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Scaling of Decoherence Effects in Quantum Computers 3 0 B. J. DALTON 0 †‡ 2 Centre for Atom Optics and Ultrafast Spectroscopy, † n Swinburne University, Hawthorn, Victoria 3122, a Australia J Department of Physics, University of Queensland, 7 ‡ 1 St Lucia, Queensland 4072, Australia 3 February 1, 2008 v 1 7 Abstract 0 9 The scaling of decoherence rates with qubit number N is studied for 0 a simplemodel of aquantumcomputerin thesituation where Nis large. 2 The two state qubits are localised around well-separated positions via 0 trapping potentials and vibrational centre of mass motion of the qubits / h occurs. Coherent one and two qubit gating processes are controlled by p external classical fields and facilitated by a cavity mode ancilla. Deco- - herence due to qubit coupling to a bath of spontaneous modes, cavity t n decay modes and to the vibrational modes is treated. A non-Markovian a treatment of the short time behaviour of the fidelity is presented, and u expressionsforthecharacteristicdecoherencetimescalesobtainedforthe q casewherethequbit/cavitymodeancilla isinapurestateandthebaths : v are in thermal states. Specific results are given for the case where the i cavity mode is in the vacuum state and gating processes are absent and X the qubitsare in (a) theHadamard state (b) the GHZ state. r a 1 Introduction ⊂ Quantumcomputersareexpectedtohaveakeyadvantageoverclassicalcomput- ers in terms of computational complexity, an advantage based on the idealised features of parallelism, entanglement and unitary quantum computation processes. For certainalgorithms [1, 2], the number ofcomputationalsteps needed usingquantumcomputersshouldincreasemuchmoreslowlywithinputnumber sizethanforclassicalcomputers,enablingcomputationsthatwereinfeasibleus- ing classical computers to be carried out. In the case of the quantum search algorithm for example, the number of steps increases with the square root of the number of items, rather thanlinearly as in classicalsearching[2]. However, the physical system whose quantum states define the N qubit system and the 1 quantum devices involved in the gating processes both interact with the environment. Suchinteractionschangethedensityoperatordescribingthequantum computer state from a pure to a mixed state, with coherences between states evolvedfromdifferentinputstatesbeingpartiallyorcompletelydestroyed. This process of decoherence is the enemy of quantum computation, though methods such as quantum error correction [3, 4], decoherence free sub-spaces [5, 6, 7, 8] and dynamical suppression of decoherence [9, 10] could be used to minimize its effects. Utilizing the computational complexity advantage of quantum computers requires suitably large qubit numbers (ca 105 qubits may be needed for searching or factoring algorithms with error correction [11] for input numbers of practical importance), so the scaling with the number of qubits of the decoherence rates in comparison to the coherent processing rates is important in determining the size limits of useful quantum computers [12, 13]. This paperextends previouswork[14,15,16,17,18,19,20]treatingscaling effects. Decoherence effects are studied for a simple model of anN qubit quantum computer, involving N two state qubit systems (see figure 1). The qubits are localised around well-separated positions via trapping potentials, and the centre of mass (CM) vibrational motions of the qubits are treated. Coherent one and two qubit gating processes are controlled by time dependent localised classical electromagnetic (EM) fields and magnetic fields that address specific qubits. The two qubit gating processesare facilitated by a cavity mode ancilla, which permits state interchange between qubits. The magnetic fields are used to bring specific qubits into resonancewith the classicalEMfields orthe cavity mode. The two state qubits are coupled to a bath of EM field spontaneous emission (SE) modes, and the cavity mode is coupled to a bath of cavity decay modes. For large N the numerous vibrational modes of the qubits also act as a reservoir, coupled to the qubits, the cavity mode and the SE modes. The system-environmentcouplinginteractionsincludeelectricdipolecouplingofthe qubitstotheSEmodes,quasi-modecouplingofthecavitymodetocavitydecay EM field modes and Lamb-Dicke coupling of qubits to CM vibrational modes and each of: (a) the SE modes (b) the cavity mode (c) the classical EM gating fields (d) the classical magnetic gating field. The system-environment coupling interactions considered include both σ phase destroying and σ population Z X destroying terms, though the latter are more important in this model. The modelissimilarto thoseofthe ion-trap[21], neutralatom[22]andcavityQED [23, 24, 25] varieties. The object is to study decoherence effects in quantum computers for the situationwherethenumberofqubitsN becomeslarge. Decoherenceeffectswill be specifiedby the fidelity, which measureshowclose is the actualbehaviourof thedensityoperatorforthequbitssystem(includingtheancilla)toitsidealised behaviour due to coherent gating processes only. The initial quantum state for the quantum computer will be assumed to be pure and the reservoirstates will be assumed to be thermal. Previous work [14, 15, 16, 17, 18, 19, 20] indicates that the decoherence time t can decrease inverselywith N (qubits decohering D independently) or N2 (qubits decohering collectively), so it can become very shortforthecasewhereN islargeandmaybecomecomparabletothereservoir 2 correlation time t . In the case of atomic spontaneous emission relaxation, t C C 10−17 s, and t 10−8s for one qubit could become t 10−18s for 105 D D ∼ ∼ ∼ qubits in the case of collective decoherence. However,one aspect of the present study is whether the qubits decohere independently or not, so we do not arbi- trarily assume either case. As maintaining coherence between all states of the quantum computer is important for its operation, we are interested in the behaviour of all density matrix elements, and not just those for a smaller density matrix describing a single qubit, even though the latter may have long decoherence times and satisfy Markovian equations. Treating the density operator forthe fullN qubitsystemviathe standardBorn-Markoffmasterequation(see for example, [26, 27]) requires that all the interaction picture density matrix elements do not change significantly during the reservoir correlation times. As the latter are just the time scales over which two-time correlation functions of reservoir operators involved in the system-reservoir interaction decay to zero, we can see that the reservoir correlation times are independent of the number of qubits. Since the generaldecrease in decoherence time with increasing qubit numbersindicates that some density matrix elements mustchangeovershorter and shorter time scales, then using Markovian master equations for studying decoherence therefore becomes questionable if the number of qubits becomes large enoughIn the present paper we do not assume that Markovianbehaviour will necessarily occur, and use methods that do not rely on the standardBorn- Markoffmaster equation. If the behaviour does turn out to be Markovianthen our approach is still correct. However as we will see, in certain cases the behaviour is not consistent with the Markoff approximation, so the necessity for our choice of non-Markovian methods is justified a posteriori. Several authors [28, 29, 30, 31] have also emphasisedthe importance of studying the short time regime for macroscopic and mesoscopic systems (such as quantum computers with large qubit numbers) using non-Markovian methods. One such approach [29,30]involvesusingeigenstatesofthesystemoperatorsinvolvedinthesystem- reservoirinteraction(pointer basis),anothertreatment[32]developsshort-time expressions for the linear entropy (or impurity). Here the approach used is similar to that in [33] and uses the Liouville-von Neumann equations to ob- tain short-time expansions for the fidelity. This leads to expressions for the characteristic decoherence time scales for the general case with gating EM and magnetic fields present, and at non-zero temperature, and with the qubit and ancilla system in an arbitrary pure state. The treatment enables situations where independent or collective decoherence occur to be distinguished. In this initial paper, the decoherence time scales will be evaluated for specific initial quantumcomputerstatessuchas: (a)theequalsuperpositionofallqubitstates (Hadamardstate)and(b)thegeneralisedGHZstates,withthecavitymodean- cilla in the vacuum state and gating processes ignored. The effects of gating processesandtherelationshipoftheresultstothoseobtainableforlongertimes using Markovianmaster equation methods will be treated later. In Section 2 the general theory used will be outlined. Results for the deco- herencetimescalesarepresentedinSection3andSection4containsasummary of the paper. 3 2 Theory 2.1 Hamiltonian The total Hamiltonian is written as the sum: H =H +H +H +V +V (1) S C B S I The Hamiltonian for the qubit system and cavity mode ancilla is given in termsofqubitatomicspinoperatorsσi andcavitymodeannihilation,creation Z operators b,b† by: 1 H = ~ω σi +~ω b†b (2) S 2 0 Z b Xi Here the qubits all have transition frequency ω and ω is the cavity mode 0 b frequency. Thecavitymodefacilitatesstatetransferbetweendifferentqubits in two qubit gating process. In the case of a large number of charged qubits, the CM vibrationalmodes are too closelyspacedto enable the in-phase vibrational mode tobe usedinthis role,asinthe caseofion-trapquantumcomputers[21]. The array of qubits confined in their trapping potentials is rather analogous to alowdensitysolid. Also,becausethecavitymodeenergyquantumislarge,this ancillaisnotsubjecttothermaleffectsasarethelowfrequencyCMvibrational modes. The Hamiltonian for the centre of mass vibrational motion of the qubit system is given in the harmonic approximation as: 1 1 H = p2 + Vαβδr δr (3) C 2m iα 2 ij iα jβ Xiα iXjαβ = ~ν A† A (4) K K K XK In these equations the qubit CM momentum operators are p and the centre iα of mass of each qubit vibrates with displacements δr about well-separated iα positions r defined by trapping potentials (α,β = x,y,z). The vibrational i0 mode K has angularfrequencyν andphononannihilation,creationoperators K A ,A† . ThevibrationalHamiltonianH appliestothegeneralcasewherethe K K C qubitsvibratecollectively,suchasintheion-trapcasewherethechargedqubits interactvialong-rangeCoulombpotentials. Ifthequbitsvibrateindependently, suchasforneutralqubitswithdipole-dipolecouplingsignored,thenthecoupling matrix is given by Vαβ = δ δ V, and the vibrational modes K are specified ij ij αβ iα. These all have the same frequency ν. The Hamiltonian for the bath of spontaneous emission and cavity decay modes is of the form: H = ~ω a†a + ~ξ b†b (5) B k k k k k k Xk Xk 4 Here the SE mode k has angular frequency ω and annihilation, creation oper- k atorsa ,a†, and the cavitydecay mode k has angularfrequencyξ andannihi- k k k lation, creation operators b ,b†. These modes are described in the quasi-mode k k picture [34]. The coherent coupling (or gating) processes for the qubit system are described by the Hamiltonian: 1 V = (~Ω σi +Hc)+ ~(∆ ∆ )σi S i X 2 i1− i0 Z Xi Xi + (~g σi b+Hc) (6) i X Xi In order, the terms are the coupling of the qubits to the classical EM field, to the magnetic field and to the cavity mode. Each qubit can be addressed by localised classical EM fields E and magnetic fields B to facilitate one and two qubit gating processes. The two qubit gating process involves the creation and annihilation of a single photon in the cavity mode, which acts as an ancilla. The quantities Ω ,∆ and g are coupling constants. The coupling constants i ia i Ω ,∆ are both spatially localised and time dependent in accordance with the i ia sequence of gating processes involved for a particular algorithm. Theinteractionofthequbitsystemandancillawiththebathsandthequbits centre of mass vibrational degrees of freedom is given by: V = (~giσi a +Hc)+ (~b†(u b +w b†)+Hc) I k X k k k k k Xik Xk + (~ni σi a [A +A† ]+Hc) kK X k K K XikK + (~pi σi b[A +A† ]+Hc) K X K K XiK + (~Θi σi [A +A† ]+Hc) K X K K XiK 1 + ~(mi (1 σi )+mi (1+σi ))[A +A† ] (7) 2 0K − Z 1K Z K K XiK The first two terms are the electric dipole coupling of the qubits to SE modes and the quasi-mode coupling of the cavity mode to the cavity decay modes. These terms involve coupling constants gi,u and w . The remaining terms k k k are Lamb-Dicke couplings with the various fields, allowing for the vibrational motion of the qubits CM around their equilibrium positions r . The third i0 term allows for coupling to the SE modes, the fourth to the cavity mode, the fifth to the classical EM field and the sixth to the magnetic field. These terms involve coupling constants ni ,pi ,Θi and mi . Since this paper studies kK K K 0K short time scale effects, the standard rotating wave approximation (RWA) has not been made. Other possible interactions, such as those due to Roentgen currents [35, 36] for neutral and charged qubit systems, and those due to ionic 5 currents [35, 36] for chargedqubit systems, were also investigated but found to be relatively small in comparison to the interactions listed above. 2.1.1 Expressions Detailed definitions or commutation rules for the qubit, cavity mode, CM vibrational modes and bath modes operators are as follows: σi = (1 0 + 0 1) σi = i(1 0 0 1) X | ih | | ih | i Y − | ih |−| ih | i σi = (1 1 0 0) (8) Z | ih |−| ih | i [b,b†] = 1 [A ,A†]=δ K L KL [a ,a†] = δ [b ,b†]=δ (9) k l kl k k kl ThequbitsarerepresentedbyPaulispinoperators,whereasbosonicannihilation andcreationoperatorsapply for the cavitymode (quantized involume V ), the b CM vibrational modes, the SE modes and the cavity decay modes (quantized in volume V). The CM displacements δr (α=x,y,z) are related to the vibrational nor- iα mal coordinates via a unitary matrix S in the form: ~ δr = S (A +A† ), (10) iα iα;Kr2mν K K XK K where the matrix S is determined from the eigenvalue equations VαβS =mν2 S . (11) ij jβ;K K iα;K Xjβ These equations also give the vibrational frequencies ν . For the case where K 1 the qubits vibrate independently Siα;K =δiα;K and ν =(V/m)2. 6 Explicit expressions for the coupling constants are: ω Ω = i c (d e )α expi(k r ω t) (12) i − Xc r2ǫ0~V 10· c c c· i0− c ∆ = [m B]/~ a=0,1 (13) ia aa − · ω g = i b (d e )exp(ik r ) (14) i − r2ǫ0~Vb 10· b b· i0 ω gi = i k (d e )exp(ik r ) (15) k − r2ǫ0~V 10· k · i0 ω ~ ni = k (d e )(k S ) kK r2ǫ0~V r2mνK 10· k · iK × exp(ik r ) (16) i0 × · ω ~ pi = b (d e )(k S ) K r2ǫ0~Vbr2mνK 10· b b· iK × exp(ik r ) (17) b i0 × · ω ~ Θi = c (d e )(k S ) K Xc r2ǫ0~V r2mνK 10· c c· iK × α expi(k r ω t) (18) c c i0 c × · − ∂(m B) ~ miaK = − ( ∂Raa· )0r2mν Siβ;K /~ a=0,1. (19) Xβ iβ K Here d is the electric dipole matrix element, m (a = 0,1) are the magnetic 10 aa dipole matrix elements for each qubit. The polarisation unit vectors for the classicalfield,thecavitymode,theSEmodeorcavitydecaymodekaree ,e ,e c b k respectively, and k ,k ,k give the wave vectors. The classical EM gating field c b couldbe thoughtofasbeingrepresentedbyamulti-modecoherentstate,where α isthecoherentstateamplitudeforthemodec. Allthemodeshaveessentially c the same polarisation vector, wave vector and frequency, though the frequency andwavevectorbandwidthsaresufficientlywidetoproducetherequiredspatio- temporal localisation. The equivalent classical electric field at the equilibrium positionofthe iqubitis givenbyE=i ~ωc e α expi(k r ω t)+cc. cq2ǫ0V c c c· i0− c P 2.2 Dynamics The total density operator W for the full system of qubits, qubits CM motion, cavity mode and baths satisfies the Liouville-von Neumann equation: ∂W i~ =[H,W] (20) ∂t Initial conditions are assumed of an uncorrelated state for qubits and ancilla, bath and CM vibrational motion given by:. W(0)=ρ (0)ρ (0)ρ (0) (21) S B C 7 and for the bath and CM in thermal states the average value of V is zero. I Tr V ρ (0)ρ (0)=Tr ρ (0)ρ (0)V =0 (22) BC I B C BC B C I In the present paper the initial state for the qubits and ancilla will be assumed in a pure state ψ , so that ρ (0)= ψ ψ . | Si S | Sih S| The dynamicsofthe quantumcomputer isdescribedbythe reduceddensity operator ρ for the qubits and ancilla subsystem defined by: S ρ =Tr W (23) S BC The exactevolutionallowingfor the coherentcouplingandthe interactionwith the bath and the CM vibrationalmodes is embodied in the time dependence of ρ . Noassumptionismadeherethatρ satisfiesaMarkovianmasterequation, S S thoughitmaydosofortimescaleswhicharelargecomparedtobathcorrelation times. The coherent evolution of the qubits and ancilla subsystem is described by the reduced density operator ρ , which is chosen to coincide with the exact S0 reduced density operator at time zero, and is then allowed to evolve under the coherent coupling term V only. Hence: S ∂ρ i~ S0 = [H +V ,ρ ] (24) ∂t S S S0 ρ (0) = ρ (0) (25) S0 S Thus the interaction between the qubits/ancilla and the baths and CM motion isconsideredtobeswitchedoffhere,henceρ evolvesaccordingtoaLiouville- S0 von Neumann equation.. 2.3 Decoherence Decoherence effects are specified by the fidelity, defined as: F =Tr (ρ ρ ) (26) S S0 S The fidelity specifies how close the actual evolution of qubits and ancilla is to the idealisedcoherentevolution. The time dependence of the fidelity is entirely duetothedecoherenceeffectscausedbythebathandCMinteractionswiththe qubits and cavity mode ancilla. As in [33] the short-time behaviour of the fidelity with the qubits in a pure statecanbe expressedasapowerseriesinthetime elapsedandexplicitexpres- 8 sions obtained for the first few time constants involved. Thus: t t2 F(t) = 1 ( ) ( )+.. (27) − τ − 2τ2 1 2 ~ = 0 (28) τ 1 ~2 = Tr ψ V (0)2 ψ ρ (0)ρ (0) 2τ2 BCh S| I | Si B C 2 Tr ψ V (0)ψ 2ρ (0)ρ (0) (29) − BCh S| I | Si B C ∆V (0)2 (30) I S BC ≡ hh i i The initial value of the fidelity is unity. The times τ ,τ ,..specify characteris- 1 2 tic decoherence times for the qubit and ancilla system, their inverses defining decoherencerates. ForthermalbathandCMstates,onlyτ isinvolvedinspec- 2 ifyingshorttimedecoherence. Forthe timeconstantτ ,the expressioninvolves 2 the average of the square of the fluctuation ∆V (0) = V (0) V (0) of the I I I S −h i zero time interaction operator V (0). The squared fluctuation operator is first I averaged over the initial qubit and ancilla pure state ψ , then the result is | Si averagedover the bath and CM initial state ρ (0)ρ (0). B C 3 Results 3.1 General case The evaluation of τ for the general case with gating EM and magnetic fields 2 both present, and at non-zero temperature, and with the qubit and ancilla system in an arbitrary pure state, leads to expressions for τ that involve: 2 (a) Expectation values for the state ψ of one qubit operators σi, two qubit operators σiσj (i=j), (α,β =X,Z|)S.i α α β 6 (b) Expectation values of cavity operators b,b†,bb†,b†b,b2,b†2. (c) Expectation values of products of these qubit and cavity operators. (d) Sums over bath modes and centre of mass vibrational modes of products of pairs of the coupling constants gi,w ,ni ,pi ,Θi ,mi , the products k k kK K K aK being weightedby factorsinvolving the thermally averagedphotonandphonon quantum numbers n ,m ,N for the SE modes, cavity decay modes and vi- k k K brational modes. The thermally averaged quantum numbers are given by the Planck function. The general case will be studied in further work. 3.2 Case of spontaneous emission for stationary qubits, no cavity mode Inthiscasethecavitymodeisignored,thequbitsareassumedstationarysothat CMvibrationalmodesareexcluded(asareallLamb-Dickeinteractions),andthe only coupling constantsincluded aregi, givenby g expik r . The qubit pure k k · i0 9 stateis φ . Thisspecialsituationhasbeenstudiedin[33],wheretemperature | Qi effects and criteria for the qubits decohering independently or collectively were examined. For this case the decoherence time τ is given by: 2 1 = ( σi σj σi σj ) gi gj∗(2n +1) (31) 2τ2 h X Xi−h Xih Xi k k k 2 Xij Xk ~ω = ∆σi ∆σj g 2cosk d coth k (32) h X Xi | k| · ij 2k T Xij Xk B where .. = φ ..φ ,∆Ω = Ω Ω and d = r r . This result is the h i h Q| | Qi −h i ij i0 − j0 same as in [33]. As in[33], if∆k is the bandwidth andk the meanof a normalisedGaussian modelforthefunction g 2coth ~ωk ,thentheconditionforindependentdeco- | k| 2kBT herence(whereonlytermswithi=j contributetoτ )requiresthat∆kd 1 2 ij ≫ foranypairofqubits(d = d ). Collectivedecoherence(wheretermsfromall ij ij | | pairs i= j of qubits contribute to τ ) requires that ∆kd 1 and kd 1. 2 ij ij 6 ≪ ≪ Further discussion about independent and collective decoherence is given be- low for the case of zero temperature, but based on evaluating the sum over SE modes k for a three dimensional model. 3.3 Case of no gating processes, zero temperature and cavity in vacuum state The case where no gating is taking place, the ancilla is back in its original no photon state 0 and the temperature is zero, provides a simple case of the A | i general results, enabling the dependence on the qubit state to be examined. Here ψ = φ 0 , where the state of the qubit system is φ . This case is | Si | Qi| iA | Qi ofsomephysicalinterest,since it correspondsto states producedafter idealised coherentgatingprocesseshaveoccurred. Toorapidadecoherenceforsuchstates would be of concern for the implementation of quantum computers. 3.3.1 Case of correlated qubit states 1 = ( σi σj σi σj )G 2τ2 h X Xi−h Xih Xi ij 2 Xij + σi σj K +L (33) h X Xi ij Xij G = gi gj∗+ ni nj∗ (34) ij k k kK kK Xk XkK K = pi pj∗ (35) ij K K XK L = w w∗ (36) k k Xk 10