Universal protection of unitary evolution from slow noise: dynamical control pushed to the extreme
aa r X i v : . [ qu a n t - ph ] M a y Universal prote tion of unitary evolution from slow noise:dynami al ontrol pushed to the extremeG. Bensky , E. Brion , F. Carlier , V.M. Akulin , and G. Kurizki Weizmann Institute of S ien e, Department of Chemi al Physi s, Rehovot 76100, Israel and Laboratoire Aimé Cotton, CNRS, Campus d'Orsay, 91405, Orsay, Fran eAbstra tWe propose a te hnique that allows to simultaneously perform universal ontrol of the evolutionoperator and ompensate for the (cid:28)rst order ontribution of an arbitrary Hermitian onstant noise.We show that, at least, a three-valued Hamiltonian is needed in order to prote t the system againstany su h noise. This te hnique is illystrated by an expli it algorithm for a ontrol sequen e that isapplied to numeri ally design a safe two-qubit gate.1. INTRODUCTIONWithin the last de ades, quantum ontrol has emerged as one the most fruitful (cid:28)elds inboth theoreti al and experimental physi s [1, 2℄. In parti ular, it is of ru ial importan ein quantum omputation [3℄. To pro ess the information stored in the omputer state, onemust indeed be able to generate any pres ribed unitary evolution operator on the omputer(cid:22) or, at least a universal set of su h operators, una(cid:27)e ted by quantum errors arising fromthe intera tion with the environment.Di(cid:27)erent strategies have been developed to ontrol the evolution of losed quantumsystems, in luding optimal ontrol approa hes [4, 5, 6, 7, 8℄ and algebrai methods[9, 10, 11, 12℄. To deal with open quantum systems, s hemes have been designed to or-re t/avoid the undesired e(cid:27)e ts due to the environment. Quantum error- orre ting odes(QEC) [3, 13, 14, 15, 16℄ and approa hes based on the quantum Zeno e(cid:27)e t [17, 18℄ useredundan y of en oding as a way to re over information after the errors o ur. Topologi alprote tion [19℄ takes advantage of the symmetries of the system to safely store informa-tion in so- alled De oheren e Free Subspa es [20, 21, 22℄. An alternative to QEC that issubstantially less resour e-intensive is dynami al de oupling (DD) [23, 24, 25℄. In DD oneapplies a su ession of short and strong pulses to the system, designed to strobos opi allyde ouple it from the environment. Similar in spirit to DD, but more general, is the methodwe term here (cid:16)dynami al ontrol by modulation(cid:17) (DCM), wherein one may apply to thesystem a sequen e of arbitrarily-shaped pulses whose duration may vary anywhere from thestrobos opi limit to that of ontinuous dynami al modulation [18, 26, 27, 28, 29, 30℄. Inthe DCM approa h, the de oheren e rate is governed by a universal expression, in the formof an overlap between the bath-response and modulation spe tra. In su h methods, it ishowever not lear a priori whether one an, at the same time, prote t the system from noiseand perform universal ontrol of its evolution operator.In this paper we investigate this question by asking whether it is possible to performany arbitrarily hosen evolution of a quantum system while ompensating for all Hermitianstati noises. To be more spe i(cid:28) , our goal here is to show how to design a time-dependent ontrol Hamiltonian whi h an, at the same time, impose a hosen evolution to the systemand eliminate the (cid:28)rst order a tion of any Hermitian stati noise. We show that, ontraryto stri t evolution ontrol problems, this obje tive annot be a hieved with only two-valued2amiltonians but requires the use of at least three ontrol operators. We go on to showthat even a null third operator, ausing the system to evolve under the a tion of noise only,is enough. Inspired by a previous result [31℄, we propose a new algorithm able to omputethe appropriate ontrol sequen e for any given desired evolution. As an appli ation, thisalgorithm was run to design a safe
CN OT gate.The paper is stru tured as follows. We (cid:28)rst set the problem to solve and give the ex-pli it onditions the evolution matrix must ful(cid:28)ll. Then we show that these onditions areimpossible to meet by a two-valued ontrol Hamiltonian. Slightly modifying the two-stagepro edure by allowing for extra steps during whi h the ontrol Hamiltonian is set to zero,we des ribe an algorithm able to ompute an appropriate prote ted ontrol sequen e. Anappli ation to a two-qubit gate is (cid:28)nally proposed.II. CONDITIONS FOR FIRST-ORDER NOISE ELIMINATIONLet us onsider an N -level quantum system whose Hamiltonian onsists of a ontrol-lable part, denoted by H c ( t ) , and an unknown but stati noise ontribution, whi h anbe written as a linear ombination N = P i ~ ε i G i of the Hermitian tra eless generators { G i , i = 1 , . . . , N − } of su ( N ) . The evolution matrix satis(cid:28)es the dynami al equation ı ~ ∂U ( t ) ∂t = ( H c ( t ) + N ) U ( t ) (1) U (0) = I. (2)Upon transforming to the intera tion pi ture relative to H c , one isolates the evolution e U ( t ) due to the noise only: de(cid:28)ning U c ( t ) ≡ T exp (cid:26) ı ~ Z t H c ( s ) ds (cid:27) (3)the evolution indu ed by the ontrol Hamiltonian alone, where T denotes the hronologi alprodu t, one sets e U ( t ) ≡ U † c ( t ) U ( t ) , whi h satis(cid:28)es ı ~ ∂ e U∂t = (cid:2) U † c ( t ) N U c ( t ) (cid:3) e U ( t ) (4) = "X i ~ ε i U † c ( t ) G i U c ( t ) U ( t ) (5) e U (0) = I. (6)3he (cid:28)rst order ontribution of the noise to the evolution is thus given by the se ond termin the Dyson expansion of e U ( t ) for the a umulated a tion, that is e U (1) ( t ) = X i ε i Z t U † c ( s ) G i U c ( s ) ds. (7)Our goal is to design a ontrol Hamiltonian H c ( t ) , su h that, at the end of the ontrolsequen e, say at time T c , the evolution operator takes an arbitrarily pres ribed value U d ∈ SU ( N ) while the (cid:28)rst order ontribution of any onstant noise vanishes. We thus require U c ( T c ) = U d (8)and, for any set of onstants { ε i } , e U (1) ( T c ) = X i ε i Z T c U † c ( s ) G i U c ( s ) ds = 0 , (9)that is ∀ i, Z T c U † c ( s ) G i U c ( s ) ds = 0 . (10)III. ALTERNATING TWO-VALUED OPERATOR SEQUENCELet us now fo us on the two-valued alternating perturbation approa h. Namely, the ontrol Hamiltonian H c ( t ) alternates between two values A and B for adjustable timingswhi h play the role of ontrol parameters. Formally, H c ( t ) then assumes the bilinear form H c ( t ) = ~ α ( t ) A + ~ β ( t ) B , where α ( t ) and β ( t ) are two pie ewise onstant fun tions takingthe values , and adding up to . In the sudden approximation, the overall evolutionoperator indu ed by su h a K -step ontrol sequen e has the pulsed form U c ( T c ) = e − ıT K H K × . . . × e − ıT H × e − ıT H , (11)where H k ≡ A when k is even, B when k is odd, and P Kk =1 T k = T c .Provided that A , B together with their all-order ommutators span su ( N ) (the bra ketgeneration ondition), it an be shown [31℄ that it is possible to design K ∝ N su h ontroltimings { T , . . . , T K } for whi h eq.(8) is met, i.e. su h that e − ıT K H K × . . . × e − ıT A × e − ıT B = U d . (12)4t turns out, however, that, ex ept for the parti ular ase U d = I , eq.(10) an-not be satis(cid:28)ed for all noises. As we shall now show, there indeed always exists a D -dimensional un orre table subspa e (1 ≤ D ≤ N − spanned by the noise operators C n ∈ N ≡ [ A + B, ( B − A ) n ] . To prove this, let us (cid:28)rst introdu e the operators U ≡ I, U K ≥ i ≥ ≡ e − ıT i H i × . . . × e − ıT A × e − ıT B (13)whi h allow us to write U c ( t ) = e − ı ( t − P ik =1 T k ) H i +1 U i (14) ( for P i T k ≤ t < P i +11 T k ) . (15)Let us further de(cid:28)ne the superoperators ˆ H i , whose a tion is given, on any operator X , by ˆ H i X ≡ [ H i , X ] . We an thus write e + εH i Xe − εH i = P ∞ k =0 ε k k ! ˆ H ki X . Sin e [ A, ( B − A ) n ] =[ B, ( B − A ) n ] = [ H i , ( B − A ) n ] for every i , we an set C n = ˆ H i ( B − A ) n .Let us now evaluate the (cid:28)rst order e(cid:27)e t of C n a ording to eq.(10): Z T c U † c ( s ) C n U c ( s ) ds (16) = X i Z T i dsU † i e ısH i ˆ H i ( B − A ) n e − ısH i U i (17) = X i U † i Z T i ds ∞ X k =0 ( − ıs ) k k ! ˆ H k +1 i ( B − A ) n U i (18) = ı X i U † i (cid:2) e ıT i H i ( B − A ) n e − ıT i H i − ( B − A ) n (cid:3) U i (19) = ı X i U † i +1 ( B − A ) n U i +1 − ı X i U † i ( B − A ) n U i (20) = ıU † ( T c )( B − A ) n U ( T c ) − ı ( B − A ) n . (21)The (cid:28)rst order ontribution we have just obtained does not depend on the spe i(cid:28) timingparameters T i 's but only on the (cid:28)nal operation U ( T c ) ; it is moreover nonzero for any U d = I . Finally, a ording to the Cayley-Hamilton theorem, the matrix ( B − A ) an els its hara teristi polynomial, whi h implies that ≤ dim [Span ( { ( B − A ) n , n ∈ N } )] ≤ N − .As a onsequen e the dimension D ≡ dim [Span ( { C n , n ∈ N } )] of the un orre table subspa esatis(cid:28)es ≤ D ≤ N − .Let us now examine how the two-operator Hamiltonian s heme an be modi(cid:28)ed so thatboth onditions eq.(8,10) are met. Suppose we have a sequen e of timings { t , . . . , t K } . Now5 and U n ≥ are the evolution operator at the end of the n th step of the ontrol sequen eas de(cid:28)ned in eq.(13), while U c ( t ) is the evolution operator at time t as de(cid:28)ned in eq.(15).Following the method des ribed in [31℄, the last N − timings { t K − N +2 , . . . , T K } an be hosen su h that they a hieve the evolution U d · U † K − N +1 , thus satisfying the onditionof eq.(8). The (cid:28)rst order ontribution of any noise G i an be expli itly expressed by thea umulated a tion G i ≡ K X n =1 U † n g i,n U n , (22)where g i,n ≡ R T n e − ısH n G i e + ısH n ds .Let us, at ea h ommutation between A and B, allow for the waiting time τ l =1 ,...,K duringwhi h no perturbation is applied. This amounts to adding a third value C = 0 to the ontrolHamiltonian H c ( t ) . We see that the overall evolution operator remains un hanged, whilethe (cid:28)rst order ontribution of any noise G i is added the term P Kn =1 τ n U † n G i U n , whi h is alinear fun tion of the waiting times.Let us hoose the timings { t , . . . , t K − N +1 } su h that the oe(cid:30) ients of the waiting timesspan the entire N · ( N − spa e. Generally this an be a hieved by randomly hoosingthe timings. Thus, by solving a simple set of N · ( N − linear equations of the form G i + P Kn =1 τ n F i,n = 0 ( F i,n ≡ U † n G i U n ) to (cid:28)nd the waiting times τ , . . . , τ K , one an eliminatethe (cid:28)rst order ontribution of all the noises G i added during the A , B ontrol sequen es. Inpra ti e, sin e the waiting times τ i must be positive, this requires a slightly larger K andthe use of linear programming methods or other minimization te hniques.IV. ALGORITHM FOR A FOUR-STATE SYSTEMThis algorithm was applied to a model four-state system, represented in Fig. 1, whi h an be used to store and pro ess two qubits of information. The four states orrespondto two di(cid:27)erent angular momenta l = 0 , : | i ≡ | l = 0 , m l = 0 i , | i ≡ | l = 1 , m l = − i , | i ≡ | l = 1 , m l = 0 i and | i ≡ | l = 1 , m l = 1 i .This system is subje t to a resonant ele tri and a stati magneti (cid:28)elds: the π - omponentof the ele tri (cid:28)eld ouples | i to | i while the σ + , − - omponents ouple | i to | i , and | i to | i , respe tively; the x, y - omponent of the magneti (cid:28)eld ouples | i to | i and | i to | i while its z - omponent shifts | i and | i out of resonan e. Finally, in the rotating wave6pproximation (RWA), the total Hamiltonian of the system assumes the form H c = e − σ − + e σ + e + σ + + b ⊥ Λ ⊥ + b z Λ z (23) σ − ≡ | i h | + | i h | (24) σ ≡ | i h | + | i h | (25) σ + ≡ | i h | + | i h | (26) Λ ⊥ ≡ | i h | + | i h | + | i h | + | i h | (27) Λ z ≡ | i h | − | i h | (28)where e − , , + and b ⊥ ,z are (cid:28)ve independent parameters, proportional to the ele tri andmagneti (cid:28)eld amplitudes, respe tively. By hoosing two di(cid:27)erent sets (cid:8) e A − , , + , b A ⊥ ,z (cid:9) and (cid:8) e B − , , + , b B ⊥ ,z (cid:9) of su h parameters, we an de(cid:28)ne two values for the Hamiltonian H c = A, B whi h may be used as our alternating perturbations.After verifying that A and B satisfy the bra ket generation ondition, we (cid:28)rst al ulate the K = N = 16 timings n t (0) k =1 ,...,K o whi h realize the identity matrix, by the method des ribedin [31℄, based on statisti al properties of the roots of the identity. By the Newton iterativemethod, we then ompute the K = 16 timings { t k =1 ,...,K } whi h perform the transformation CN OT N − = CN OT , where
CN OT ≡ .Finally, we look for the K ( N −
1) = 240 waiting times τ l =1 ,...,K ( N − whi h mini-mize all possible noise operators to zero. By applying the omplete -step ontrolsequen e, onsisting of the ( N −
1) = 15 repetitions of the sequen e { t k =1 ,...,K =16 } plus the K ( N −
1) = 240 waiting periods { τ l } with zero ontrol after ea h step, we an thus imposea safe CN OT gate on the system in the presen e of any noise that varies slower than the ontrol sequen e.V. DISCUSSIONIt is plausible that the onditions eq.(8,10) an be satis(cid:28)ed by an arbitrary three-valuedHamiltonian, with C = 0 . We have looked for an appropriate sequen e of timings with su ha Hamiltonian by dire t optimization of a y le of KN operations and found satisfa tory7 = l = ! = l = , m l = ! ! + ! " ! " ! z ! z ! = l = , m l = ! ! ! = l = , m l = ! ! = l = , m l = ! Figure 1: The two-level atomi system ( l = 0 , ). The arrows show the di(cid:27)erent ouplings due tothe resonant ele tri (cid:28)eld (operators σ − , , + ) and the stati magneti (cid:28)eld (operators Λ ⊥ ,z ).numeri al solutions up to N = 16 . This approa h is however slower than the s hemedes ribed above.There are several open issues regarding the universal ontrol method proposed here.First, it is important to treat higher orders in the perturbation expansion of the noise and,in parti ular, explore the feasibility of a ontrol Hamiltonian an eling the se ond order ontribution of the noise, i.e. the integrals Z T c dt Z t dsU † c ( t ) G m U c ( t ) U † c ( s ) G n U c ( s ) (29)for any ( m, n ) . Se ond, the issue of time-dependent noises is important. The present methodholds for noise that slowly varies with time, ompared to the ontrol sequen e. The approa hmust hange altogether if fast noises a(cid:27)e t the system. Finally, it is imperative to establishwhether redundan y an be ombined with dynami ontrol te hniques to safely pro ess theinformation, as, for example, in an ensemble of identi al systems.8o on lude, we have raised the question whether universal ontrol of evolution may beperformed while ompensating for the (cid:28)rst order ontribution of arbitrary onstant Hermi-tian noise, by means of an alternating perturbation pro edure. We have demonstrated thatthis an not be a hieved by a two-valued ontrol Hamiltonian: in that ase there alwaysexists a subspa e of un orre table noises. If, however, we allow for waiting times, duringwhi h the system is only subje t to noise, our obje tive be omes feasible. This has beendemonstrated by an expli it algorithm that yields the appropriate ontrol sequen e, as testedon the ase N = 4= 4