Quantitative Evaluation of Decoherence and Applications for Quantum-Dot Charge Qubits
aa r X i v : . [ qu a n t - ph ] M a y Quantitative Evaluation of Decoherence andApplications for Quantum-Dot Charge Qubits
Leonid Fedichkin and
Vladimir Privman
Center for Quantum Device Technology, Department of Physics andDepartment of Electrical and Computer Engineering, Clarkson University,Potsdam, New York 13699–5721, USA
Abstract
We review results on evaluation of loss of information in quantum registers dueto their interactions with the environment. It is demonstrated that an optimalmeasure of the level of quantum noise effects can be introduced via the maxi-mal absolute eigenvalue norm of deviation of the density matrix of a quantumregister from that of ideal, noiseless dynamics. For a semiconductor quantumdot charge qubits interacting with acoustic phonons, explicit expressions for thismeasure are derived. For a broad class of environmental modes, this measure isshown to have the property that for small levels of quantum noise it is additiveand scales linearly with the size of the quantum register.
In recent years, there has been significant progress in quantum computationand design of solid-state quantum information processors [1, 2, 3, 4, 5, 6, 7, 8,9, 10, 11, 12, 13, 14, 15, 16]. Quantum computers promise enormous speed-upof computation of certain very important problems, including factorization oflarge numbers [1] and search [2]. However, practically useful quantum informa-tion processing devices have not been made yet. One of the major obstacles toscalability has been decoherence. This is due to the fact that the effect of quan-tum speed-up is crucially dependent upon the coherence of quantum registers.Therefore, understanding the dynamics of coherence loss has drawn significantexperimental and theoretical effort.In general, decoherence [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30,31, 32, 33, 34, 35, 36, 37, 38] reveals itself in most experiments with quantumobjects. It is a process whereby the quantum coherent physical system of inter-est interacts with the environment and, because of this interaction, changes itsevolution from unperturbed “ideal” dynamics. The change of the dynamics isreflected by the corresponding change of the density matrix [39, 40, 41, 42, 43]of the system. The time-dependence of the system’s density matrix should beevaluated for an appropriate model of the system and its environment. If amulti-particle quantum system is considered then the respective density matrixbecomes rather large and difficult to deal with. This occurs even for relativelysmall quantum registers containing just a few quantum bits (qubits). In this1aper, we review evaluation of decoherence effects starting from the systemHamiltonian and followed by the definition and estimation of a decoherenceerror-measure in a quantum information processing “register” composed of sev-eral qubits.The paper is organized as follows: In Section 2, we consider a specific ex-ample of a solid state nanostructure. As a representative model for a qubit, weconsider an electron in a semiconductor double quantum dot system. We derivethe evolution of the density matrix of the electron, which losses coherence dueto interaction with phonons. In Section 3, we define a measure characterizingdecoherence and show how to calculate it from the density matrix elements fora semiconductor double quantum dot system introduced earlier. Finally, in Sec-tion 4, we establish that the measure of decoherence introduced, is additive forseveral-qubit registers, i.e., the total “computational error” scales linearly withthe number of qubits.
Solid-state nanostructures attracted much attention recently as a possible ba-sis for large scale quantum information processing [44]. Most stages of theirfabrication can be borrowed from existing fabrication steps in microelectronicsindustry. Also, only microelectronics technology has demonstrated the abilityto create and control locally evolution of thousands of nano-objects, which isrequired for quantum computation. There were several proposals for semicon-ductor qubits, reviewed, e.g., in [24]. In particular, the encoding of quantuminformation in the position of the electron was investigated in [45, 46, 47, 48, 49].In [50] it was argued that an electron in a typical quantum dot will loose co-herence very fast which will prevent it from being a good qubit. However, thisproblem can be resolved with sophisticated designs of quantum-dot arrange-ments, e.g., arrays of several quantum dots, if properly designed [51], can forma coherent quantum register. It was also shown that a symmetric layout of justtwo quantum dots can strongly diminish decoherence effects due to phonons andother environmental noises [52, 53, 54].Recent successful observations [55, 56, 57, 58, 59] of spatial evolution of anelectron in symmetric semiconductor double dot systems have experimentallyconfirmed that such a system is capable of maintaining coherence at least ontime scales sufficient for observation of several cycles of quantum dynamics.In the above experiments measurements were performed at very low substratetemperatures of few tens of mK, in order to avoid additional thermally activatedsources of decoherence. Theoretical results on the influence of the temperatureon the first-order phonon relaxation rates in double dot systems were presentedin [60, 61].In view of the above experimental advances, we have chosen a single elec-tron in semiconductor double quantum dot system, whose dynamics is affectedby vibrations of the crystal lattice, as a representative example of a quantumcoherent system interacting with the environment. In the range of parameters2 P (cid:72) Figure 1: Double well potential.corresponding to experiments [55, 57, 58, 59] phonons dominate decoherence.Of course, for different systems or for similar systems in different ranges of ex-ternal conditions some other sources of decoherence may prevail, for example,noise due to hopping of charge carries on nearby traps, studied in [62, 63], ordue to the electron-electron interaction [64].Semiconductor double quantum dot creates three-dimensional double wellconfinement potential for electron in it. Let us denote the line connecting centersof the dots as the x -axis. Then the electron confining potential along x , isschematically shown in Fig. 1. The nanostructure is composed of two quantumdots with a potential barrier between them. Parameters of the structure areproperly adjusted so that two lower energy levels of spatial quantization lie veryclose to each other compared to the external temperature and to the distancesto higher energy levels. Therefore hopping of the electron to higher levels issuppressed. The electron is treated as a superposition of two basis states, | i and | i , corresponding to “false” and “true” in Boolean logic, ψ = αψ + βψ . (1)It should be noted that the states that define the “logical” basis are not theground and first excited states of the double-dot system. Instead, ψ (the “0”state of the qubit) is chosen to be localized at the first quantum dot and, ina zeroth order approximation, be similar to the ground state of that dot if itwere isolated. Similarly, ψ (the “1” state) resembles the ground state of the3econd dot (if it were isolated). This assumes that the dots are sufficiently (butnot necessarily exactly) symmetric. We denote the coordinates of the potentialminima of the dots (dot centers) as vectors R and R , respectively. Theseparation between the dot centers is L = R − R . (2)The Hamiltonian of an electron interacting with a phonon bath consists ofthree terms H = H e + H p + H ep . (3)The electron term is H e = − ε A ( t ) σ x − ε P ( t ) σ z , (4)where σ x and σ z are Pauli matrices, whereas ε A ( t ) and ε P ( t ) can have time-dependence, as determined by unitary single-qubit quantum gate-functions to beimplemented for specific quantum algorithm. This can be achieved by adjustingthe potential on the metallic nanogates surrounding the double-dot system. Forconstant ε A and ε P , the energy splitting between the electron energy levels is ε = q ε A + ε P . (5)The Hamiltonian term of the phonon bath is described by H p = X q ,λ ¯ hω q b † q ,λ b q ,λ , (6)where b † q ,λ and b q ,λ are the creation and annihilation operators of phonons,respectively, with the wave vector q and polarization λ . We approximate theacoustic phonon spectrum as isotropic one with a linear dispersion ω q = sq, (7)where s is the speed of sound in the semiconductor crystal.In the next few paragraphs we show that the electron-phonon interactioncan be expressed as H ep = X q ,λ σ z (cid:16) g q ,λ b † q ,λ + g ∗ q ,λ b q ,λ (cid:17) , (8)with the coupling constants g q ,λ determined by the geometry of the double-dot and the properties of the material. The derivation follows [53, 54]. Thepiezoacoustic electron-phonon interaction [65] is given by H ep = i X q ,λ s ¯ h ρsqV M λ ( q ) F ( q )( b q + b †− q ) , (9)4here ρ is the density of the semiconductor, V is volume of the sample, and forthe matrix element M λ ( q ), one can derive M λ ( q ) = 12 q X ijk ( ξ i q j + ξ j q i ) q k M ijk . (10)Here ξ j are the polarization vector components for polarization λ , while M ijk express the electric field as a linear response to the stress, E k = X ij M ijk S ij . (11)For a crystal with zinc-blende lattice, like GaAs, the tensor M ijk has only thosecomponents non-zero for which all three indexes i , j , k are different; furthermore,all these components are equal, M ijk = M . Thus, we have M λ ( q ) = Mq ( ξ q q + ξ q q + ξ q q ) . (12)The form factor F ( q ) accounting for that we are working with electronswhich are not usual plane waves, is given by F ( q ) = X j,k c † j c k Z d rφ ∗ j ( r ) φ k ( r ) e − i q · r , (13)where c k , c † j are the annihilation and creation operators of the basis states k, j = 0 ,
1. In quantum dots formed by a repulsive potential of nearby gates, anelectron is usually confined near the potential minima, which are approximatelyparabolic. Therefore the ground states in each dot have Gaussian shape φ j ( r ) = e −| r − R j | / a a / π / , (14)where 2 a is a characteristic size of the dots.We assume that the distance between the dots, L = | L | , is sufficiently largecompared to a , and that the different dot wave functions do not strongly overlap, (cid:12)(cid:12)(cid:12)(cid:12) Z d rφ ∗ j ( r ) φ k ( r ) e − i q · r (cid:12)(cid:12)(cid:12)(cid:12) ≪ , for j = k. (15)In other words tunneling between the dots is small, as is the case for the re-cently studied experimental structures [55, 66, 67, 68], where the splitting dueto tunneling, measured by ε A , was just several tens of µ eV, while the electronquantization energy in each dot was at least several meV.For j = k , we obtain Z d rφ ∗ j ( r ) φ j ( r ) e − i q · r = 1 a π / Z d re −| r − R j | /a e − i q · r e − i q · R j e − a q / . (16)The resulting form factor is F ( q ) = e − a q / e − i q · R ( c † c e i q · L / + c † c e − i q · L / ) , (17)where R = ( R + R ) /
2. Therefore F ( q ) = e − a q / e − i q · R [cos( q · L / I + i sin( q · L / σ z ] , (18)where I is the identity operator. Only the last term in (18) represents aninteraction affecting the qubit states. It leads to a Hamiltonian term of theform (8), with coupling constants g q ,λ = − s ¯ h ρqsV M e − a q / − i q · R × ( ξ e e + ξ e e + ξ e e ) sin( q · L / , (19)where e k = q k /q .The general form of qubit evolution controlled by the Hamiltonian term (4)is time dependent. Decoherence estimates for some solid-state systems withcertain shapes of time dependence of the system Hamiltonian were reportedrecently [38, 69, 70]. However, such estimations are rather sophisticated. Toavoid this difficulty we observe that all single-qubit rotations which are requiredfor quantum algorithms can be successfully performed by using two constant-Hamiltonian gates without loss of quantum speed-up, e.g., by amplitude rotationgate and phase shift gate [71]. To implement these gates one can keep the Hamil-tonian term (4) constant during the implementation of each gate, adjusting theparameters ε A and ε P as appropriate for each gate and for the idling qubit inbetween gate functions. In the next paragraph we initiate our considerationof decoherence during the implementation of the NOT amplitude gate. Thenconsider π -phase shift gate later in the section.The quantum NOT gate is a unitary operator which transforms the states | i and | i into each other. Any superposition of | i and | i transforms accordingly,NOT ( α | i + β | i ) = β | i + α | i . (20)The NOT gate can be implemented by properly choosing ε A and ε P in theHamiltonian term (4). Specifically, with constant ε A = ε (21)and ε P = 0 , (22)the “ideal” NOT gate function is carried out, with these interaction parameters,over the time interval τ = π ¯ hε . (23)6he major source of quantum noise for double-dot qubit subject to the NOT-gate type coupling, is relaxation involving energy exchange with the phononbath (i.e., emission and absorption of phonons). Here it is more convenient tostudy the evolution of the density matrix in the energy basis, {| + i , |−i} , where |±i = ( | i ± | i ) / √ . (24)Then, assuming that the time interval of interest is [0 , τ ], the qubit densitymatrix can be expressed [41] in the energy basis as ρ ( t ) = ρ th ++ + (cid:2) ρ ++ (0) − ρ th ++ (cid:3) e − Γ t ρ + − (0) e − (Γ / − iε/ ¯ h ) t ρ − + (0) e − (Γ / iε/ ¯ h ) t ρ th −− + (cid:2) ρ −− (0) − ρ th −− (cid:3) e − Γ t . (25)This is a standard Markovian approximation for the evolution of the densitymatrix. For large times, this type of evolution would in principle result in thethermal state, with the off-diagonal density matrix elements decaying to zero,while the diagonal ones approaching the thermal values proportional to theBoltzmann factors corresponding to the energies ± ε/
2. However, here we areonly interested in such evolution for a relatively short time interval, τ , of a NOTgate. The rate parameter Γ is simply the sum [41] of the phonon emission rate, W e , and absorption rate, W a , Γ = W e + W a . (26)The probability for the absorption of a phonon due to excitation from theground state to the upper level is w λ = 2 π ¯ h |h f | H ep | i i| δ ( ε − ¯ hsq ) , (27)where | i i is the initial state with the extra phonon with energy ¯ hsq and | f i isthe final state, q is the wave vector, and λ is the phonon polarization. Thus,we have to calculate W a = X q ,λ w λ = V (2 π ) X λ Z d q w λ . (28)For the interaction (8) one can derive w λ = 2 π ¯ h | g q ,λ | N th δ ( ε − ¯ hsq ) , (29)where N th = 1exp(¯ hsq/k B T ) − T , and k B is the Boltzmannconstant. 7he coupling constant in (19) depends on the polarization if the interactionis piezoelectric. For longitudinal phonons, the polarization vector has Cartesiancomponents, expressed in terms of the spherical-coordinate angles, ξ k = e = sin θ cos φ, ξ k = e = sin θ sin φ, ξ k = e = cos θ, (31)where e j = q j /q . For transverse phonons, it is convenient to define the twopolarization vectors ξ ⊥ i and ξ ⊥ i to have ξ ⊥ = sin φ, ξ ⊥ = − cos φ, ξ ⊥ = 0 , (32) ξ ⊥ = − cos θ cos φ, ξ ⊥ = − cos θ sin φ, ξ ⊥ = sin θ. (33)Then for longitudinal phonons, one obtains [54] w k = πρsV q M e − a q / (34) × θ cos θ sin φ cos φ sin ( qL cos θ/ . For transverse phonons, one gets w ⊥ = πρsV q M e − a q / ( − θ cos θ sin φ cos φ + sin θ cos φ sin φ ) sin ( qL cos θ/ , (35) w ⊥ = πρsV q M e − a q / ( − θ cos θ cos φ + sin θ cos θ sin φ ) sin ( qL cos θ/ . (36)By combining these contributions and substituting them in (28), one can obtainthe probability of absorption of a phonon for all polarizations, W a piezo = M πρs ¯ hL k exp (cid:16) − a k (cid:17) exp (cid:16) ¯ hskk B T (cid:17) − × n ( kL ) + 5 kL h kL ) − i cos ( kL )+ 15 h − kL ) i sin ( kL ) o , where k = ε ¯ hs (38)is the wave-vector of the absorbed phonon.Finally, the expressions for the phonon emission rates, W e , can be obtainedby multiplying the above expression, (37), by ( N th + 1) /N th .The π phase gate is a unitary operator which does not change the absolutevalues of the probability amplitudes of a qubit in the superposition of the | i and8 i basis states. instead it increases the relative phase between the probabilityamplitudes by π angle. Consequently, superposition of | i and | i transformsaccording to Π ( α | i + β | i ) = α | i − β | i . (39)Over a time interval τ , the π gate can be carried out with constant interactionparameters, ε A = 0 (40)and ε P = ε = π ¯ hτ . (41)Charge qubit dynamics during implementation of phase gates was investi-gated in [53]. The relaxation dynamics is suppressed during the π gate, becausethere is no tunneling between the dots. The main quantum noise then resultsdue to pure dephasing. It leads to the decay of the off-diagonal qubit densitymatrix elements, while keeping the diagonal density matrix elements unchanged.The qubit density matrix can be represented in this regime as [72, 73] ρ ( t ) = ρ (0) ρ (0) e − B ( t )+ iεt/ ¯ h ρ (0) e − B ( t ) − iεt/ ¯ h ρ (0) , (42)with the spectral function, B ( t ) = 8¯ h X q ,λ | g q ,λ | ω q sin ω q t hω q k B T = V ¯ h π Z d q X λ | g q ,λ | q s sin qst hqs k B T . (43)For the piezoelectric interaction, the coupling constant g q ,λ was obtained in(19), and expression for the spectral function is B ( t ) = M π ¯ hρs Z ∞ q dq Z π sin θdθ Z π dϕ × X λ ( ξ λ e e + ξ λ e e + ξ λ e e ) q exp( − a q / × sin ( qL cos θ ) sin qst hqs k B T . (44)In summary, in this section we obtained the leading-order expressions for thesemiconductor double-dot qubit density matrix in the presence of decoherencedue to piezoelectric interaction with acoustic phonons during implementationof amplitude and phase gates. 9
Quantification of Decoherence
Quantum information processing at the level of qubits and few-qubit registers,assumes near coherent evolution, which is at best achievable at short to inter-mediate times. Therefore attention has recently shifted from large-time systemdynamics in the regime of onset of thermalization, to almost perfectly coherentdynamics at shorter times. Since many quantum systems proposed as candidatesfor qubits for practical realizations of quantum computing require estimation oftheir coherence, quantitative characterization of decoherence is crucially impor-tant for quantum information processing [4, 5, 6, 46, 47, 48, 49, 50, 52, 55, 60,61, 66, 67, 68, 71, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90,91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102]. A single measure characterizingdecoherence is highly desirable for comparison of different qubit designs. Besidesthe evaluation of single qubit performance one also has to analyze scaling of de-coherence as the register size (the number of qubits involved) increases. Directquantitative calculations of decoherence of even few-qubit quantum registers arenot feasible. Therefore, a practical approach has been to explore quantitativemeasures of decoherence [100], develop techniques to calculate such measures atleast approximately for realistic one- and two-qubit systems [53, 54], and thenestablish scaling (additivity) [101, 102]) for several-qubit quantum systems.In this section, we outline different approaches to define and quantify de-coherence. We argue that a measure based on a properly defined as a certainoperator norm of deviation of the density matrix from ideal, is the most appro-priate for quantifying decoherence in quantum registers.We consider several approaches to generally quantifying the degree of deco-herence due to interactions with environment. We first mention the approachbased on the asymptotic relaxation time scales. The entropy and idempotency-defect measures are then reviewed. The fidelity measure of decoherence is con-sidered next. Finally, we introduce our operator norm measure of decoherence.Furthermore, we discuss an approach to eliminate the initial-state dependenceof the decoherence measures.Markovian approximation schemes typically yield exponential approach tothe limiting values of the density matrix elements for large times [40, 41, 42].For a two-state system, this defines the time scales T and T , associated, re-spectively, with the approach by the diagonal (thermalization) and off-diagonal(dephasing, decoherence) density-matrix elements to their limiting values. Moregenerally, for large times we approximate deviations from stationary values ofthe diagonal and off-diagonal density matrix elements as ρ kk ( t ) − ρ kk ( ∞ ) ∝ e − t/T kk , (45) ρ jk ( t ) ∝ e − t/T jk ( j = k ) . (46)The shortest time among T kk is often identified as T . Similarly, T can bedefined as the shortest time among T n = m . These definitions yield the charac-teristic times of thermalization and decoherence (dephasing).10nfortunately the exponential behavior of the density matrix elements in theenergy basis is applicable only for large times, whereas for quantum computingapplications, the short-time behavior is usually relevant [31]. Moreover, whilethe energy basis is natural for large times, the choice of the preferred basisis not obvious for short and intermediate times [31, 72]. Therefore, the timescales T and T have limited applicability for evaluating coherence in quantumcomputing.An alternative approach is based on the calculation of the entropy [39] ofthe system, S ( t ) = − Tr ( ρ ln ρ ) , (47)or the first order entropy (idempotency defect) [103, 104, 105], s ( t ) = 1 − Tr (cid:0) ρ (cid:1) . (48)Both expressions are basis independent, have a minimum at pure states andeffectively describe the degree of the state’s “purity.” Any deviation from apure state leads to the deviation from the minimal values, 0, for both measures, S pure state ( t ) = s pure state ( t ) = 0 . (49)Unfortunately, entropy measures the deviation from pure-state evolution ratherthan deviation from a specific ideal evolution.The fidelity measure, considered presently, has been widely used. If theHamiltonian of the system and environment is H = H S + H B + H I , (50)where H S is the internal system dynamics, H B gives the evolution of envi-ronment (bath), and H I describes system-bath interaction, then the fidelitymeasure [106, 107] can be defined as, F ( t ) = Tr S [ ρ ideal ( t ) ρ ( t ) ] . (51)Here the trace is over the system degrees of freedom, and ρ ideal ( t ) representsthe pure-state evolution of the system under H S only, without interaction withthe environment ( H I = 0). In general, the Hamiltonian term H S governing thesystem dynamics can be time dependent. For the sake of simplicity throughoutthis review we consider constant H S over time intervals of quantum gates, cf.Section 2. In this case ρ ideal ( t ) = e − iH S t ρ (0) e iH S t . (52)More sophisticated scenarios with qubits evolving under time dependent H S were considered in [38, 69, 70].The fidelity provides a measure of decoherence in terms of the differencebetween the “real,” environmentally influenced evolution, ρ ( t ), and the “ideal”evolution, ρ ideal ( t ). It will attain its maximal value, 1, only provided ρ ( t ) =11 ideal ( t ). This property relies on the added assumption the ρ ideal ( t ) remains aprojection operator (pure state) for all times t ≥ ρ ideal ( t ) = (cid:18) (cid:19) , (53) ρ ( t ) = (cid:18) − e − Γ t e − Γ t (cid:19) , (54)and the fidelity is monotonic, F ( t ) = e − Γ t . (55)Note that the requirement that ρ ideal ( t ) is a pure-state (projection operator),excludes, in particular, any T > ρ (0) = ρ ideal ( t ) = (cid:18) / / (cid:19) , (56)which is not a projection operator. The spontaneous-decay density matrix isthen ρ ( t ) = (cid:18) − ( e − Γ t /
2) 00 e − Γ t / (cid:19) . (57)The fidelity remains constant F ( t ) = 1 / , (58)and it does not provide any information of the time dependence of the decayprocess.Let us now consider the operator norms [108] that measure the deviationof the system from the ideal state, to quantify the degree of decoherence, asproposed in [100, 101, 102]. Such measures do not require the initial densitymatrix to be pure-state. We define the deviation according to σ ( t ) ≡ ρ ( t ) − ρ ideal ( t ) . (59)We can use, for instance, the eigenvalue norm [108], k σ k λ = max i | λ i | , (60)or the trace norm, k σ k Tr = X i | λ i | , (61)etc., where λ i are the eigenvalues of the deviation operator (59). Since densityoperators are Hermitian and bounded, their norms, as well the norm of the12eviation, can be always defined and evaluated by using the expressions shown,avoiding the more formal mathematical definitions. We also note that k A k = 0implies that A = 0.The calculation of these norms is sometimes simplified by the observationthat σ ( t ) is traceless. Specifically, for two-level systems, we get k σ k λ = q | σ | + | σ | = 12 k σ k Tr . (62)For our example of the two-level system undergoing spontaneous decay, thenorm is k σ k λ = 1 − e − Γ t . (63)The measures considered above quantify decoherence of a system providedthat its initial state is given. However, in quantum computing, it is impracticalto keep track of all the possible initial states for each quantum register, thatmight be needed for implementing a particular quantum algorithm. Further-more, even the preparation of the initial state can introduce additional noise.Therefore, for evaluation of fault-tolerance (scalability), it will be necessary toobtain an upper-bound estimate of decoherence for an arbitrary initial state.To characterize decoherence for an arbitrary initial state, pure or mixed, weproposed [100] to use the maximal norm, D , which is determined as an operatornorm maximized over all the initial density matrices(the worst case scenarioerror estimate), D ( t ) = sup ρ (0) (cid:18) k σ ( t, ρ (0)) k λ (cid:19) . (64)For realistic two-level systems coupled to various types of environmentalmodes, the expressions of the maximal norm are surprisingly elegant and com-pact. They are usually monotonic and contain no oscillations due to the internalsystem dynamics. Most importantly, in the next section we will establish the additivity property of the maximal norm of deviation measure.Here we conclude by presenting the expressions for this measure for the twogates for the semiconductor double-dot system introduced in preceding section.The qubit error measure, D , was obtained from the density matrix deviationfrom the “ideal” evolution by using the operator norm approach [100]. Afterlengthy calculations, one gets [53] relatively simple expressions for the NOTgate, D NOT = 1 − e − Γ τ e − ε/k B T , (65)and for the π gate, D π = 12 h − e − B ( τ ) i , (66)where all the parameters were defined in Section 2. A realistic “general” noiseestimate per typical quantum-gate cycle time τ , could be taken as the larger ofthese two expressions. 13 Additivity of the Decoherence Measure
In the study of decoherence of several-qubit systems, one has to consider thedegree to which noisy environments of different qubits are correlated [73, 101,109]. Furthermore, if all constituent qubits are interacting with the same bath,then there are methods to reduce decoherence without quantum error correction,by instead encoding the state of one logical qubit in a decoherence-free subspaceof the states of several physical qubits [51, 73, 110, 111, 112]. In this section,we will consider several-qubit system and assume the “worst case scenario,”i.e., that the qubits experience uncorrelated noise, and each is coupled to aseparate bath. Since analytical calculations for several qubits are impractical,we have to find some “additivity” properties that will allow us to estimate theerror measure for the whole system from the error measures of the constituentqubits. For a general class of decoherence processes, including those occurring insemiconductor qubits considered in Section 2, we argue that maximal deviationnorm measure introduced in Section 3 is additive.The decoherence dynamics of a multiqubit system is rather complicated. Theloss of quantum coherence results also in the loss of two-particle and several-particle entanglements in the system. The higher order (multi-qubit) entangle-ments are “encoded” in the far off-diagonal elements of the multi-qubit registerdensity matrix, and therefore these quantum correlations will decay at least asfast as the products of the decay factors for the qubits involved, as exemplifiedby several explicit calculations [36, 113, 114, 115]. This observation supportsthe conclusion that at large times the rates of decay of coherence of the qubitswill be additive.However, here we seek a different result. We look for additivity propertywhich is valid not in the regime of the asymptotic large-time decay of quantumcoherence, but for short times, τ , of quantum gate functions, when the noiselevel, namely the value of the measure D ( τ ) for each qubit, is relatively small. Inthis regime, we will establish [101]: even for strongly entangled qubits — whichare important for the utilization of the power of quantum computation — theerror measures D of the individual qubits in a quantum register are additive.Thus, the error measure for a register made of similar qubits, scales up linearlywith their number, consistent with other theoretical and experimental observa-tions [106, 116, 117].Thus, to characterize decoherence for an arbitrary initial state, pure ormixed, we use the maximal norm, D , which was defined (64) as an operatornorm maximized over all the possible initial density matrices. One can showthat 0 ≤ D ( t ) ≤
1. This measure of decoherence will typically increase mono-tonically from zero at t = 0, saturating at large times at a value D ( ∞ ) ≤
1. Thedefinition of the maximal decoherence measure D ( t ) looks rather complicatedfor a general multiqubit system. However, it can be evaluated in closed form forshort times, appropriate for quantum computing, for a single-qubit (two-state)system. We then establish an approximate additivity that allows us to estimate D ( t ) for several-qubit systems as well.The evolution of the reduced density operator of the system (51) and the14ne for the ideal density matrix (52) can be formally expressed [71, 96, 97] inthe superoperator notation as ρ ( t ) = T ( t ) ρ (0) , (67) ρ ( i ) ( t ) = T ( i ) ( t ) ρ (0) , (68)where T , T ( i ) are linear superoperators. The deviation matrix can be expressedas σ ( t ) = h T ( t ) − T ( i ) ( t ) i ρ (0) . (69)The initial density matrix can decomposed as follows, ρ (0) = X j p j | ψ j ih ψ j | , (70)where P j p j = 1 and 0 ≤ p j ≤
1. Here the wavefunction set | ψ j i is not assumedto have any orthogonality properties. Then, we get σ ( t, ρ (0)) = X j p j h T ( t ) − T ( i ) ( t ) i | ψ j i h ψ j | . (71)The deviation norm can thus be bounded, k σ ( t, ρ (0)) k λ ≤ (cid:13)(cid:13)(cid:13)h T ( t ) − T ( i ) ( t ) i | φ ih φ | (cid:13)(cid:13)(cid:13) λ . (72)Here | φ i is defined according to (cid:13)(cid:13)(cid:13)h T − T ( i ) i | φ ih φ | (cid:13)(cid:13)(cid:13) λ = max j (cid:13)(cid:13)(cid:13)h T − T ( i ) i | ψ j ih ψ j | (cid:13)(cid:13)(cid:13) λ . (73)For any initial density operator which is a statistical mixture, one can alwaysfind a density operator which is pure-state, | φ ih φ | , such that k σ ( t, ρ (0)) k λ ≤k σ ( t, | φ ih φ | ) k λ . Therefore, evaluation of the supremum over the initial densityoperators in order to find D ( t ), see (64), can be done over only pure-state densityoperators, ρ (0).Consider briefly strategies of evaluating D ( t ) for a single qubit. We canparameterize ρ (0) as ρ (0) = U (cid:18) P
00 1 − P (cid:19) U † , (74)where 0 ≤ P ≤
1, and U is an arbitrary 2 × U = (cid:18) e i ( α + γ ) cos θ e i ( α − γ ) sin θ − e i ( γ − α ) sin θ e − i ( α + γ ) cos θ (cid:19) . (75)Then, one should find a supremum of the norm of deviation (60) over all thepossible real parameters P , α , γ and θ . As shown above, it suffices to consider15he density operator in the form of a projector and put P = 1. Thus, one shouldsearch for the maximum over the remaining three real parameters α , γ and θ .Another parameterization of the pure-state density operators, ρ (0) = | φ ih φ | ,is to express an arbitrary wave function | φ i = P j ( a j + ib j ) | j i in some convenientorthonormal basis | j i , where j = 1 , . . . , N . For a two-level system, ρ (0) = (cid:18) a + b ( a + ib )( a − ib )( a − ib )( a + ib ) a + b (cid:19) , (76)where the four real parameters a , , b , satisfy a + b + a + b = 1, so that themaximization is again over three independent real numbers. The final expres-sions (65) and (66) for D ( t ), for our selected single-qubit systems considered inSection 2, are actually quite compact and tractable.In quantum computing, the error rates can be significantly reduced by usingseveral physical qubits to encode each logical qubit [51, 110, 111]. Therefore,even before active quantum error correction is incorporated [87, 88, 89, 90, 91,92, 93, 94, 95], evaluation of decoherence of several qubits is an important, butformidable task. Here our aim is to prove the approximate additivity of D q ( t ),including the case of the initially entangled qubits, labeled by q , whose dynamicsis governed by H = X q H q = X q ( H Sq + H Bq + H Iq ) , (77)where H Sq is the Hamiltonian of the q th qubit itself, H Bq is the Hamiltonian ofthe environment of the q th qubit, and H Iq is corresponding qubit-environmentinteraction. We consider a more complicated (for actual evaluation) diamondnorm [71, 96, 97], as an auxiliary quantity used to establish the additivity ofthe more easily calculable operator norm D ( t ).The establishment of the upper-bound estimate for the maximal deviationnorm of a multiqubit system, involves several steps. We first derive a bound forthis norm in terms of the diamond norm. Actually, for single qubits, in severalmodels the diamond norm can be expressed via the corresponding maximaldeviation norm. At the same time, the diamond norm for the whole quantumsystem is bounded by sum of the norms of the constituent qubits by using acertain specific stability property of the diamond norm, K ( t ). This norm isdefined as K ( t ) = k T − T ( i ) k ⋄ = sup ̺ k{ [ T − T ( i ) ] ⊗ I } ̺ k Tr . (78)The superoperators T , T ( i ) characterize the actual and ideal evolutions accord-ing to (67), (68). Here I is the identity superoperator in a Hilbert space G whosedimension is the same as that of the corresponding space of the superoperators T and T ( i ) , and ̺ is an arbitrary density operator in the product space of twicethe number of qubits.The diamond norm has an important stability property, proved in [71, 96,97], k B ⊗ B k ⋄ = k B k ⋄ k B k ⋄ . (79)16ote that (79) is a property of the superoperators rather than that of the op-erators.Consider a composite system consisting of two subsystems S , S , with thenoninteracting Hamiltonian H S S = H S + H S . (80)The evolution superoperator of the system will be T S S = T S ⊗ T S , (81)and the ideal one T ( i ) S S = T ( i ) S ⊗ T ( i ) S . (82)The diamond measure for the system can be expressed as K S S = k T S S − T ( i ) S S k ⋄ = k ( T S − T ( i ) S ) ⊗ T S + T ( i ) S ⊗ ( T S − T ( i ) S ) k ⋄ ≤ k ( T S − T ( i ) S ) ⊗ T S k ⋄ + k T ( i ) S ⊗ ( T S − T ( i ) S ) k ⋄ . (83)By using the stability property (79), we get K S S ≤ k ( T S − T ( i ) S ) ⊗ T S k ⋄ + k T ( i ) S ⊗ ( T S − T ( i ) S ) k ⋄ = k T S − T ( i ) S k ⋄ k T S k ⋄ + k T ( i ) S k ⋄ k T S − T ( i ) S k ⋄ = k T S − T ( i ) S k ⋄ + k T S − T ( i ) S k ⋄ = K S + K S . (84)The inequality K ≤ X q K q , (85)for the diamond norm K ( t ) has thus been obtained. Let us emphasize thatthe subsystems can be initially entangled. This property is particularly usefulfor quantum computing, the power of which is based on qubit entanglement.However, even in the simplest case of the diamond norm of one qubit, thecalculations are extremely cumbersome. Therefore, the use of the measure D ( t )is preferable for actual calculations.For short times, of quantum gate functions, we can use (85) as an approxi-mate inequality for order of magnitude estimates of decoherence measures, evenwhen the qubits are interacting. Indeed, for short times, the interaction effectswill not modify the quantities entering both sides significantly. The key pointis that while the interaction effects are small, this inequality can be used for strongly entangled qubits.The two deviation-operator norms considered are related by the followinginequality k σ k λ ≤ k σ k Tr ≤ . (86)Here the left-hand side follows fromTr σ = X j λ j = 0 . (87)17herefore the ℓ th eigenvalue of the deviation operator σ that has the maximumabsolute value, λ ℓ = λ max , can be expressed as λ ℓ = − X j = ℓ λ j . (88)Thus, we have k σ k λ = 12 (2 | λ ℓ | ) ≤ | λ ℓ | + X j = ℓ | λ j | = 12 X j | λ j | = 12 k σ k Tr . (89)The right-hand side of (86) then also follows, because any density matrix hastrace norm 1, k σ k Tr = k ρ − ρ ( i ) k Tr ≤ k ρ k Tr + k ρ ( i ) k Tr = 2 . (90)From the relation (90) it follows that K ( t ) ≤ . (91)By taking the supremum of both sides of the relation (89) we get D ( t ) = sup ρ (0) k σ k λ ≤
12 sup ρ (0) k σ k Tr ≤ K ( t ) , (92)where the last step involves technical derivation details [101] not reproducedhere. In fact, for a single qubit, calculations for typical qubit models [101] give D q ( t ) = 12 K q ( t ) . (93)Since D is generally bounded by (or equal to) K/
2, it follows that the multiqubitnorm D is approximately bounded from above by the sum of the single-qubitnorms even for the initially entangled qubits, D ( t ) ≤ K ( t ) ≤ X q K q ( t ) = X q D q ( t ) , (94)where q labels the qubits.For specific models of decoherence of the type encountered in Section 2, aswell as those formulated for general studies of short-time decoherence [100], astronger property has been demonstrated by deriving additional bounds notreviewed here [101], namely that the noise measures are actually equal, for lowlevels of noise, D ( t ) = X q D q ( t ) + o X q D q ( t ) ! . (95)Thus, in this section we considered the maximal operator norm suitable forevaluation of decoherence for a quantum register consisting of qubits immersed18n noisy environments. We established the approximate additivity property ofthis measure of decoherence for multi-qubit registers at short times, for whichthe level of quantum noise is low, and the qubit-qubit interaction effects aresmall, but without any limitation on the initial entanglement of the qubit reg-ister.In conclusion, we surveyed the theory of evaluation of quantum noise ef-fects for quantum registers. Maximal deviation norm was proposed for errorestimation and its expressions were presented for a realistic model of semicon-ductor double-dot qubit interacting with acoustic phonons. Maximal deviationnorm has a unique additivity property which facilitates error rate estimationfor several-qubit registers. Acknowledgments
We are grateful to A. Fedorov, D. Mozyrsky, D. Solenov, I. Vagner, and D. Tolkunovfor collaborations and instructive discussions. This research was supported bythe National Science Foundation, grant DMR-0121146.
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