Effects of uncertainties and errors on Lyapunov control
aa r X i v : . [ qu a n t - ph ] O c t Effects of uncertainties and errors on Lyapunov control
X. X. Yi , , B. Cui , Chunfeng Wu , and C. H. Oh School of Physics and Optoelectronic TechnologyDalian University of Technology, Dalian 116024 China Centre for Quantum Technologies and Department of Physics,National University of Singapore, 117543, Singapore
Lyapunov control (open-loop) is often confronted with uncertainties and errors in practical appli-cations. In this paper, we analyze the robustness of Lyapunov control against the uncertainties anderrors in quantum control systems. The analysis is carried out through examinations of uncertaintiesand errors, calculations of the control fidelity under influences of the certainties and errors, as wellas discussions on the caused effects. Two examples, a closed control system and an open controlsystem, are presented to illustrate the general formulism.
PACS numbers: 03.65.-w, 03.67.Pp, 02.30.Yy
I. INTRODUCTION
Quantum control is the manipulation of the temporalevolution of a system in order to obtain a desired targetstate or value of a certain physical observable, realizingit is a fundamental challenge in many fields [1–3], in-cluding atomic physics [4], molecular chemistry [5] andquantum information [6]. Several strategies of quantumcontrol have been introduced and developed from classi-cal control theory. For example, optimal control theoryhas been used to assist in control design for molecular sys-tems and spin systems [7, 8]. A learning control methodhas been presented for guiding the control of chemicalreactions [5]. Quantum feedback control approaches in-cluding measurement-based feedback and coherent feed-back have been used to improve performance for severalclasses of tasks such as preparing quantum states, quan-tum error correction and controlling quantum entangle-ment [9, 10]. Robust control tools have been introducedto enhance the robustness of quantum feedback networksand linear quantum stochastic systems [11, 12].Control systems are broadly classified as either closed-loop or open-loop. An open-loop control system is con-trolled directly, and only, by an input signal, whereasa closed-loop control system is one in which an inputforcing function is determined in part by the system re-sponse. Among the open-loop controls, Lyapunov controlhas been proven to be a sufficient control to be analyzedrigourously, moreover, this control can be shown to behighly effective for systems that satisfy certain sufficientconditions that roughly speaking are equivalent to thecontrollability of the linearized system.Lyapunov control for quantum systems in fact use afeedback design to construct an open-loop control. Inother words, Lyapunov control is used to first design afeedback law which is then used to find the open-loopcontrol by simulating the closed-loop system. Then thecontrol is applied to the quantum system in an open-loopway. From the above description of Lyapunov control,we find that the Lyapunov control includes two steps:(1) for any initial states and a system Hamiltonian (as- sumed to be known exactly), design a control law, i.e.,calculate the control field by simulating the dynamicsof the closed-loop system, (2) apply the control law tothe control system in an open-loop way. Although someprogress has been made, more research effort is necessaryin Lyapunov control, especially, the robustness of quan-tum control systems has been recognized as a key issuein developing practical quantum technology. In this pa-per, we study the effect of uncertainties and errors onthe performance of Lyapunov control. The uncertaintiescome from initial states and system Hamiltonian, and er-rors may occur in applying the control field (control law).Through this study, we show the robustness of Lyapunovcontrol against uncertainties and errors. In particular,the relation between the uncertainties and the fidelity isestablished for a closed two-level control system and anopen four-level control system.This paper is organized as follows. In Sec.II, we intro-duce the Lyapunov control and formulate the problem. Ageneral formulism is given to examine the robustness ofthe Lyapunov control. In Sec. III, we exemplify the gen-eral formulation in Sec.II through a closed and an openquantum control systems. Concluding remarks are givenin Sec. IV.
II. PROBLEM FORMULATION
A control quantum system can be modeled in differ-ent ways, either as a closed system evolving unitarilygoverned by a Hamiltonian, or as an open system gov-erned by a master equation. In this paper, we restrictour discussion to a N -dimensional open quantum system,and consider its dynamics as Markovian. The discus-sion is applicable for closed systems, since closed systemis a special case of open system with zero decoherencerates. Therefore we here consider a system that obeysthe Markovian master equation (¯ h = 1 , throughout thispaper), ˙ ρ = − i [ H, ρ ] + L ( ρ ) (1)with L ( ρ ) = 12 M X m =1 λ m ([ J m , ρJ † m ] + [ J m ρ, J † m ]) , and H = H + F X n =1 f n ( t ) H n , where λ m ( m = 1 , , ..., M ) are positive and time-independent parameters, which characterize the deco-herence and are called decoherence rates. Furthermore, J m ( m = 1 , , ..., M ) are the Lindblad operators, H isthe free Hamiltonian and H n ( n = 1 , , ..., F ) are controlHamiltonians, while f n ( t ) ( n = 1 , , ..., F ) are controlfields. Equation (1) is of Lindblad form, this means thatthe solution to Eq. (1) has all the required propertiesof physical density matrix at any times. Since the freeHamiltonian can usually not be turned off, we take non-stationary states ρ D ( t ) as target states that satisfy,˙ ρ D ( t ) = − i [ H , ρ D ( t )] . (2)The control fields { f n ( t ) , n = 1 , , , ... } can be estab-lished by Lyapunov function. Define V ( ρ D , ρ ), V ( ρ D , ρ ) = Tr( ρ D ) − Tr( ρρ D ) , (3)we find V ≥ V = − F X n f n ( t )Tr { ρ D [ − iH n , ρ ] } − Tr[ ρ D L ( ρ )] . (4)For V to be a Lyapunov function, it requires ˙ V ≤ V ≥ . If we choose a n such that f n ( t )Tr { ρ D [ − iH n , ρ ] } + Tr[ ρ D L ( ρ )] = 0, and f n ( t ) =Tr { ρ D [ − iH n , ρ ] } for n = n , then ˙ V ≤ . With thesechoices, V is a Lyapunov function. Therefore, the evolu-tion of the open system with Lyapunov control governedby the following nonlinear equations[15]˙ ρ ( t ) = − i [ H + X n f n ( t ) H n , ρ ( t )] + L ( ρ ) ,f n ( t ) = Tr { [ − iH n , ρ ] ρ D } , for n = n ,f n ( t ) = − Tr[ ρ D L ( ρ )]Tr { ρ D [ ρ, iH n ] } , and˙ ρ D ( t ) = − i [ H , ρ D ( t )] (5)is stable in Lyapunov sense at least. In Eqs (2) and (3),we have identified ρ D ( t ) with target states, this meansthat if a quantum system is driven into the target states,it will be maintained in these states under the action ofthe free Hamiltonian. However, in practical applications,it is inevitable that there exist errors and uncertainties inthe free Hamiltonian, in the initial states and in the con-trol fields. These uncertainties and errors would disturbthe dynamics and steer the system away from the target state. In the following, we suppose that the uncertaintiescan be approximately described as perturbations δH inthe free Hamiltonian, and as deviations δρ in the initialstate as well as fluctuations δf n ( n may take 1 , , , ... )in the control fields. Then the actual final state ρ R ( t ) ofthe control system starting from ( ρ + δρ ) governed byEq. (5) with ( H + δH ) and [ f n ( t ) + δf n ( t )] instead of H and f n ( t ) would be different from ρ D ( t ) . We quan-tify the difference between the target states ρ D ( t ) andthe practical states ρ R ( t ) by using the fidelity defined by F ( ρ D , ρ R ) = Tr q ρ D ρ R ρ D . For a Lyapunov control with negative gradient of Lya-punov function in the neighborhood of target states, thecontrolled system state will be attracted to and main-tained in the target state, when there are no uncertaintiesand errors. With uncertainties and errors, the problemof robustness of the control system is not trivial, becausethe Lyapunov-based feedback design for the control lawwould induce nonlinearity in the control system. TheLaSalle invariant principle[14] tells that the autonomousdynamical system Eq.(5) converges to an invariant setdefined by E = { ρ in : ˙ V = 0 } , which is equivalent to f n ( t ) = 0 , n = 1 , , , ... by Eq.(5). This set is in generalnot empty and the final state will be in it. From Eqs. (4)and (5) we find that the invariant set is an intersectionof all sets E n ( n = 1 , , , ..., n = n ), each satisfies, E n = { ρ in,n : Tr( ρ d H n ρ in,n − H n ρ d ρ in,n ) = 0 } , n = n . (6)Since the control fields are proportional to ˙ V , the errorsin the control fields would change the invariant set. Theuncertainties in the initial state affect the invariant setin the same way, and the uncertainties in the free Hamil-tonian change the target sets ρ d ( t ), leading to an invari-ant set different from that without uncertainties. In thenext section, we will illustrate and exemplify the effectof errors and uncertainties on the fidelity through simpleexamples. III. ILLUSTRATION
In this section, we first introduce a Lyapunov controlon a closed two-level quantum system, then we studythe robustness of this Lyapunov control by examiningthe effects of uncertainties and errors on the fidelity ofcontrol. Next, we extend this study into open systems byconsidering a dissipative four-level system and steering itto a target state in its decoherence-free subspace (DFS).We start with a closed two-level system described bythe Hamiltonian, H = ω σ z + f ( t ) σ x ≡ H + H , (7)where H = ω σ z denotes the free Hamiltonian of thesystem, H = f ( t ) σ x is the control Hamiltonian witha control field f ( t ) . We define one of the eigenstates of H , say the ground state | g i , as the target state, the δ x δ z F i de li t y δ φ [ π ] δ β [ π ] F i de li t y FIG. 1: (Color online) Fidelity of Lyapunov control versusthe uncertainties in the free Hamiltonian (left) and in theinitial states (right). ω = 4 (in arbitrary units) and φ = β = π are chosen for this plot. Lyapunov function in Eq. (3) for this closed system isthen V ( | g i , | Φ( t ) i ) = 1 − |h g | Φ( t ) i| . (8)For closed system, the Liouvillian L ( ρ ) vanishes, thus wedo not need to choose a control field f n ( t ) in Eq. (5)to cancel the drift term. The only control field f ( t ) thatcan be derived from Eq. (5) is, f ( t ) = 2Im( h g | σ x | Φ( t ) ih Φ( t ) | g i ) . (9)Here | Φ( t ) i represents states at time t starting from aninitial states | Φ(0) i = cos β | e i + sin β e iφ | g i , under the action of the Hamiltonian H without any un-certainties and errors. We further suppose that the un-certainties in the free Hamiltonian H can be describedas a perturbation, δH = δ x σ x + δ z σ z , (10)and the uncertainties in the initial state | Φ(0) i can becharacterized by replacing β and φ with ( β + δβ ) and( φ + δφ ), respectively. We describe the errors in thecontrol fields f ( t ) as fluctuations δ ( t ) · f ( t ) with randomnumber δ ( t ). With these descriptions, the practical con-trol system can be described by, i ¯ h ∂∂t | ψ ( t ) i R = [ H + δH + f ( t ) H (1 + δ ( t ))] | ψ ( t ) i R , (11)with initial condition | Φ(0) + δ Φ(0) i = cos( β + δβ ) | e i +sin( β + δβ ) e i ( φ + δφ ) | g i . We have performed numerical simulations for Eq. (11),selected results are presented in figures 1 and 2. Figure1 shows the control fidelity as a function of uncertain-ties ( δ x , δ y ) in the free Hamiltonian and uncertainties( δβ , δφ ) in the initial state. Two observations can bemade from the figures. (1) The control fidelity rapidlydepends on the uncertainties δ z σ z , whereas it is not sen-sitive to δ x σ x , (2) the control fidelity is an oscillatingfunction of δβ and δφ with different periods. These ob-servations indicate that the Lyapunov control on closed F i de li t y Time(a)(b)(c)
FIG. 2: (Color online) Fidelity of Lyapunov control as a func-tion of time. This plot shows the effects of fluctuations in thecontrol field f ( t ) on the fidelity. (a), (b) and (c) are for differ-ent types of fluctuations. (a) The fluctuation δ ( t ) was takenfrom ( −
1) to zero; (b) from ( −
1) to (+1); and (c) from 0to (+1). All fluctuations are taken randomly. The other pa-rameters chosen are the same as in Fig. 1. There are nouncertainties in the free Hamiltonian and in the initial states. systems is robust against the uncertainties that commutewith the control Hamiltonian, while it is fragile with theother uncertainties in the free Hamiltonian. This claimis confirmed by Fig. 2, where the effect of fluctuationsin the control field on the control fidelity is shown. Onecan clearly see from figure 2 that there are almost no ef-fects for the fluctuations with zero mean on the fidelity.This can be understood as follows. Since the fluctuationsis randomly chosen for the control fields, the net effectintrinsically equals to an average over all fluctuations,which must be zero for fluctuation with zero mean.Now we turn to another example that shows the ro-bustness of Lyapunov control on open quantum systems.We borrow the model in Ref.[16] shown in Fig.3, wherea four-level system coupling to two external lasers andbeing subject to decoherence has been considered. TheHamiltonian of this system has the form, H = X j =0 ∆ j | j ih j | + ( X j =1 Ω j | ih j | + h.c. ) , (12)where Ω j ( j = 1 ,
2) are coupling constants. Withoutloss of generality, in the following the coupling constantsare parameterized as Ω = Ω cos φ and Ω = Ω sin φ withΩ = p Ω + Ω . The excited state | i is not stable, itdecays to the three stable states with rates γ , γ and γ respectively. We assume this process is Markovian andcan be described by the Liouvillian, L ( ρ ) = X j =1 γ j ( σ − j ρσ + j − σ + j σ − j ρ − ρσ + j σ − j ) (13)with σ − j = | ih j | and σ + j = ( σ − j ) † . It is not difficult tofind that the two degenerate eigenstates | D i = cos φ | i− sin φ | i , | D i = | i , of the free Hamiltonian H form a FIG. 3: The schematic energy diagram. A four-level systemwith two degenerate stable states | i and | i in external laserfields. The two degenerate states are coupled to the excitedstate | i by two separate lasers with coupling constants Ω and Ω , respectively. While the stable state | i is isolatedfrom the other levels. The excited state | i decays to | j i ( j = 1 , ,
3) with decay rate γ j . DFS. Now we show how to control the system to a desiredtarget state (e.g., | D i ) in the DFS. For this purpose, wechoose the control Hamiltonian H c = X j =1 f j ( t ) H j (14)with H is a 4 by 4 matrix with all elements equal to 1. H = | D ih D | + | D ih D | , H = | ih D | + | D ih | . Weshall use Eq. (5) to determine the control fields { f n ( t ) } ,and choose | Ψ(0) i = sin β cos β | i + cos β cos β | i + cos β sin β | i + sin β sin β | i (15)as initial states for the numerical simulation, where β , β and β are allowed to change independently. The initialstate written in Eq.(15) omits all (three) relative phasesbetween the states | i , | i , | i and | i in the superposi-tion, and satisfies the normalization condition. f ( t ) hereis specified to cancel the drift term Tr[ L ( ρ ) ˆ A ] in ˙ V , thismeans that f ( t ) = − i Tr( L ( ρ ) ˆ A )Tr([ ˆ A,H ] ρ ) , f ( t ) and f ( t ) are de-termined by Eq.(5).We examine how the uncertainties in the free Hamil-tonian and initial states as well as the errors in the con-trol fields f n ( t ) , ( n = 1 , , , ... ) affect the fidelity of thecontrol. These effects can be illustrated by numericalsimulations on Eqs(12,13,14), with the free Hamiltonian H , the initial state | Φ(0) i and the control fields f n ( t ) re-placed by ( H + δH ), | Φ(0) + δ Φ(0) i and f n ( t )(1 + δ n ),respectively. Here, δH = ∆ x ( | ih | + | ih | ) + ∆ z ( | ih | + | ih | ) , | Φ(0) + δ Φ(0) i = sin( β + δβ ) cos β | i + cos( β + δβ ) cos( β + δβ ) | i + cos( β + δβ ) sin( β + δβ ) | i + sin( β + δβ ) sin β | i , (16) δ β δ β F i de li t y
0 0.3 0.5 0 0.50.20.951 ∆ x ∆ z F i de li t y FIG. 4: (Color online) The fidelity of control as a function ofuncertainties in the free Hamiltonian (left) and in the initialstate (right). The other parameters chosen are Ω = 5 , φ = π , β = π , β = π , β = π , γ = γ = γ = γ, κ = 1 , ∆ = 4 , ∆ = ∆ = 2 and γ = 1 . δβ and δβ are in unites of π. Time t F i de li t y (a)(a)(b)(b) FIG. 5: (Color online) Fidelity of Lyapunov control versustime t . The fluctuations δ n ( n = 1 , , , ... ) range from (-1) to(+1) for the upper panel, while from ( −
1) to 0 (or 0 to (+1))for the lower panel. (a) in both panels denotes the probabilityin state | D i , while (b) in state | D i . The other parameterschose are the same as in Fig. 4. and δ n are random numbers ranging from − x , ∆ z , δβ , δβ ) and errors ( δ n , n = 1 , , , ... ) are presented infigures 4 and 5. Figures 4 and 5 tell us that the Lya-punov control on open system with the target state | D i is robust against the uncertainties in the initial state, andthe fidelity is above 95% when the uncertainties in thefree Hamiltonian is bounded by 0.5 (in units of γ ). TheLyapunov control is also robust gainst the fluctuations inthe control fields f n ( t ) as figure 5 shows. We note thatthe effects of fluctuations with zero mean are differentfrom that with non-zero mean. This can understood asan average results taken over all fluctuations. IV. CONCLUDING REMARKS
To summarize, we have examined the robustness ofLyapunov control in quantum systems. The robustnessis characterized by the fidelity of the quantum state to thetarget state. Uncertainties in the free Hamiltonian and inthe initials states as well as the errors in the control fieldsdiminish the fidelity of control. The relation between theuncertainties (errors) and the fidelity is established fora closed two-level control system and an open four-levelcontrol system. These results show that the Lyapunovcontrol is robust against the type of uncertainties whichcommute with the control Hamiltonian, while it is fragileto the others. The fidelity is not sensitive to zero mean random fluctuations (white noise) in the control fields,but it really decreases due to the non-zero (positive ornegative) mean fluctuations.This work is supported by NSF of China under grant Nos61078011 and 10935010, as well as the National ResearchFoundation and Ministry of Education, Singapore underacademic research grant No. WBS: R-710-000-008-271. [1] D. Dong and I.R. Petersen, arXiv:0910.2350.[2] H.M. Wiseman and G.J. Milburn, Quantum Measure-ment and Control, Cambridge, England: Cambridge Uni-versity Press, 2010.[3] H. Rabitz, New Journal of Physics , 105030 (2009).[4] S. Chu, Nature , 206 (2002).[5] H. Rabitz, R. de Vivie-Riedle, M. Motzkus and K.Kompa, Science , 824 (2000).[6] M.A. Nielsen and I.L. Chuang, Quantum Computationand Quantum Information, Cambridge, England: Cam-bridge University Press, 2000.[7] N. Khaneja, R. Brockett and S.J. Glaser, Phys. Rev. A ,032308 (2001).[8] D. DAlessandro and M. Dahleh, IEEE Transactions onAutomatic Control , 866 (2001).[9] H.M. Wiseman and G.J. Milburn, Phys. Rev. Lett. ,548 (1993).[10] A.C. Doherty, S. Habib, K. Jacobs K, H. Mabuchi andS.M. Tan, Phys. Rev. A , 053803(2006).[12] M.R. James, H.I. Nurdin and I.R. Petersen, IEEE Trans-actions on Automatic Control
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