TTrap-free manipulation in the Landau-Zener system
Alexander Pechen
1, 2, ∗ and Nikolay Il’in Department of Chemical Physics, Weizmann Institute of Science, Rehovot 76100, Israel Steklov Mathematical Institute of Russian Academy of Sciences, Gubkina 8, Moscow 119991, Russia (Dated: November 12, 2018)The analysis of traps, i.e., locally but not globally optimal controls, for quantum control systemshas attracted a great interest in recent years. The central problem that has been remained open is todemonstrate for a given system either existence or absence of traps. We prove the absence of trapsand hence completely solve this problem for the important tasks of unconstrained manipulation ofthe transition probability and unitary gate generation in the Landau-Zener system—a system witha wide range of applications across physics, chemistry and biochemistry. This finding provides thefirst example of a controlled quantum system which is completely free of traps. We also discuss theimpact of laboratory constraints due to decoherence, noise in the control pulse, and restrictions onthe available controls which when being sufficiently severe can produce traps.
PACS numbers: 02.30.Yy, 32.80.QkKeywords: Quantum control landscapes, Landau-Zener system
I. INTRODUCTION
Manipulation by atomic and molecular systems is animportant branch of modern science with applicationsranging from optimal laser driven population transfer inatomic systems out to laser assisted control of chemicalreactions [1]. High interest is directed towards controlof the Landau-Zener (LZ) system—a two-state quantumsystem whose unitary evolution under the action of thecontrol ε ( t ) (e.g., shaped laser field) is governed by theequation˙ U εt = − i(∆ σ x + ε ( t ) σ z ) U εt , U εt =0 = I (1)where ∆ > σ x and σ z are the Pauli matrices. Thecase ε ( t ) = εt with constant ε was studied by Landau,Zener, St¨uckelberg, and Majorana [2]. This system hasbeen widely applied in physics, chemistry, and biochem-istry, e.g., for describing transfer of charge along with itsenergy [3], photosynthesis [4], atomic and molecular colli-sions, processes in plasma physics [5], Bose-Einstein con-densate [7], experimental realizations of qubits, etc. [6, 8–13].Controlled manipulation by a quantum system can beformulated as finding global maxima of a suitable objec-tive J ( ε ) associated to the system. For example, maxi-mizing the probability of transition from the initial state | i (cid:105) to a target final state | f (cid:105) at a final time T can bedescribed by maximizing J ( ε ) = P i → f = |(cid:104) f | U εT | i (cid:105)| . Acontrol which attains a local maximum of J can be foundeither numerically using the model of the system or ex-perimentally. In both circumstances, the first step of acommon procedure is to apply a trial pulse ε and obtainthe outcome J ( ε ), either numerically or measuring it inthe laboratory. The second step is to make various small ∗ Electronic address: [email protected]; URL: modifications of ε and find ε which produces maximumincrease in J . Then ε is used as a new trial pulse andthe procedure is repeated until no significant increase isproduced or a maximum number of iterations is reached.Of crucial practical importance is to know whether J ( ε ) has traps , i.e. local maxima with the values less thanthe global maximum, as necessary to properly choose be-tween local (e.g., gradient) and global optimization meth-ods [14–19]. Traps can strongly influence on both theo-retical and experimental quantum control studies—theydetermine the level of difficulty of controlling the systemand can significantly slow down or even completely pre-vent finding globally optimal controls. Whereas the anal-ysis of traps in manipulation by quantum systems has at-tracted high attention [20–28], no examples of trap-freequantum systems have been known. Only partial the-oretical results have been obtained stating the absenceof traps at special regular controls. This finding doesnot at all exclude the absence of traps that makes theproblem open since even a single trap may produce sig-nificant difficulties for the optimization if it has a largeattracting domain [29]. In this work we show that the LZsystem is trap-free and hence, for example, the only ex-trema of J ( ε ) = P i → f for this system are global maximaand minima. This finding provides the first example ofa trap-free quantum control system where unconstrainedlocal manipulations are always sufficient to find best con-trol pulses. Sufficiently strong constraints on the controlsmay destroy this property and in the end we discuss pos-sible limitations for the analysis due to decoherence, noisein the control pulses and limited tunability of the controlstrength and time scales in laboratory experiments. II. TRAPS AND CONTROL LANDSCAPES
Formally, a control field ε ( t ) is a trap for the objec-tive J ( ε ) if it is a local maximum, i.e., a maximum withthe value less than the global maximum, J ( ε ) < J max = a r X i v : . [ qu a n t - ph ] A p r max ε J ( ε ) (in this work we consider as control goal maxi-mizing the objective; if the goal is to minimize the objec-tive then traps are local minima). Answering the ques-tion whether traps exist for a given control problem iscrucial for determining proper algorithms and our abili-ties for finding optimal control fields. In the absence oftraps, local search algorithms should generally be ableto find globally optimal controls (exceptions may occurif the initial control is chosen exactly at a saddle point,where the gradient of the objective is zero). If the ob-jective has traps (perhaps even a single trap with largeattracting domain) then local search procedures may con-verge to local maxima instead of attaining a desired glob-ally optimal control and more sophisticated global searchmethods should be exploited for a successful optimiza-tion.Traps are critical points, i.e., the gradient ∇ J ε = 0 atany trap. Critical points for control objectives J ( ε ) = P i → f were studied in seminal works [20], where the ab-sence of traps was suggested. The suggestion was drawnfrom the proof that any function of the form f ( U ) =Tr[ U ρ U † O ] ( ρ is a positive matrix and O is Hermitian)defined on the unitary group U ( n ), where n is the systemdimension, has as extrema only global maxima, globalminima, and saddles and has no traps. (Extrema of tracefunctions over unitary and orthogonal groups were stud-ied in other contexts by J. von Neumann [30], R. Brock-ett [31], S. Glaser et al [32], etc.) Then, under the con-trollability condition which assumes that any U ∈ U ( n )can be generated by some control, this result was used toconclude the absence of traps for the underlying objectivefunctional J ( ε ). Later it was shown that the conclusionof the absence of traps requires an additional assumptionthat the map χ : ε → U εT is non-degenerate [21], meaningthat arbitrary infinitesimal variations of ε produce vari-ations of U εT in all directions on U ( n ) [33, 34]. While thecontrollability condition is relatively easy to verify [35],checking the non-degeneracy assumption turned out tobe a hard problem. Moreover, critical controls violatingthis assumption were found [24, 25], and even second-order traps—critical controls which are not global max-ima and where the Hessian H = δ J/ ( δε ) is negativesemidefinite were shown to exist under rather general as-sumptions [26]. (Second-order traps are not necessarilylocal maxima but effectively they are traps for local algo-rithms exploiting at most second order local informationabout the objective; see Chapter 20 of [36] for a gen-eral discussion of the non-degeneracy and second orderoptimality conditions.) These findings led to reconsid-eration of the conclusion of absence of traps. Some nu-merical simulations suggested that the condition of non-degeneracy might be generally satisfied or at least its vi-olation does not produce multiple traps [27], while otherindicated possible trapping behavior [25, 29]. However,numerical search is limited and the extent to which theseruns span the full space of quantum control possibilitiesis questionable [29]. Hence the problem of proving eitherexistence or absence of traps has been remained open. III. ABSENCE OF TRAPS FOR THELANDAU-ZENER SYSTEM
Our main result is that the only critical points of anyobjective of the form J ( ε ) = f ( U εT ), where U εT satis-fies (1) and f ( U ) is any function on the special unitarygroup SU (2) which has no local extrema, are global max-ima, global minima, and the zero control field ε ( t ) = 0.Important examples of such objectives include • Transition probability J i → f ( ε ) = |(cid:104) f | U εT | i (cid:105)| This objective is maximized by a control whichcompletely transfers the initial state | i (cid:105) into the de-sired final state | f (cid:105) . • Expectation of a system observable OJ O ( ε ) = Tr[ U εT ρ U ε † T O ]Here O is a Hermitian matrix representing the ob-servable and ρ is the initial system density matrix.The objective is maximized by a control which max-imizes quantum-mechanical average of O at time T . • Generation of a unitary process WJ W ( ε ) = 14 | Tr( W † U εT ) | Here W is the unitary matrix representing a desiredsystem evolution or a desired quantum gate, forexample Hadamard gate. Maximum of this objec-tive is achieved by a control such that U T = e iφ W ,where φ is arbitrary (generally unphysical) phase.Factor 1 / ε J W ( ε ) = 1. Proof of the main result.
We will consider first J ( ε ) = J i → f ( ε ). For brevity, we will sometimes omitthe superscript ε in U εt and U εT , and without loss of gen-erality set ∆ = 1. Gradient of J ( ε ) = |(cid:104) f | U εT | i (cid:105)| for theLZ system has the form [26] ∇ J ε ( t ) = 2 (cid:61) (cid:16) (cid:104) i | U † T | f (cid:105)(cid:104) f | U T U † t σ z U t | i (cid:105) (cid:17) (2)It can be written as ∇ J ε ( t ) = L ( U † t σ z U t ) = l ( t ), where L : su (2) → R is the linear map on the Lie algebra oftraceless Hermitian 2 × L ( A ) =2 (cid:61) [ (cid:104) i | U † T | f (cid:105)(cid:104) f | U T A | i (cid:105) ], and l ( t ) is a real-valued function.If ε is a critical control field, then l ( t ) ≡ l (cid:48) ( t ) = l (cid:48)(cid:48) ( t ) = 0. These derivatives can becomputed to be l (cid:48) ( t ) = L ( − iU † t [ σ x + ε ( t ) σ z , σ z ] U t ) = − L ( U † t σ y U t ) l (cid:48)(cid:48) ( t ) = − L ( − iU † t [ σ x + ε ( t ) σ z , σ y ] U t )= − L ( U † t σ z U t ) + 4 ε ( t ) L ( U † t σ x U t )Thus the condition l (cid:48)(cid:48) = l (cid:48) = l = 0 for any t such that ε ( t ) (cid:54) = 0 takes the form L ( U † t σ x U t ) = L ( U † t σ y U t ) = L ( U † t σ z U t ) = 0 (3)The matrices U † t σ x U t , U † t σ y U t , U † t σ z U t are linearly inde-pendent traceless Hermitian 2 × su (2) and hence (3) implies L ( A ) = 0 for any A ∈ su (2).Let | i ⊥ (cid:105) be the state which is orthogonal to | i (cid:105) . Taking A = | i (cid:105)(cid:104) i ⊥ | + | i ⊥ (cid:105)(cid:104) i | and A (cid:48) = i ( | i (cid:105)(cid:104) i ⊥ | − | i ⊥ (cid:105)(cid:104) i | ) gives L ( A ) = 0 ⇒ (cid:61) (cid:16) (cid:104) i | U † T | f (cid:105)(cid:104) f | U T | i ⊥ (cid:105) (cid:17) = 0 L ( A (cid:48) ) = 0 ⇒ (cid:60) (cid:16) (cid:104) i | U † T | f (cid:105)(cid:104) f | U T | i ⊥ (cid:105) (cid:17) = 0Thus (cid:104) i | U † T | f (cid:105)(cid:104) f | U T | i ⊥ (cid:105) = 0, i.e. either (cid:104) i | U † T | f (cid:105) = 0 or (cid:104) f | U T | i ⊥ (cid:105) = 0. The former case corresponds to the globalminimum of the objective ( J = 0) and the latter to itsglobal maximum ( J = 1). These are the only allowedcritical controls except of ε ( t ) ≡
0. This finishes theproof of the main result for J i → f ( ε ).The analysis above immediately implies that if a lin-ear map L : su (2) → R satisfies L ( U † t σ z U t ) = 0 then L ≡
0. Now we will show that it means that the map χ : ε → U εT is non-degenerate everywhere outside of ε ( t ) ≡
0. Since we consider objectives produced by func-tions on SU (2) which therefore invariant with respect tothe overall phase of U εT , we can identify U εT with the cor-responding element of SU (2). Small variations around U εT can be represented as ˜ U εT = U εT e δw ≈ U εT (1 + δw ),where δw = − i (cid:82) T U † t σ z U t δε ( t ) dt . For the map χ tobe non-degenerate, ˜ U εT should span a neighborhood of U εT that in turn requires δw to span su (2). If δw doesnot span su (2), then there exists A ∈ su (2) , A (cid:54) = 0such that ( A, δw ) ≡ Tr( A † δw ) = 0 for all δε and hence L A ( U † t σ z U t ) := Tr( A † U † t σ z U t ) = 0. This is possible onlyif A = 0 and hence the map can not be degenerate.Therefore our result immediately implies the absence oftraps at any ε (cid:54) = 0 for any objective functional J ( U εT )which has no traps if considered as a function on SU (2).This includes important objectives J O = Tr[ U εT ρ U ε † T O ]for maximizing expectation of a system observable O and J W = (1 / | Tr( W † U εT ) | for optimal generationof a unitary process W (e.g., for unitary gate genera-tion). These objectives appear to be trap-free for theLZ system since functions f O ( U ) = Tr[ U ρ U † O ] and f W ( U ) = (1 / | Tr( W † U ) | have no local maxima on SU (2) [20].The control ε ( t ) ≡ l (cid:48)(cid:48) ( t ) = 0 for ε ≡ L ( U † t σ x U t ) = 0. This control is however not atrap for example for J i → f as shown by direct computationin the Appendix. FIG. 1: (Color online) The control landscape of J → ( a , a )for the LZ system controlled by piecewise constant controls( N = 2, T = 10, ∆ = 1). The landscape possesses multipletraps (local maxima). IV. DISCUSSION
Now we discuss important limitations for the presentanalysis. No real-world system will perfectly evolve ac-cording to Eq. (1) and three general kinds of deviationsfrom the ideal situation include decoherence effects, de-viations of the actual control from the intended one dueto noise or imperfections of the laboratory setup, andlimited tunability of the control strength and time scalesin laboratory experiments. While we consider the sys-tem as evolving according to the Schr¨odinger equationwith unitary evolution, in real circumstances it can ex-perience additional influence of the environment whichcauses the dynamics to be non-unitary. We also assumethat any shape of the control ε ( t ) is available, whereastypical pulses are either piecewise constant or finite sumsof cosines and sines at certain fixed frequencies. Theseassumptions are common for the first step of control land-scape analysis which deals with the ideal situation ofnoiseless unconstrained controls. The next step upon es-tablishment of the ideal landscape properties is to studythe effects of possible deviations, which we discuss belowfor the LZ system.The requirements on the available control fields (e.g.on their strength and time scale) necessary for the con-clusion of the absence of traps for attaining maximal ob-jective value are such that the available controls are suffi-cient to guarantee controllability of the system. Minimalcontrol time for the LZ system can be estimated using thefundamental theory of optimal control at the quantumspeed limit as T QSL ≈ ∆ E − arccos( |(cid:104) i , f (cid:105)| ), where ∆ E is the energy variance of the free Hamiltonian H = ∆ · σ x calculated on the initial state [10]. Hence our analysisapplies to any final time T (cid:38) π ∆ E − . If for a givenphysical system decoherence effects occur on a time scaleslower than ∆ E − , they can be neglected when the con-trol is implemented in the time optimal fashion. Thisshows that while finite-time [37] and decoherence [38–40]effects can be important for the LZ system, they do notmodify the trap-free landscape property as soon as finaltime T is sufficiently smaller that the relaxation time andat the same time is not too small to violate controllability FIG. 2: (Color online) Probability of trapping as a func-tion of N for piecewise constant controls ( T = 10 , ∆ = 1).For every point, 10 runs of MATLAB realization of theBroyden-Fletcher-Goldfarb-Shanno (BFGS) optimization al-gorithm where performed each starting at a random initialcontrol a = ( a , . . . , a N ) [41]. Initial control amplitudes areuniformly distributed in the range a i ∈ [ A, A ] but are allowedto escape this range during the search. The search is definedas trapped if the attained objective is less than 0 .
99. Trap-ping may occur due to the presence of local maxima and/orprincipal impossibility of attaining the objective value greaterthat 0 .
99 with available controls. Probability of trapping isestimated as a fraction of trapped runs among all 10 runs. of the systems.An extensive numerical analysis of control landscapesfor multi-level model systems with realistic laboratorycontrol fields is provided in [27]. To analyze the role oflimitations on the available control fields for the LZ sys-tem, we numerically estimate the probability of trappingwhen available controls are piecewise constant controls ofthe form ε ( t ) = (cid:80) Ni =1 a i χ [ t i ,t i +1 ] ( t ), where χ [ t i ,t i +1 ] ( t ) = 1if t ∈ [ t i , t i +1 ] and zero otherwise and a i are the controlparameters. Typically, N ≈
100 and control amplitudesare constrained within certain ranges, say a i ∈ [ − A, A ].Exact solution for piecewise constant controls can be ob-tained for example in the simplest case N = 1. Theobjective for maximizing the probability of spin flip J → = |(cid:104) | U εT | (cid:105)| by a constant control ε ( t ) = a canbe computed to be J → ( a ) = sin ( T √ a ) / (1 + a ).Its traps (local maxima) are given by solutions of theequation tan( T √ a ) = T √ a ; the correspond-ing objective values are J → ( a ) = T / (1 + T + T a ).Control landscapes for a two-dimensional control space( N = 2) are more complex. Fig. 1 shows as an exam-ple the control landscape of J → ( a , a ). The landscapehas multiple local maxima showing that significant re-strictions on the control space in an originally trap-freesystem may produce traps. Fig. 2 provides the numeri-cally estimated probability of trapping for piecewise con-stant controls as a function of N . The probability oftrapping becomes negligible already for N = 10–15 thatmeans that limitations on the number of components N of available laboratory control fields have minor effectalready for N (cid:38)
10 and hence should be negligible for realistic case N ≈ ε ( t ) be an optimal control in the ideal situation ofabsence of noise. In the presence of a random noise ξ ( t ),the actual control will fluctuate as ε ( t ) = ε ( t ) + (cid:37) ( t ) ξ ( t ),where (cid:37) ( t ) = 1 for additive noise and (cid:37) ( t ) = ε ( t ) formultiplicative noise. A weak noise modifies the averagedobjective as E [ J ( ε )] ≈ J ( ε )+ 12 T (cid:90) T (cid:90) H ( t, t (cid:48) ) (cid:37) ( t ) (cid:37) ( t (cid:48) ) E [ ξ ( t ) ξ ( t (cid:48) )] dtdt (cid:48) where H ( t, t (cid:48) ) = δ Jδε ( t ) δε ( t (cid:48) ) is the Hessian of the ob-jective computed at the optimal control field ε and E [ ξ ( t ) ξ ( t (cid:48) )] is the autocorrelation function of the noise.Since Hessian is negative semidefinite at the maximum,the noise generally decreases the average fidelity. Theobjective for additive (AWN) and multiplicative (MWN)white noise with autocorrelation function E [ ξ ( t ) ξ ( t (cid:48) )] = σδ ( t − t (cid:48) ), where σ is the variance of the noise amplitudedistribution, takes the forms E AWN [ J ( ε )] ≈ J ( ε ) + σ T (cid:90) H ( t, t ) dt E MWN [ J ( ε )] ≈ J ( ε ) + σ T (cid:90) H ( t, t ) | ε ( t ) | dt The last term in these equations is the noise-induced de-crease −D ( ε , σ, T ) of the objective (such that E ( J ) ≈ J ( ε ) − D with D ≥ J = J i → f can be shown to be H ( t, t ) = − |(cid:104) i | U † t σ z U t | i ⊥ (cid:105)| so that |H ( t, t ) | ≤
2. Therefore D ( ε , σ, T ) for J = J i → f is majorized by D AWN ( ε , σ, T ) ≤ σ T D MWN ( ε , σ, T ) ≤ σ E where E = (cid:82) T | ε ( t ) | dt is the total energy of the pulse.The diagonal of the Hessian for the objective J W is H ( t, t ) = − D AWN = σ T and D MWN = σ E . It then follows that in both casesthe influence of a weak AWN can be minimized by us-ing time optimal controls, while minimizing weak MWNcan be done by selecting less energetic pulses among alloptimal pulses.If the ideal landscape has multiple global optima withdifferent H ( t, t ), then the noise induced decrease of ob-jective can be different at different optima that can pro-duce traps in the non-ideal landscape. Weak decoherenceoperates similarly to weak noise and can also producetraps in the ideally trap-free landscape [20]. These de-viations from the ideal situation should be avoided toreveal the trap-free landscape property by either oper-ating in the time optimal regime or using weak optimalcontrols to combat MWN. Strong noise and strong deco-herence that can significantly modify the landscape areout of scope of this discussion. Conclusions. —This work shows that unconstrainedmanipulation in the Landau-Zener system is free of trapsand hence unconstrained local search for optimal con-trols is always able to find best optima. The impact onthis result of laboratory limitations due to decoherence,noise in the actual control pulses, and restrictions on theavailable control fields is discussed.
Acknowledgments
A. Pechen acknowledges support of the Marie Curie In-ternational Incoming Fellowship within the 7th EuropeanCommunity Framework Programme. N. Il’in is partiallysupported by the Russian Foundation for Basic Research.This research is made possible in part by the historicgenerosity of the Harold Perlman family and by the Min-istry of Education and Science of the Russian Federation,project 8215.
Appendix
Here we prove that the control ε ( t ) ≡ J ( ε ) = J i → f ( ε ). The evolution operator produced by ε ( t ) = 0has the form U t = e − itσ x . Therefore V t := U † t σ z U t =cos(2 t ) σ z + sin(2 t ) σ y and the gradient of the objective is ∇ J ε =0 ( t ) = cos 2 t · L ( σ z ) + sin 2 t · L ( σ y )If ε ( t ) = 0 is a critical point, then ∇ J ε =0 ( t ) = 0 for any t ∈ [0 , T ], and hence L ( σ z ) = L ( σ y ) = 0. If α := L ( σ x ) =0, then L ≡ su (2) and similarly to the proof of themain result we conclude that ε = 0 is not a trap.Now consider the case α (cid:54) = 0. In this case | i (cid:105) and | f (cid:105) are such that ε = 0 is neither a global maximum norglobal minimum. The evolution operator produced by asmall variation of the control δε can be represented as U δεT = e − iT σ x W T , where W T satisfies˙ W t = − i δε ( t ) V t W t , W = I The operator W T can be computed up to the second order in δε as W T = I + A + A + o ( (cid:107) δε (cid:107) ) A = − i (cid:90) T dtδε ( t ) V t ,A = − (cid:90) T dt (cid:90) t dt δε ( t ) δε ( t ) V t V t This gives the perturbation expansion for the objective(here | f (cid:48) (cid:105) = ei T σ z | f (cid:105) ) J ( δε ) = |(cid:104) f (cid:48) | I + A + A + . . . | i (cid:105)| = |(cid:104) f (cid:48) | i (cid:105)| + δJ ( δε ) + δJ ( δε ) + o ( (cid:107) δε (cid:107) )where δJ ( δε ) = 2 (cid:60) ( (cid:104) f (cid:48) | i (cid:105)(cid:104) f (cid:48) | A | i (cid:105) ) δJ ( δε ) = |(cid:104) f (cid:48) | A | i (cid:105)| + 2 (cid:60) ( (cid:104) f (cid:48) | i (cid:105)(cid:104) f (cid:48) | A | i (cid:105) )Hence the variation of the objective satisfies (note that |(cid:104) f (cid:48) | i (cid:105)| = J (0)) δJ = J ( δε ) − J (0) = δJ ( δε ) + δJ ( δε ) + o ( (cid:107) δε (cid:107) )If ε = 0 is a critical control, then δJ ( δε ) = 0 for any δε . We will show the existence of controls δε and δε such that δJ ( δε ) and δJ ( δε ) have opposite signs. Itis sufficient to choose δε and δε to satisfy (cid:104) f (cid:48) | A | i (cid:105) = 0,e.g. (cid:90) T dtδε i ( t ) cos 2 t = (cid:90) T dtδε i ( t ) sin 2 t = 0 , i = 1 , . Since V t V t = cos 2( t − t ) + i σ x sin 2( t − t ), we have δJ ( δε ) = 2 (cid:90) T dt (cid:90) t dt δε ( t ) δε ( t ) (cid:16) J (0) cos 2( t − t )+ α sin 2( t − t ) (cid:17) = 2 α (cid:90) T dt (cid:90) t dt δε ( t ) δε ( t ) sin 2( t − t )Assuming for simplicity that T ≥ π , we take δε ( t ) = χ [0 ,π ] ( t ) and δε ( t ) = cos(4 t ) χ [0 ,π ] ( t ), where χ [0 ,π ] ( t ) isthe characteristic function of the interval [0 , π ]. Then δJ ( δε ) = − πα and δJ ( δε ) = πα/
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