasing time is considered here. tem can be described by| an exitended Hubbard Hamil- The readout of qubits can be realized via microwave ir- tonian Hˆ =(E +µ) nˆ T (cˆ† cˆ +hc)+ radiation of the resonatorby probing the transmitted or os i,σ i,σ− σ L,σ R,σ reflected photons. The main decoherence processes are U inˆi,↑nˆi,↓+W σP,σ′nˆL,σnˆR,σ′+P∆ σ(nˆL,σ−nˆR,σ) forPcˆ (cˆ† )annihPilating(creating)anPelectroninquan- dissipationofthe striplineresonator,charge-basedrelax- i,σ i,σ ation and dephasing of the semiconductor DDMs, and tum doti L,R with spin σ , ,nˆ =cˆ† cˆ a ∈{ } ∈{↑ ↓} i,σ i,σ i,σ spindephasinglimitedbyhyperfineinteractionswithnu- number operator, and ∆ an energy offset yielded by the clei. By numerical analysis we show all gate operations external electric field along x axis shown in Fig. 1(b). andmeasurementscanbeimplementedwithinthecoher- The first term corresponds to on-site energy E plus os 2 site-dependent field-induced corrections µ. The second ( a ) ' ( b ) 2# term accounts for i j electron tunneling with rate T, z x and the third term i↔s the on-site charging cost U to put .$ 0 &" twoelectronswithoppositespininthesamedot,andthe L= 1 '!-)*+,#$ % fourthtermcorrespondstointer-siteCoulombrepulsion. Inthebasis (1,1)S , (0,2)S ,theHamiltoniancanbe ./ 4µm &"#$% , !"#$% {| i | i} deduced as !" '!()*+, ! #$% Hˆd =−∆|(0,2)Sih(0,2)S|+T |(1,1)Sih(0,2)S|+hc. 10mm !" &" (1) Withtheenergyoffset∆,degenerateperturbationthe- oryinthetunnelingT revealsanavoidedcrossingatthis FIG. 2: (a) Schematic of DDMs, biased with an energy off- set ∆, capacitively coupled to the stripline resonator. The balanced point between (1,1)S and (0,2)S with an | i | i coupling can be switched on and off via the external elec- energy gap ω = √∆2+4T2, and the effective tunneling tric field along x axis. The stripline resonator is driven by a between the left and right dots with the biased energies classical fieldalong xaxis. (b)Energyspectrumofthebared ∆ is changed from T to T′ =ω/2. We choose the super- (solidlines)anddressed(dashedlines)statesinthedispersive positions of the singlet states as our qubit states: regime. 0 ((1,1)S (0,2)S )/√2; | i≡ | i−| i 1 ((1,1)S + (0,2)S )/√2. (2) Controllably changing ∆ allows for adiabatic passage to | i≡ | i | i past the charge transition, with (0,2)S as the ground The essential idea is to use an effective electric dipole state if ∆ T achieved. First w|e turnion the external moment of the singlet states (1,1)S and (0,2)S of a electric fiel≫d along x axis and prepare the two electrons | i | i DDM coupled to the oscillating voltage associated with ofDDMsinthestate (0,2)S byalargeenergyoffset∆. a stripline resonator shown in Fig. 2(a). We consider a We change ϑ adiabati|cally toiπ/4 by tuning the electric stripline resonator with length L, the capacitance cou- field, andtheninitialize the qubits inthe qubitstate 0 . pling of the resonatorto the dot Cc, the capacitance per The SWAP operation [12], where a qubit state| iis unit length C0, the total capacitance of the double-dot swapped with a photonic state of the resonator, can be Ctot, and characteristic impedance Z0. The fundamen- used to implement transmission of qubits. If there is no tal mode frequency of the resonator is ω0 = π/LZ0C0. photonintheresonator,withthedetuningδ = ω ω0 = The resonator is coupled to a capacitor Ce for writing 0 and evolution time π/g, a qubit is mappe|d−to t|he and reading the signals. Neglecting the higher modes photonic state in the resonator (α 0 +β 1 ) 0 of the resonator and working in the rotating frame with 0 (α 0 +β 1 ) . Then we switch| oiff the|ciou|pilrinesg−b→e- res the rotating wave approximation, we obtain an effective t|wieen|tihis qu|biit and resonator and switch on that be- interaction Hamiltonian as tween the desired qubit and resonator via the local elec- tricfieldsalongxaxis. Afterthesameevolutiontime,the Hˆ =g(aˆσˆ +hc) (3) int + previous qubit state is transmitted to the desired qubit with aˆ (aˆ†) the annihilation (creation) operator of the via the interaction with the resonator. resonator field, σˆ = 1 0, σˆ = 0 1, and the effec- + − | ih | | ih | tive coupling coefficient IV. A UNIVERSAL SET OF GATES 1 C π c g = e sin2θ (4) 2 LCtotC0rZ0 Single-qubit gates including bit-flip and phase gates, and an entangling two-qubit gate can be implemented with θ = 1tan−1(2T). 2 ∆ via resonator-associated interaction with a stripline res- Theinteractionbetweentheresonatorandqubitstates onator. We consider a DDM interacts with a stripline is switchableviatuning the electricfieldalongxaxis. In resonatorfield, which is driven by a strong classicalfield the case of the energy offset yielded by the electric field along x axis ∆ 0, we obtain the maximum value of the coupling ≈ between the resonator and singlets in double dots. That Hˆ (t)=Ω(aˆ†e−iωdrt+hc) (5) is so called the optimal point. Whereas ∆ T, θ tends dr ≫ to 0, the interaction is switched off. withtheRabifrequencyΩ,andthefrequencyoftheclas- sical field ω substantially detuned from the resonator dr frequencyω . Inthe rotating frameatthe frequency ω III. INITIALIZATION AND 0 dr for a single qubit and the field, we obtain TRANSPORTATION ω ω Ω Hˆ =(ω ω )aˆ†aˆ+ − drσˆ g(aˆ†σ +hc)+ Rσˆ Initialization of qubit states can be implemented by 1q 0− dr 2 z− − 2 x an adiabatic passage between the two singlet states [11]. (6) 3 with the effective Rabi frequency Ω = 2Ωg . awayfromtheoptimalpointaspresentedin[13]. Inthat R ω0−ωdr In the dispersiveregime δ g, we obtainthe effective case the interaction is out of the strong coupling regime ≫ Hamiltonian from Eq. (6) as even with the maximum coupling, e.g. gT < 1 with T 2 2 the extended dephasing time. ω+g2/δ ω Ω Hˆ =(ω ω )aˆ†aˆ+ − drσˆ + Rσˆ . (7) x 0 dr z x − 2 2 V. READOUT Bychoosingω =ω+g2/δ,the Hamiltonian(7)evolves dr as a rotation around the x axis. The gate operates on To perform a measurement of qubits, a drive of fre- thetimescalingt 1/Ω . TheseRabioscillationshave x R already been observ∼ed experimentally in [6]. quency ωdr modeled by Eq. (5) is sent through the res- onator. Inthedispersiveregime,theenergygapbetween In another case the drive is sufficiently detuned from ethffeecqtiuvbeitHa|ωmi−ltoωndira|n≫fromΩREq[1.5(]6,)we obtain a different tqhuebidtrienssethdestsatatetes |00ir,eswahnilde |t1hiereseniesrgωy0 −gagp2/ωδ0 +forg2th/δe | i for the state 1 shown in Fig. 2(b). The matrix ele- Hˆ =(ω ω )aˆ†aˆ+ Ω′Rσˆ , (8) ment of the H|amiiltonian Hˆdr corresponding to a bit-flip z 0 dr z − 2 from the state 1 is suppressed, and depending on the | i qubitbeing inthe states 0 or 1 the transmissionspec- which generates rotations around z axis at a rate Ω′ = | i | i R trumwillpresentapeak ofwidthκ (the resonatordecay ωop+ergat2i/oδn−isωtdzr∼+112/ωΩ−Ω′Rω2R.dr. The time scaling of this gate rthateer)easotnωa0to−rfgr2e/qδueonrcωy0is+±gg22//δκ.δT,ahnisddtihsepeprusilvleisppuolwl eorf Since we can switch on and off the coupling between dependent and decreases in magnitude for photon num- the resonator and any DDM by tuning the local electric bers inside the resonator[17]. Via microwaveirradiation fieldsalongxaxis,forthecaseoftwoidenticalDDMs si- of the resonator by probing the transmitted or reflected multaneouslycoupledtotheresonator,withoutthedrive, photons, the readout of qubits can be realized and com- in the dispersive regime, we obtain the effective Hamil- pletedonatimescalingt =1/γ ,whereγ =8n¯(g2)21 m φ φ δ κ tonian for the system from Eq. (3) is the dephasing rate due to quantum fluctuations of the number of photon n¯ within the resonator. g2 1 g2 Hˆ =(ω + σˆi)aˆ†aˆ+ (ω+ ) σˆi 2q 0 δ z 2 δ z iX=1,2 iX=1,2 VI. DECOHERENCE g2 + (σˆ1σˆ2 +hc). (9) δ + − Now we analyze the dominant noise source of our sys- In the rotating frame at the frequency ω, the evolu- temincludingthecharge-baseddephasingandrelaxation, tionoperationofthetwo-qubitsystemdominatedbythe the spin phase noise due to hyperfine coupling and the above Hamiltonian after tracing out the field state (as- photonloss. Couplingtoaphononbathcausesrelaxation sume there is one photon in the resonator) of the charge system in a time T . The characteristic 1 charge dephasing with a rate T−1. The time-ensemble- g2 3 2 U =exp i t σˆi √iSWAP (10) averageddephasingtimeT∗ islimitedbyhyperfineinter- 2q n− δ 2iX=1,2 zo actions with nuclear spins.2The decay of the resonatorκ isconsideredasanotherdominantsourceofdecoherence. which provides an entangling two-qubit gate—root of For the charge relaxation time T , the decay is caused 1 SWAP gate on a time scaling t2q that satisfies g2t2q/δ = by coupling qubits to a phonon bath. With the spin- π/4. boson model, the perturbation theory gives an overall Hence we have built a universal set of gates for quan- errorrate from the relaxationand incoherent excitation, tum computing with semiconductor DDMs coupled to a with which one can estimate the relaxation time T 1 stripline resonator field. Compared to the previous pro- 1µs [12]. ∼ tocols [13, 16], we drive the resonator instead of driv- The charge dephasing T rises from variations of 2 ing qubits directly to implement single-qubit gates, in the energy offset δ(t) = δ + ǫ(t) with ǫ(t)ǫ(t′) = which no addressingqubits individually is required. The dωS(ω)eiω(t−t′), which is caused by the lohw frequeincy feasibility of single-qubit gates has already been proved Rfluctuationoftheelectricfield. Thegatebiasofthequbit in[6]experimentally. Forthetwo-qubitgate,werealizeit drifts randomly when an electron tunnels between the withtheoff-resonantinteractionbetweenbothqubitsand metallic electrode. Due to the low frequency property, resonator and release the requirement for driving qubits theeffectofthe1/f noiseonthequbitisdephasingrather with a strong classical field [13, 16]. Compared to the than relaxation. At the zero derivative point, compared previous protocol [13], the system works in the strong to abaredephasing time T =1/ dωS(ω), the charge coupling regime and the second order dephasing time is b q considered. Driving qubits introduces large energy dif- dephasingisT ωT2neartheoptiRmalpointδ =0. The 2 ∼ b ference between the potentials which moves the system baredephasingtimeT 1nswasobservedin[18]. Then b ∼ 4 the charge dephasing is estimated as T 10 100ns. ting a qubit to a photonic qubit in the resonator is 2 ∼ − Using quantum control techniques, such as better high- about t = π/g 4ns. Readout of qubits takes the tr ≈ andlow-frequencyfilteringofelectronicnoise,T exceed- time t 0.02ns in the case n¯ = 1 with the pa- b m ≈ ing 1µs was observed [8], which suppresses the charge rameters ω ,ω,ω ,g,Ω /2π = 10,5,5,0.125,10 GHz. 0 dr { } { } dephasing. The operating time for the single-qubit rotation along The hyperfine interactions with the gallium arsenide x axis is tx 1/ΩR 0.3ns with the above parame- ∼ ≈ host nuclei causes nuclear spin-related dephasing T∗. ters. The single-qubit rotation along z axis takes a time 2 The hyperfine field can be treated as a static quan- tz ∼ 1/Ω′R ≈ 0.03ns with the parameters different from tity, because the evolution of the random hyperfine field aboveto obtainthe desiredevolutionofthe system,that is several orders slower than the electron spin dephas- isω/2π 1MHz(therestaresame). Thetwo-qubitgate ≈ ing. In the operating point, the most important deco- in(9)canbe realizedonthe time scalingt2q whichsatis- herence due to hyperfine field is the dephasing between fiesg2t2q/δ =π/4andiscalculatedast2q 8nswiththe ≈ the singlet state (1,1)S and one of the triplet state parameters g,δ /2π= 0.125,1 GHz. Hereforthetwo- | i { } { } T . By suppressing nuclear spin fluctuation, the de- qubitgatewechoosethetunnelingT 18µeVwhichwas | 0i ≈ phasingtime canbe obtainedbyquasi-staticapproxima- recently realized in [21]. Thus, all these operating times tion as T∗ =1/gµ ∆Bz , where ∆Bz is the nuclear are less than the minimum decoherence time. 2 Bh nirms n hyperfine gradient field between two coupled dots and Now we analyze the effect on gate operations due to rms means a root-mean-square time-ensemble average. noise. The variations of the energy gap ∆(t) caused by A measurement of the dephasing time T∗ 10ns was the fluctuation of the electric field would lead to un- 2 ∼ demonstrated in [8]. wanted phase to the desired gate operations. The quality factorQ ofthe superconducting resonator We use the two-qubit gate in Eq. (9) as an ex- in the microwave domain can be achieved 106 [19]. In ample. With the time dependent fluctuations δλ(t) practice, the local external magnetic field 100mT re- of the effective coupling coefficient λ = g2/δ, the ducesthelimitofthequalityfactortoQ ∼104 [20]. The evolution operator of the system becomes U′ = ∼ 2q tdiimsseipaabtioounto1fµsthweitrhestohneaptoarraκm=eteωrs0/ωQ=lea2dπs th1e0GdHecza.y U2qexp −i 0t2qdtδλ(t)(32 i=1,2σˆzi +σˆ+1σˆ−2 +σˆ−1σˆ+2) , 0 × where t(cid:8)he unRwanted phasePφ = t2qdtδλ(t). The dis(cid:9)- 0 tribution of the unwanted phasRe becomes Gaussian distribution because λ is in Gaussian distribution. With VII. FEASIBILITY the parameters above, we numerically calculate the variances of the unwanted phase Var(φ) 5 10−3π. Now we analyze the feasibility of the proposal with a For single-qubit σ gate, the un∼wan×ted phase x gate-defined double-dot device as an example which is is txdtδΩ (t), while for σ gate, that becomes fabricated using a GaAs/AlGaAs heterostructure grown 0 R z tzRdtδΩ′ (t). With the same method, we can calculate by molecular beam epitaxy with a two-dimensionalelec- 0 R Rthe variance of the phases. tron gas 100nm below the surface, with density 2 1011cm2 [8]. When biased with negative voltages, th×e From the analysis, we show that even the dephasing occurs over the gate operation, we can still implement a patterned gates create a double-well potential shown in universal set of gates with high fidelities. For example, Fig.1(b). Thequantum-mechanicaltunnelingT between with the parameters we show above the fidelity for the the two quantum dots is about T 0 10µeV. The ≃ − entangling two-qubit gate is about 0.9946, 0.9952 for σ striplineresonatorcanbefabricatedwithexistinglithog- x gate and 0.9961 for σ gate. raphy techniques [19]. The qubit can be placed within z theresonatorformedbythetransmissionlinetostrongly suppress the spontaneous emission. The stripline res- onator in coplanar waveguides with Q 104 have al- VIII. SUMMARY ∼ ready been demonstrated in [20]. The diameter of the quantum dot is about 400nm, and the correspondingca- If a quantum computer is built, intractable prob- pacitance of the double-dot C is about 200aF. The lems such as factorization would be solved efficiently, tot capacitive coupling of the resonator to the dot is about withenormousramificationsforcommunicationsecurity. C 2C = 400aF. In practice, for ω = 2π 10GHz, Semiconductor DDMs quantum computer, which would c tot 0 ≈ × Z = 50Ω, L λ = 3cm, the coupling coefficient capitalizeonchipfabricationtechnologyandcouldbehy- 0 ∼ g 2π 125MHz is achievable by the numerical esti- bridizedwithexistingcomputers,isthepreferredmethod ∼ × mationsin4. The frequencyandcouplingcoefficientcan for quantum computation. Here we propose scalable be tuned by changing LC . The external magnetic field quantumcomputingwithelectricallycontrolledsemicon- 0 along z axis is about Bz = 100mT to make sure the en- ductor spins of DDMs coupled to a microwave stripline ergysplittingEz =gµBBz betweenthetwotripletstates resonatoron a chip. Quantum informationis encoded in T± is larger than ω ω0. thesingletstatesofDDMs. Initializationofqubitscanbe | i ∼ With these parameters we can estimate the time scal- realized with an adiabatic passage. With the switchable ing for quantum computing. The time for transmit- coupling to the resonator, we can implement a universal 5 set of quantum gates on any qubit. Although in general acterized through exact numerical simulations that in- charge qubits have less coherent life time compared to corporate various sources of experiment noise and these spin qubits, the generation and measurement methods results demonstrate the practicality by way of current are much simpler and faster, which makes our protocol experimental technologies. competitive with spin qubits in the context of circuit- based quantum computing. Because of the switchable couplingbetweenthedouble-dotpairsandtheresonator, wecanapplythisentanglinggateonanytwoqubitswith- IX. ACKNOWLEDGEMENTS outaffectingothers,whichisnottrivialforimplementing scalablequantumcomputingandgeneratinglargeentan- gled state. The fidelities of the gates in our protocol are ThisworkhasbeensupportedbyNationalNaturalSci- studied including all kinds of major decoherence, with ence Foundation of China, Grant No. 10944005, South- promising results for reasonably achievable experimen- eastUniversityStartupfund,NSERC,MITACS,CIFAR, tal parameters. The feasibility of this scheme is char- QuantumWorks and iCORE. [1] L. Grover, Phys.Rev.Lett. 79 (1997) 325-328. [12] J.M. Taylor, M.D. Lukin,arXiv: cond-mat/0605144. [2] P.W.Shor,Proc.35thAnnualSymp.onFound.ofComp. [13] Z.R. Lin et al., Phys. Rev.Lett. 101 (2008) 230501. Sci. (Los Alamitos, CA: IEEE Computer Society Press, [14] D.P. DiVincenzo, Fortschr. Phys. 48 (2000) 771-783. 1994) 124-130. [15] A. Blais et al., Phys.Rev. A 75 (2007) 032329. [3] D. Loss, D.P. DiVincenzo, Phys. Rev. A 57 (1998) 120- [16] S.B. Zheng,Phys. Rev.A 66 (2002) 060303(R). 126. [17] A. Blais et al., Phys.Rev. A 69 (2004) 062320. [4] B.E. Kane,Nature 393 (1998) 133-137. 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