Control of noisy quantum systems: Field theory approach to error mitigation
CControl of noisy quantum systems: Field theory approach to error mitigation
Rafael Hipolito and Paul M. Goldbart
School of Physics, Georgia Institute of Technology, 837 State Street, Atlanta, Georgia 30332 (Dated: October 17, 2018)We consider the basic quantum-control task of obtaining a target unitary operation (i.e., a quan-tum gate) via control fields that couple to the quantum system and are chosen to best mitigateerrors resulting from time-dependent noise, which frustrate this task. We allow for two sources ofnoise: fluctuations in the control fields and fluctuations arising from the environment. We addressthe issue of control-error mitigation by means of a formulation rooted in the Martin-Siggia-Rose(MSR) approach to noisy, classical statistical-mechanical systems. To do this, we express the noisycontrol problem in terms of a path integral, and integrate out the noise to arrive at an effective,noise-free description. We characterize the degree of success in error mitigation via a fidelity metric,which characterizes the proximity of the sought-after evolution to ones that are achievable in thepresence of noise. Error mitigation is then best accomplished by applying the optimal control fields,i.e., those that maximize the fidelity subject to any constraints obeyed by the control fields. Tomake connection with MSR, we reformulate the fidelity in terms of a Schwinger-Keldysh (SK) pathintegral, with the added twist that the “forward” and “backward” branches of the time-contourare inequivalent with respect to the noise. The present approach naturally and readily allows theincorporation of constraints on the control fields—a useful feature in practice, given that constraintsfeature in all real experiments. We illustrate this MSR-SK reformulation by considering a modelsystem consisting of a single spin- s freedom (with s arbitrary), focusing on the case of 1 /f noisein the weak-noise limit. We discover that optimal error-mitigation is accomplished via a universalcontrol field protocol that is valid for all s , from the qubit (i.e., s = 1 /
2) case to the classical (i.e., s → ∞ ) limit. In principle, this MSR-SK approach provides a transparent framework for addressingquantum control in the presence of noise for systems of arbitrary complexity. I. INTRODUCTION
The ability to control the fate of quantum systemswhose dynamics are subject to noisy influences is a criti-cal factor in numerous settings, including, notably, quan-tum information processing [1–3]. A typical character-istic of such systems is that in order to control themit is necessary to couple them to external time-varyingfields that cannot themeselves be perfectly controlled andwhich, therefore, inevitably introduce classical noise intothe system. This noise, present in the external fields,alters the time-evolution of the quantum system of inter-est (i.e., the system we wish to control), typically push-ing the end result away from the intended target. Theexternal control fields enter the Hamiltonian governingthe dynamics of the quantum system by way of couplingdirectly to the system’s degrees of freedom, and thuscoupling these sources of noise directly to it. The dy-namics, including the contribution from the noisy exter-nal fields, can be described via a Hamiltonian containingexternal stochastic parameters in addition to terms de-scribing internal system dynamics. Although the prob-abilistic properties of the stochastic parameters can bedetermined, it is generally impossible to predict the val-ues realized in any instance of the control attempt. Evenin the absence of these external control fields, quantumsystems are still generally subject to external sources ofnoise, because no system is truly isolated from the en-vironment. Environmental noise is detrimental to thecontrol mission because it inevitably leads to the unre-mediable loss of information from the quantum system,due to entanglement between system and environment degrees of freedom, leading to quantum decoherence [4].In the light of these remarks it is of value to identifyand understand how to design schemes capable of mit-igating the effects of noise to the greatest extent possi-ble. Several approaches have been developed to addressthis issue, including dynamical decoupling [5], dynami-cal control by modulation [6, 7] and, more recently, thefilter function approach used in Refs. [8, 9]). The aim ofthe present Paper is to develop an alternative approachto the task of quantum control in the presence of noise.The approach is rooted in the Schwinger-Keldysh (hence-forth SK) path-integral framework [10, 11], which has fre-quently been invoked in treatments of quantum dynami-cal systems that are not in thermodynamic equilibrium;see, e.g., Refs. [12, 13]. The SK framework is especiallywell suited for providing a transparent account of the ef-fects of quantum fluctuations (i.e., quantum noise ) viainterference between quantum fields that propagate ‘for-ward’ and ‘backward’ in time. Although the frameworkwas originally developed with closed quantum systemsin mind, it can readily be modified to account for openquantum systems that are coupled to external fields andsources of noise.The core idea behind the present approach is as fol-lows. We consider noisy quantum systems for which wepossess: (i) a complete characterization of the noise-freesystem via the specification of its freedoms and a Hamil-tonian that governs them; and (ii) a complete statistical characterization of the noise that perturbs the dynamicsfrom its noise-free form, in the form a probability distri-bution for the history of the noise parameters. We takethe goal of the control process to be to guide the system a r X i v : . [ qu a n t - ph ] D ec as accurately as possible, i.e., to impart upon it (as ac-curately as possible) some predetermined unitary trans-formation, without being informed about the instance ofthe noise-parameter history. We characterize the perfor-mance of the control process—i.e., its ability to impart apredetermined target operation upon the quantum sys-tem of interest—via an overlap metric or fidelity , whichis designed to assess the accuracy of the process, aver-aged over the noise-parameter history weighted by itsknown distribution. The guiding of the system is accom-plished via a time-dependent Hamiltonian that we selectfrom some menu; typically the menu is incomplete, in thesense that only certain operators are regarded as beingavailable and the time-dependence of the classical vari-ables that characterize these operators is restricted bythe kinds of constraints present in real experiments. Aninevitable consequence of the present framework is thatthe guiding that will be ascertained will be more accu-rate for some noise histories and target unitaries thanfor others. A strength, however, is that it need only bedetermined once, for any given noisy system and noisedistribution.We formulate the task of controlling a generic noisyquantum system via an optimization problem, in whichone seeks the control-field history that maximizes thefidelity, i.e., the measure of success alluded to earlier;see also Refs. [8, 9] (although the present definition ofthe fidelity differs slightly from those used in these ref-erences). The path-integral formulation of the fidelity(which is the object of primary interest to us) takes aform that is close to the conventional SK path integralbut has the following twist: The ‘forward-in-time’ and‘backward-in-time’ branches are asymmetrical with re-spect to the external noise (both environmental and dueto the controls), which is present in the former branch butabsent from the latter. This is a direct consequence ofthe definition we have chosen for the fidelity. By contrast,the internal quantum noise , which originates within thesystem of interest still appears symmetrically in the twobranches, as it does in the usual SK path integral. TheSK formulation of control is similar in spirit to the ap-proach introduced by Martin-Siggia-Rose (MSR) [14] tostudy classical statistical dynamics. Once we have con-structed the path integral for our SK-type formulation ofthe fidelity, we integrate out the environmental noise andthus arrive at an effective description that is completelydeterministic (i.e., noise free), to which we can apply thetools available from field theory, such as diagrammaticexpansion and even nonperturbative methods.Via our approach, we show that the optimization pro-cedure used to maximize the fidelity naturally gives riseto an action principle, which leads to equations of motionfor the control-field history. The solution of these equa-tions is a continuous deformation of the control schemerelevant to the noise-free case. As one continuously in-creases the strength of the noise, the optimal controlscheme continuously deforms, parametrically, away fromthe noise-free scheme. The noise-free scheme typically presents an arbitrary number of schemes to choose from.By adopting any one of these schemes and continuouslyincreasing the strength of the noise, we sweep through acontinuous family of control schemes, with members ofthe family being parametrized by the noise strength. Dif-ferent choices of noise-free schemes typically correspondto different sets of winding numbers, as we show below.Each family can then be labeled by its set of windingnumbers, which are invariant under these continuous de-formations. In common with more familiar action princi-ples, the present approach has the advantage of allowingthe straightforward implementation of many classes ofconstraints, including those naturally appearing in ex-periments, via the Lagrange multiplier technique. Thepath-integral formalism also provides a natural startingpoint for developing a semiclassical approach to the taskof controlling noisy quantum systems, in which quantumeffects are introduced as a refinement of an underlyingclassical process. The literature is extensive on control-ling noisy classical systems, as it is on the control oftwo-level quantum systems (i.e., qubits), which consti-tute the extreme quantal case. Inter alia , the formalismwe develop here serves to bridge the gap between thesetwo extremes—a regime that has been left relatively un-explored, to date.We illustrate our approach by analyzing some concreteexamples in detail. We specifically develop the formalismfor a single quantum spin ˆ S , keeping the spin quantumnumber s arbitrary. We couple the spin to external con-trol fields in order to achieve a target operation, and allowthe spin to be under the influence of noise (from the envi-ronment or the control fields or both) with the statisticsof the noise presumed known. Although we develop theformalism with this specific system in mind, we note thatit can be readily generalized to more elaborate systems,including interacting systems such as spin chains and ul-tracold atomic gases. We also note that the SK path inte-gral, if formulated in terms of coherent states, provides anatural starting point for a semiclassical expansion. Thisis a useful feature, if one is interested in studying controlproblems for systems where the semiclassical expansiongives an excellent approximation to the full quantum dy-namics (e.g., ultracold bosonic gases [15]).Continuing with the case of a single spin, we use ourformalism to find the fidelity and the corresponding op-timized control fields for various choices of noise distri-butions, with a special focus on 1 /f noise sources, whicharise in many systems of interest [16]. For the specificcase of a single spin, we find results for arbitrary s , rang-ing from the qubit limit ( s = 1 /
2) up to the classicallimit s → ∞ .The Paper is organized as follows. In Sec. II we givea discussion of the general setting that we shall be con-cerned with, and also review the various questions thatwe shall be exploring in detail. In Sec. III we formulatethe control problem in terms of a modifed SK approach,and apply it to the case of a single spin ˆS of arbitraryspin quantum number s in the presence of noise. We alsoconstruct the expression for the fidelity and find the op-timal control scheme for the case of weak 1 /f noise andconstraints being imposed on the strength of the controlfields. In Sec. IV we summarize our results and brieflydiscuss future directions. The technical details for thederivation of our formalism are mostly relegated to ap-pendices for the sake of the clarity of the presentation,and will be referred to where relevant. II. ELEMENTS
Our task is to control a given quantum system —specifically, we wish to complete a predetermined unitarytransformation , which we call the target unitary operator U T , upon the quantum state. The state is not necessar-ily known ahead of time. In order to accomplish this,we invoke external time-dependent control fields , whichcouple directly to the system degrees of freedom and areused to steer the quantum system in such a way that, atthe end of some transit time τ , the quantum system hasevolved to a final state that is equivalently described by U T in the sense that the net effect of the two is the same.The quantum system we wish to control is not gener-ally isolated. It is coupled to an external environmentand is thus subject to environmental noise. In addition,there may be fluctuations in the control fields themselves(i.e., originating in the devices used to generate thesefields), which provide another source of noise. The neteffect originating from all sources of noise will generallyinterfere with the control scheme. Assuming one choosesthe control fields such that one obtains a unitary equiva-lent to U T at the end of the transit time τ in the absenseof noise , these same control fields will generally give riseto a unitary transformation that differs from U T .The question we address can be stated as follows: givenmany histories of sets of control fields to choose from,each of which would give rise to U T in the absence ofnoise, which particular set gives rise to a unitary trans-formation that is closest to U T ? In the presence of noise,it is not possible to predict how the quantum system thatwe aim to control will evolve in each given instance of thenoise field history. The best we can do is determine witha statistical measure that tells us how close we are toreaching our stated goal, which is to have unitary evolu-tion that is as close as possible to U T at the end of thetransit time τ .The simplest such measure, viz., the fidelity (whichwe define below), reports how close we get on average ,i.e., after averaging over all sources of noise. In otherwords, we choose a single set of control fields; then, foreach instance of noise, we find the corresponding unitarytransformation at the end of the transit time τ ; we av-erage these unitary transformations over all instances ofnoise histories; and, finally, we compare the resulting av-eraged unitary transformation with U T . We define thebest single set of control field histories — what we aresearching for — to be the set from which one obtains the (averaged) transformation that lies as close as possi-ble to the one resulting from U T . We rigorously defineall of the quantities of interest below, as we develop ourmethodology.Before developing our methodology, let us spend sometime to obtain a better quantitative understanding of theproblem at hand in terms of the various Hamiltonianterms involved. A. Hamiltonian
Consider a quantum system described by a Hamilto-nian H ( t ), which can be written as a sum of several parts: H ( t ) = H c ( t ) + H s ( t ) + H e ( t ) + (cid:15) se H se ( t ); (1)we allow all parts of the Hamiltonian to be explicitly timedependent, and H c ( t ), H s ( t ), H e ( t ), and H se ( t ) are re-spectively taken to be the control Hamiltonian, systemHamiltonian, environmental Hamiltonian, and system-environment coupling Hamiltonian. The parameter (cid:15) se determines the strength of the system-environment cou-pling, which we take to be weak.Let us denote by { ˆ q i } the quantum degrees of freedomof the system (i.e., the part that we wish to control ), andby ˆ Q i the degrees of freedom of the environment. Thevarious Hamiltonian terms have the following character-istics: H c ( t ) = (cid:80) i c i ( t ) ˆ q i couples the control fields c i ( t )directly to the quantum degrees of freedom ˆ q i , which weare interested in controlling; H s ( t ) = H s ( t, ˆ q i ) determinesthe internal system dynamics; H e ( t ) = H e ( t, ˆ Q i ) deter-mines the dynamics of the environment; and H se ( t ) = H se ( t, ˆ q i , ˆ Q i ) couples the system and environment to oneanother. From the perspective of RG, the terms thatlinearly couple the system and environment degrees offreedom are the most relevant (see, e.g. Ref. [17]). Tak-ing the system-environment coupling to be weak (i.e., (cid:15) se small), and temporarily ignoring all other terms in H se ( t )we get H se ( t ) ∼ = (cid:88) ij ˆ q i η ij ( t ) ˆ Q j . (2)Next, by using the path integral language we can pro-ceed to integrate out the environment degrees of free-dom, and thus obtain an effective Hamiltonian, in whichonly the system degrees of freedom remain. Carrying outthis procedure, and assuming that one can treat the en-vironmental degrees of freedom using a semiclassical ap-proximation, we obtain an effective system-environmentHamiltonian (cid:101) H se ( t ) which in general has the followingform: (cid:101) H se ( t ) (cid:39) (cid:88) i ˜ η i ( t ) ˆ q i , (3)where the histories { ˜ η i ( · ) } are stochastic , with a historiesfunctional probability density P e [ { ˜ η i ( · ) } ] (where { ˜ η i ( · ) } designates the set of ˜ η i ( · ) for all i ), which in principle canbe determined from the state of the environment and theenvironment Hamiltonian H e .In actual experiments, there is also some degree of un-certainty associated with the control fields c i ( t ) them-selves, such as fluctuations in the fields due to noise gen-erated in the experimental apparatus responsible for thegeneration of these fields. In general, we can write c i ( t ) = ˜ c i ( t ) + (cid:15) c δc i ( t ) , (4)where ˜ c i ( t ) is the control field as given in the absenceof any fluctuations, the fluctuations δc i ( t ) are describedby some probability distribution P c ( { δc i ( · ) } ), and (cid:15) c isa parameter describing the strength of fluctuations. Asexperimental measurements are sensitive to the net con-tribution of noise, we combine the environmental noiseand noise due to fluctuations in the control fields into asingle effective noise term, H n ( t ), given by (cid:15)H n ( t ) = (cid:15) (cid:88) i n i ( t ) ˆ q i (5)where (cid:15) ≡ (cid:15) se + (cid:15) c and n i ( t ) ≡ (cid:15) se (cid:15) se + (cid:15) c ˜ η i ( t ) + (cid:15) c (cid:15) se + (cid:15) c δc i ( t ) , (6)so that { n i ( · ) } are effective stochastic fields described bya (classical) effective probability distribution P [ { n i ( · ) } ],which can in principle be determined via experiment. Inwhat follows, we assume that we know P [ { n i ( · ) } ].Let us replace ˜ c i by c i , where c i is now understoodto be the control field. We can now specify the effectiveHamiltonian, which we will refer to as H (cid:15) ( t ), so as toremind us of its dependence on (cid:15) [note this Hamiltonianis different from Eq. (1)] H (cid:15) ( t ) = H s ( t, ˆ q i ) + (cid:88) i (cid:0) c i ( t ) + (cid:15)n i ( t ) (cid:1) ˆ q i ≡ H s ( t ) + H c ( t ) + (cid:15)H n ( t ) . (7)This effective Hamiltonian, H (cid:15) , contains the full quantumdescription of the system degrees of freedom ˆ q i , which arecoupled to both the stochastic fields n i ( t ) and the control fields c i ( t ), as well as to one another via the internalsystem dynamics as described by H s . B. Fidelity
We now construct the fidelity , i.e., a measure of suc-cess in effecting the specified unitary transformations,averaged over the noise history. At the end of the tran-sit time τ , our aim is to have the system evolution tobe as close as possible to U T . The evolution operator U (cid:15) ( t, { c i ( · ) } , { n i ( · ) } ) corresponding to H (cid:15) , cf. Eq. (7), isgiven by U (cid:15) ( τ, { c i ( · ) } , { n i ( · ) } ) = T exp (cid:20) − i (cid:90) τ d t H (cid:15) ( t ) (cid:21) (8) where the notation reminds us that U (cid:15) is a function of τ as well as a functional of the control and effectivenoise fields. We shall often use the shorthand U (cid:15) ( t ),provided there is no risk of confusion. The quantity U ( t ) ≡ U (cid:15) ( t ) | (cid:15) =0 corresponds to the evolution operatorin the absence of noise. Recall that at the transit time τ ,we have U ( τ ) = U T (i.e. the target unitary). We electto define the fidelity F as follows: F ( { c i ( · ) } ) = (cid:104) Tr U † T U (cid:15) ( τ ) (cid:105) n (9a)= (cid:104) Tr U † ( τ ) U (cid:15) ( τ ) (cid:105) n (9b)where the brackets (cid:104)· · · (cid:105) n denote averaging over { n i ( · ) } ,the trace operation is taken over the entire Hilbert spaceof the system, normalized by the dimension of the Hilbertspace so that Tr 1 l = 1. Observe that the fidelity obeys0 ≤ |F| ≤ U (cid:15) = U T (i.e., perfect control is achieved regard-less of initial state). The fidelity is a functional of thecontrol fields { c i ( · ) } . In order to best accomplish thesought for unitary transformation, we solve the varia-tional problem to find the set { c i ( · ) } that maximizes F .We shall make use of the formulation given in the secondline of Eq. (9b), which turns out to be efficacious whenexpressed in terms of a path integral. In terms of theHamiltonian, the fidelity is given by F [ { c i ( · ) } ] = (cid:68) Tr T K e i (cid:82) τ d t H ( t ) e − i (cid:82) τ d t H (cid:15) ( t ) (cid:69) n . (10)Here the operation T K corresponds to time-ordering onthe Keldysh contour (see, e.g., Ref. [12]): the first ex-ponential factor is anti -time ordered; the second one istime-ordered in the usual way. Equation (10) can be ex-pressed in terms of a Schwinger-Keldysh path integralover a closed-time contour, but there is a twist: un-like the mere usual quantum-dynamical problems (e.g.,for isolated quantum systems), for which this techniqueis commonly applied, in the present setting there is an asymmetry between the forward and backward branchesof the time contour. Specifically, the stochastic degrees offreedom { n i ( · ) } are only present in the forward branch.As we shall see, for some purposes, it is useful to referto the fidelity amplitude A instead of the fidelity. This isdefined via A [ { c i ( · ) } , { n i ( · ) } ] = Tr T K e i (cid:82) τ d t H ( t ) e − i (cid:82) τ d t H (cid:15) ( t ) , (11)which is just the expression for the fidelity before tak-ing the average over the stochastic fields, and thus is afunctional of both { c i ( · ) } and { n i ( · ) } . In terms of A wehave F [ { c i ( · ) } ] = (cid:104)A ( { c i ( · ) } , { n i ( · ) } ) (cid:105) n (12) C. Constraints
Determining the control sequence that best mitigatesnoise amounts to finding the set of control fields { c i ( · ) } that maximize F and yet satisfy all constraints imposedon the set { c i ( · ) } together with the boundary conditionat the end of the transit time τ , viz. U ( τ ) = U T . Inany realistic situation, there will be physical limitationson { c i ( · ) } that we may consider. For instance, each c i ( t )must be of finite magnitude , and its functional depen-dence on time would be constrained by the experimentaldevices in use.There are certain classes of constraints that can be en-tirely accounted for by using Lagrange multipliers. Theseinclude, but are not limited to, holonomic constraints.Explicit examples will be worked out in the followingsections. III. SINGLE SPIN ˆS
We now spell out explicitly how the abstract formalismpresented in Sec. II applies in the case of a single spin ˆS ,for now leaving the spin quantum number s arbitrary. Inthis setting, the total Hamiltonian is given by H (cid:15) ( t ) = ( ω ( t ) + (cid:15) n ( t )) · ˆS (13)where the field ω ( t ) corresponds to the external controlfield and n ( t ) is the stochastic field, discussed in Sec. II,that represents the effect of environmental noise and alsoaccounts for any fluctuations in the control field inher-ently present in the devices used to generate it. In addi-tion, ω ( t ) is the control in the absence of such fluctua-tions. In the present example, the system Hamiltonian, H s ( t ), vanishes.Our goal, then, is to determine the ω ( t ) that maxi-mizes the fidelity F at the end of the transit time τ . Inother words, we seek ω ( t ) such that, after averaging overdifferent realizations of n ( t ), the evolution operator is asclose as possible to some prescribed target U T .As discussed in Sec. II, the stochastic fields { n i ( · ) } aregoverned by a distribution functional P [ { n i ( · ) } which ispresumed to be known. In the present case, we take P to be Gaussian with zero mean and covariance given by (cid:104) n i ( t ) n j ( t ) (cid:105) n = N ij ( t, t (cid:48) ) . (14)We restrict our attention to stochastic fields that are sta-tionary in time and time-reversal invariant, in which case N ij ( t, t (cid:48) ) = N ij ( | t − t (cid:48) | ) = N ji ( | t − t (cid:48) | ) . (15)We make use of the Schwinger-Keldysh (SK) path-integral formulation (see, e.g. Refs. [10, 11] for originalwork by Schwinger and Keldysh and Ref. [12] for a mod-ern treatment), which for the present system can be doneeither in terms of spin coherent states or bosonic coherentstates (if one makes use of the Schwinger representationfor spins), see e.g. Ref. [18]. Choosing the latter, we makeuse of the mapping between the spin operator ˆS and thetwo-component bosons ˆa † ≡ ( a † , a † ), given by ˆS = 12 ˆa † · σ · ˆa (16) where σ = ( σ x , σ y , σ z ) are the Pauli matrices. The to-tal spin quantum number s is conserved by the dynam-ics, and is related to the total number of bosons in theSchwinger representation: s = a † · a . The total Hamil-tonian then takes the form: H (cid:15) ( t ) = a † · H (cid:15) ( t ) · a , (17)where H (cid:15) = 12 ω ( t ) · σ + (cid:15) n ( t ) · σ ≡ H c ( t ) + (cid:15) H n ( t ) . (18) A. Schwinger-Keldysh (SK) path integral
We now construct the path-integral expression for thefidelity amplitude A s , Eq. (11), where s indicates the(arbitrary) spin quantum number. We make use of thecoherent state basis | α (cid:105) ≡ | α , α (cid:105) , for complex α and α , defined by a | α (cid:105) = α | α (cid:105) (see Ref. [18]), in order toevaluate the SK path integral along the SK contour. Weuse the labels α f,b for the forward-in-time and backward-in-time branches along the contour, respectively. We thusobtain the expression A s = Tr (cid:90) D α f ( t )D α b ( t ) × | α b (0) (cid:105)× e − [ | α b ( τ ) | + | α f (0) | ]+ α (cid:63)b ( τ ) · α f ( τ ) × e i (cid:82) τ d t [ α (cid:63)f · ( i∂ t −H (cid:15) ) · α f − α (cid:63)b · ( i∂ t −H c ) · α b ] (cid:104) α f (0) | , (19)where the normalized trace Tr is taken over the completeset of two-mode bosonic number states | n, s − n (cid:105) num for n = 0 , , · · · , s , satisfying the constraint (cid:104) n, s − n | ˆa † · ˆa | n, s − n (cid:105) num = s , so as to fix the quantum spin number s . For further details on the meaning of Eq. (19), as wellas explicit expressions for the measures D α b,f and otherdetails concerning the path integral, we refer the readerto App. C. Note that our approach differs from conven-tional SK approach [12] used to study nonequilibriumquantum dynamics, in that the forward and backwardbranches of the time evolution are asymmetric with re-spect to noise: it is present in the forward branch andcompletely absent in the backward branch.The expression for A s in Eq. (19) can be evaluatedwith the help of a generating functional G s [ J ], which wedefine as follows: G s [ J ] = Tr (cid:90) D α f ( t )D α b ( t ) × | α b (0) (cid:105)(cid:104) α f (0) |× e − [ | α b ( τ ) | + | α f (0) | ]+ α (cid:63)b ( τ ) · α f ( τ ) × e i (cid:82) τ d t [ α (cid:63)f · ( i∂ t −H c ) · α f − α (cid:63)b · ( i∂ t −H c ) · α b ] × e (cid:82) τ d t [ J (cid:63) · α f + α f · J ] , (20)where, as anticipated, this expression contains the noise-free Hamiltonian on both branches of the Keldysh con-tour. Moreover, we have coupled the two-component ex-ternal source J ( t ) to the forward branch only. This allowsus to calculate quantum averages involving the forwardbranch fields. Recall that noise couples exclusively tothis branch. Observe that we have G s [0] = 1 since inthe absence of a source, G s is just the usual SK partitionfunction and the asymmetry due to the noise no longerarises. Physically, this corresponds to the feature thatin the absence of noise the fidelity is exactly unity, as isnatural.In terms of G s [ J ], we may evaluate the fidelity ampli-tude as follows: A s = e − i(cid:15) (cid:82) τ d t δδ J ( t ) ·H n ( t ) · δδ J (cid:63) ( t ) G s [ J ( t )] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) J ( t )= . (21) G s [ J ] is evaluated in App. C, giving G s [ J ] = (cid:73) d z πi e (cid:82) τ (cid:82) τ d t d t (cid:48) J (cid:63) ( t ) ·G c ( z,t,t (cid:48) ) · J ( t (cid:48) ) (2 s + 1)(1 − z ) z s +1 (22)where the integral over the auxiliary complex variable z is taken over any closed contour that encircles the originonce, but does not include the pole at z = 1. The variable z plays the role of a conjugate variable to the discretespin number s — this integral can be interpreted as anintegral transform between the z representation and s representation.In the exponent of Eq. (22) we have the Green function G c ( z, t, t (cid:48) ), which is given by G c ( z, t, t (cid:48) ) ≡ (cid:18) z − z + Θ( t − t (cid:48) ) (cid:19) U c ( t, t (cid:48) ) , (23)where Θ( t ) is the Heaviside step function (zero and unityfor negative and positive arguments, respectively), and U c ( t, t (cid:48) ) ≡ T exp[ − i (cid:82) tt (cid:48) d t (cid:48)(cid:48) H c ( t (cid:48)(cid:48) )] is the unitary matrixcorresponding to the control Hamiltonian. Note that inthis expression and formula (21) for A s , for an arbitraryspin s , we only need consider the dynamics of a two levelsystem (i.e. spin 1 / A s moretransparent, we can re-express Eq. (21) in a termsof a two-component Gaussian quantum field φ ( t ) † ≡ (cid:0) φ (cid:63) ( t ) , φ (cid:63) ( t ) (cid:1) whose description is completely given interms of the expectation value (cid:104) φ ( t ) φ (cid:63) ( t (cid:48) ) (cid:105) q = G c ( z, t, t (cid:48) ) , (24)where (cid:104)· · · (cid:105) q denotes a quantum average over φ ( t ), andhigher-order averages are determined by means of Wick’stheorem. Thus, A s = (cid:73) d z πi (cid:68) e − i(cid:15) (cid:82) τ d t φ (cid:63) H n φ (cid:69) q (2 s + 1)(1 − z ) z s +1 . (25)Note the the expressions for A s given in Eqs. (25) and(21) are entirely equivalent.The quantum expectation value for the fields φ ( t ) inEq. (25) can be evaluated exactly with the use of dia-grammatic techniques, which are developed in Section 2 of App. C. Through an application of these techniques,we obtain the expression: (cid:104) e − i(cid:15) (cid:82) τ d t φ (cid:63) H n φ (cid:105) q = (1 − z ) Det[1 l − z T e − i(cid:15) (cid:82) τ d t U † c H n U c ] . (26)Inserting this expression into Eq. (25) and evaluating theintegral over z , we obtain the result for the fidelity ampli-tude A s . Recalling that A s are functionals of the control ω ( t ) and the noise n ( t ), we finally obtain A s [ ω ( · ) , n ( · )] = 12 s + 1 s (cid:88) j = − s e − ji cos − (cid:0) A / [ ω ( · ) , n ( · )] (cid:1) , (27)where we are emphasizing the functional dependence of A s on ω ( · ) and n ( · ). The quantity A / [ ω ( · ) , n ( · )], ap-pearing in Eq. (27), is the fidelity amplitude for the spin-half case. It is given by the expression A / [ ω ( · ) , n ( · )] = 12 Tr T e − i (cid:82) τ d t U † c ( t ) H n ( t ) U c ( t ) (28)in which T denotes the usual time-ordering operator, τ is the transit time, and U c ( t ) = T e − i (cid:82) t d t (cid:48) H c ( t (cid:48) ) (29)is the evolution operator corresponding to the controlHamiltonian. The derivation of Eqs. (26,27) is given inSection 2 of App. C. B. Connection with quaternions
Before continuing with our development, let us pauseto streamline our notation. To this end, we observe a con-nection with quaternions that makes the physics moretransparent. The advantage stems from the fact thatquaternions provide a unified method for treating bothgeometrical 3-vectors and unitary evolution operators, allunder the language of quaternion algebra. In what comesbelow, the reader may freely replace pure quaternions(see App. A) by vectors in all settings, except when theyoccur in exponents. In that case, the correct interpreta-tion is to replace pure quaternions by the correspondingvector dotted with − i σ , where σ = ( σ x , σ y , σ z ) is a vec-tor of Pauli matrices. The quaternion wedge and dotproducts may be freely replaced by the vector cross anddot products respectively, and unit quaternions may bereplaced by the corresponding unitary matrices, as seenbelow. For a full review of all quaternion properties andoperations used in this Paper, see App. A.Before returning to the task of finding the fidelity andoptimizing it, let us define the quantities E i ( t ), given by E i ( t ) ≡ − i U † c ( t ) σ i U c ( t ) , (30)where, recalling that U c ( t ) is the unitary correspondingto the control Hamiltonian, E i ( t ) are the Pauli matricesin the rotating frame of the control fields. Note thatthe E i ( t ) satisfy the quaternion algebra, regardless of theform of the control fields, for all times t , i.e., E i ( t ) E j ( t ) = − δ ij l + (cid:15) kij E k ( t ) , (31)where the indices i , j , and k take on the values 1, 2, and3, corresponding respectively to x , y , and z , and repeatedindices are summed over. The symbol 1 l denotes the 2 × E ( t ) ≡ l ; then evidentlywe have E ( t ) E i ( t ) = E i ( t ) E ( t ) = E i ( t ) (32)for all i . The isomorphism between the set of quanti-ties E i ( t ) and quaternions is given by identifying E i ( t )(for i = 1 , ,
3) respectively with the quaternion imag-inary units ˆ ı, ˆ , ˆ k , and E ( t ) with the real number 1.Quaternion addition is defined component-wise, whereasmultiplication is defined through Eqs. (31,32); note thatquaternion multiplication is not commutative.Given this identification, for calculational purposes itis simpler to work with quaternion quantities. Recall thatany quaternion Q can be expressed in terms of compo-nents as Q = Q i E i with (cid:8) Q i (cid:9) real and i summed overthe values 0 , , ,
3. The component Q E is referred toas the real , or scalar , part of the quaternion, and theremainder is the imaginary part, also called the vector part, of the quaternion. Quaternions having vanishingscalar part are called pure quaternions. In what followswe will almost exclusively work with pure quaternions.For additional properties of quaternions, other types ofoperations (specifically, the dot and wedge products) andthe terminology associated with quaternions, we refer thereader to App. A.We refer to the set of quaternions E i ( t ), restricting i to be 1 , , or 3 as the rotating triad , and we refer to theset e i given by e i = − iσ i , (33)as the static triad , because these are fixed in the labo-ratory frame. For convenience we represent pure quater-nions using a bold font, i.e., p , and they can be rep-resented by a restricted sum over the indices, i.e., p = (cid:80) i =1 p i E i in the rotating basis and p = (cid:80) i =1 p i e i inthe static basis. Unless indicated otherwise, from nowon we interpret repeated indices as a summation overthe set i = 1 , , E i ( t )and e i within the quaternion language. Associated withthe evolution operator U c ( t ) is a unit quaternion; for sim-plicity, we indicate unit quaternions by means of a reg-ular font. If we take the pure quaternion Ω ( t ) to repre-sent the control Hamiltonian in the quaternion language, Ω ( t ) ≡ ω i ( t ) E i ( t ), then the associated unit quaternion, corresponding to the unitary in the vector language, isgiven by u c ( t ) = T e (cid:82) t d t (cid:48) Ω ( t (cid:48) ) (34)and the relation between the rotating triad E i ( t ) and thestatic triad e i is given by E i ( t ) = ¯ u c ( t ) e i u c ( t ) (35)where ¯ u c ( t ) ≡ T e − (cid:82) t d t (cid:48) Ω ( t (cid:48) ) is the quaternion conju-gate of u c ( t ) (corresponds to U † c ( t ) in matrix language,see App. A), and the operation on the right hand side isjust quaternion multiplication. In quaternion language,Eq. (35) tells us that the e i and E i ( t ) are related via apure rotation — in other words the rotating triad corre-sponds to a rigid rotation of the static triad.Let us now continue with the evaluation of the fidelity,but now making use of the quaternion language. Equa-tion (28) can be rewritten in terms of quaternions as A / [ E ( · ) , n i ( · )] = Sc T e (cid:15) (cid:82) τ d t n ( t ) , (36)where n ( t ) = n i ( t ) E i ( t ) is now interpreted as the pure quaternion representing the stochastic field in the rotat-ing frame, A / is now interpreted to be a functional ofthe rotating triad [we take E ( t ) as shorthand for the setof E i ( t )] as well as the stochastic fields { n i ( · ) } , and the Scoperation simply takes the scalar part of the expressionfollowing it.In this formulation, the idea is to find the best rotat-ing triad E ( t ), i.e., the one that maximizes the fidelity. Itmay seem that we have not gained much from this refor-mulation. However, working with E ( t ) is in fact a muchsimpler task than working with the control field Ω ( · ) di-rectly, as Ω ( · ) is buried inside time-ordered exponentials[see Eqs. (28,29)]. As a result, in practice one usually re-sorts to approximate schemes. In contrast, the set E ( t )appears at the same level as the stochastic fields, andthus can be treated exactly. Furthermore, once we havefound the best triad history E ( t ), it is straightforward torecover the control fields Ω ( t ): Ω ( t ) = 12 (cid:0) ∂ t E k ( t ) (cid:1) E k ( t )= 12 (cid:15) ijk E i ( t ) (cid:0) E j ( t ) · ∂ t E k ( t ) (cid:1) . (37)These are the control fields in the rotating frame . Ulti-mately, the objects we are interested in are the controlfield in the laboratory frame , ω ( t ) = ω i ( t ) e i , which istrivially found via ω ( t ) = u c ( t ) Ω ( t ) ¯ u c ( t )= 12 (cid:15) ijk u c ( t ) E i ( t ) ¯ u c ( t ) (cid:0) E j ( t ) · ∂ t E k ( t ) (cid:1) = 12 (cid:15) ijk e i (cid:0) E j ( t ) · ∂ t E k ( t ) (cid:1) . (38)In other words, ω i ( t ) = ω ( t ) · e i = Ω ( t ) · E i ( t ). The labframe components of the control field can be recovereddirectly from the rotating frame triad E ( t ), without theneed to take the intermediate step of computing either Ω ( t ) or u c ( t ): ω i ( t ) = 12 (cid:15) ijk E j ( t ) · ∂ t E k ( t ) (39)Here and elsewhere, the dot denotes the quaternion dotproduct (see App. A), and repeated indices are summedover. Note that Eq. (39) is an exact relation between ω ( t ) and E ( t ).By taking the triad E ( t ) as the relevant degrees of free-dom [as opposed to the control Ω ( t )], we no longer haveto worry about the time-ordered exponential associatedwith Ω ( t ), but there still remains an overall time-orderingoperator in front of the whole expression; see Eq. (28). For a pure quaternion n ( t ), one can always find another pure quaternion m (cid:15) ( t ) such that the relation T e (cid:15) (cid:82) τ d t n ( t ) = e (cid:15) m (cid:15) ( τ ) (40)is satisfied, and we note that m (cid:15) ( t ) is a function of (cid:15) as well as time. The expression for m (cid:15) ( t ) can be found,as is commonly done in quantum mechanics from theMagnus expansion; i.e., as an expansion of log T exp( (cid:15)X )(for some quantum operator X , i.e. see Ref. [22]) for asmall parameter (cid:15) . By using the quaternion formulation,however, it is straightforward to determine how the twoquantities n ( t ) and m (cid:15) ( t ) are related to each other ex-actly ; this relationship comes in the form of a differentialequation, viz., d m (cid:15) ( t ) dt = n ( t ) − (cid:15) m (cid:15) ( t ) ∧ n ( t ) + (cid:18) − (cid:15)m (cid:15) ( t )2 cot (cid:15)m (cid:15) ( t )2 (cid:19) (cid:98) m (cid:15) ( t ) ∧ (cid:0) (cid:98) m (cid:15) ( t ) ∧ n ( t ) (cid:1) (41)where m (cid:15) ( t ) ≡ | m (cid:15) ( t ) | is the quaternion modulus, and ˆm (cid:15) ( t ) = m (cid:15) ( t ) / | m (cid:15) ( t ) | is a unit pure quaternion and, assuch, it can be interpreted as the direction of m (cid:15) ( t ) (seeApp. B for a derivation of this result). Note that in thecase of pure quaternions the quaternion wedge productacts just like the vector cross product. If Eq. (41) issolved perturbatively in (cid:15) , one recovers the Magnus ex-pansion (see App. B for details), but the real power ofEq. (41) is that it is an exact relation: the solution of thisdifferential equation is equivalent to the exact summationover all terms in the Magnus expansion.After this lengthy detour, let us return to the taskof analyzing the fidelity. Making use of the results inEqs. (36,40), we find A / [ E ( · ) , m (cid:15) ( · )] = cos (cid:18) (cid:15)m (cid:15) ( τ )2 (cid:19) (42)which, in view of its simplicity relative to the expressionin terms of { n i ( · ) } , we can use to find the exact expressionfor the fidelity amplitude for general spin s : A s ( E ( t ) , m (cid:15) ( t )) = 12 s + 1 s (cid:88) j = − s e − ij(cid:15)m (cid:15) ( τ ) . (43)This enables us to construct a simple expression for thefidelity F s = (cid:104)A s (cid:105) , using Eqs. (43,14): F s [ E ( · )] = 12 s + 1 s (cid:88) j = − s e − ( j(cid:15) ) S [ E ( · )]+ O ( (cid:15) ) (44)where S [ E ( · )] is a functional of the rotating triad, i.e., S [ E ( · )] = 12 (cid:90) τ (cid:90) τ d t d t (cid:48) N ij ( t, t (cid:48) ) E i ( t ) · E i ( t (cid:48) ) . (45)In the expression (44) for F s , we make the assumptionthat (cid:15) is small, and only consider the leading-order term in (cid:15) in the exponent. Higher order contributions can beeasily computed, see App. B for details on how this isaccomplished.It is clear from Eq. (44) that in order to maximizethe fidelity, we need only minimize the functional S [ E ( · )]— note that this condition is independent of the spinnumber s . In other words, the functional form of therotating triad E i ( t ) (and therefore the control field; seeEq. (39) ) that maximizes the fidelity is the same for allvalues of s . The fidelity itself, however, does depend on s . C. Extremals and constraints
The task that remains is to find the rotating triad E i ( t )that minimizes the functional S [ E ( · )]; cf. Eq. [45]. Welook for solutions in the form of extremals of the action,i.e., we seek E i ( t ) such that the variation δS [ E ( · )] van-ishes. Note that the variations in the triad δ E i ( t ) are notfully arbitrary. They are constrained due to the fact thatthe triad has to rotate as a rigid body. Simply put, thevariations are constrained to take the form δ E i ( t ) = δ A ( t ) ∧ E i ( t ) , (46)where δ A ( t ) is now any arbitrary pure quaternion.In addition to the rigidity constraints just stated, thereare also experimental constraints present that one shouldaccount for, such as limitations on the frequency, ampli-tude, etc. that the control fields can take. Even in theabsence of experimental considerations, it is natural toimpose such constraints, if we are to make fair compar-isons between different realizations of the control fields.For instance, if there is no limit to the amplitude of thecontrol field, one may simply pick a large enough ampli-tude such that one can achieve the target operation overa time-scale much smaller than any time-scale associatedwith the noise N ij ( t, t (cid:48) ). The greater the separation oftime scales, the smaller will be the effective action S , andthus the higher will be the fidelity F . In this case, thereis no sense in comparing strong fields with weak ones, asstrong fields always win.In order to make a sensible comparison between dif-ferent choices for the control fields, we have to put someconstraint on the solution space in which we seek trajec-tories for the triad E ( t ). As an example, from a purelyphysical standpoint, one may choose to compare trajec-tories for which the total energy output associated withthe control field is prescribed . This quantity is given bythe time-integral of the square of the control field, so onehas the following constraint [19] E out ≡ (cid:90) τ d t | Ω ( t ) | . (47)In what follows, we shall also prescribe the transit time τ . As we are fixing the energy output, if we were to make τ too small, there would not be enough time to achievea given target. Effective values for τ should be boundedfrom below in an E out dependent way to ensure that wehave access to all desired targets. We shall seek optimalcontrols in the space of fixed τ and E out .To determine the optimal controls in this constrainedspace we seek minima corresponding to the constrainedfunctional S c , determined via S c ≡ (cid:90) τ d t (cid:90) τ d t (cid:48) {N ij ( t, t (cid:48) ) E i ( t ) · E j ( t (cid:48) )+ λ δ ( t − t (cid:48) ) | Ω ( t ) | (cid:9) , (48)where λ is the Lagrange multiplier associated with theenergy output constraint, and the output power | Ω ( t ) | is determined entirely in terms of the triad E ( t ), via | Ω ( t ) | = 18 δ ik δ j(cid:96) (cid:16) E i ( t ) · ∂ t E j ( t ) − E j ( t ) · ∂ t E i ( t ) (cid:17) × (cid:16) E k ( t ) · ∂ t E (cid:96) ( t ) − E (cid:96) ( t ) · ∂ t E k ( t ) (cid:17) = 12 (cid:15) ijk E i ( t ) · (cid:16) ∂ t E j ( t ) ∧ ∂ t E k ( t ) (cid:17) . (49)Thus we see that S c is a functional of the triad E ( t ) andits first time-derivatives only. To obtain the first lineof Eq. (49) we use the expression for Ω ( t ) in Eq. (37);the second line requires a little algebra. In practice itis simpler to work with the expression as given in thesecond line, because it is of lower order in the triad andits derivatives. Note that λ | Ω ( t ) | / kinetic energy associated with the triad E ( t ), with the Lagrange multiplier λ playing the role ofinertia.The constraint present in Eq. (48) is just one type ofa constraint that we may impose on the system. We arefree to impose other types. For instance, we can replacethe Lagrange multiplier in Eq. (48) by a matrix, and even make that time dependent:12 (cid:90) τ d t ω ( t ) · λ ( t ) · ω ( t ) . This may be useful, e.g., in the situation where the con-trol fields ω ( t ) are constrained to lie in the xy plane. Inthis case, we simply let λ zz tend to infinity. We are alsofree to impose constraints on the derivatives of ω ( t ), i.e.,to impose a frequency cutoff. Indeed, we have a lot offreedom on the types of constraints we may impose. Anadvantage of this method (based on extremals) is the easewith which one can implement them.For illustration purposes, we work with with the caseof scalar, time-independent λ in Eq. (48), imposing con-straints on the total energy output. From the action S c ,we can obtain the Euler-Lagrange equation, which willgive us the condition that the extremals must satisfy.For our case, we are specifically interested in the min-ima . Setting δS c /δ A ( t ) = 0 (see Eq. (46) ) we get theEuler-Lagrange equation for the triad, λ∂ t Ω ( t ) + E i ( t ) ∧ (cid:20)(cid:90) τ d t (cid:48) N ij ( t, t (cid:48) ) E j ( t (cid:48) ) (cid:21) = 0 , (50)where, in writing down Eq. (50), we used the fact that ∂ t Ω ( t ) = 12 (cid:15) ijk E i ( t ) (cid:2) E j ( t ) · ∂ t E k ( t ) (cid:3) . (51)The form of Eq. (50) fits naturally with the fact that Ω ( t )is the angular velocity of the triad, and indicates that theinterpretation of λ | Ω ( t ) | / λ playing the role of inertia, is correct.Given the connection between Ω ( t ) and the set E i ( t )(see Eq. (37) ), we also have ∂ t E i ( t ) = E i ( t ) ∧ Ω ( t ) . (52)It is useful to define the dual triad D i ( t ), via D i ( t ) = (cid:90) τ d t (cid:48) N ij ( t, t (cid:48) ) E j ( t (cid:48) ) . (53)The solution of the set of coupled equations ∂ t Ω ( t ) = − λ E i ( t ) ∧ D i ( t ) ,∂ t E i ( t ) = E i ( t ) ∧ Ω ( t ) , (54)along with the appropriate set of boundary conditions,determines the optimal controls. Recall that we havea target operation U T , which is to be satisfied at thetransit time t = τ . For our case, U T is simply a rotationoperator, and can therefore always be written as U T = e − i θ T ˆr , · σ (55)where θ T is the angle and ˆr is the axis of rotation. Itis easy to translate this into quaternion language andfind the corresponding unit quaternion u T that gives thetarget rotation u T = e q T , (56)0where q T = q i e i is the pure quaternion corresponding to − iθ T ˆr · σ , as determined by Eq. (33). The quaternionmodulus | q T | corresponds to the angle of rotation, and ˆq T = q T / | q T | corresponds to the axis of rotation. Theboundary conditions for the equation of motion Eq. (54)are then entirely determined from the target u T , via E i (0) = e i , E i ( τ ) = ¯ u T e i u T . (57)The presence of boundary conditions specific to thetarget rotation is an inconvenience when investigatinggeneric properties of extremals. It is easy to take care of arbitrary boundary conditions once and for all by trans-forming to a suitably rotating frame , which is describedby some angular drift velocity Ω ( t ), generally an arbi-trary function of time. The rotating-frame triad (cid:101) E i ( t ) isthen given by (cid:101) E i ( t ) = u ( t ) E i ( t ) ¯ u ( t ) , (58)where u ( t ) ≡ T exp (cid:104) (cid:82) t d t (cid:48) Ω ( t (cid:48) ) (cid:105) . Likewise, for anyrotating-frame quantity (cid:101) X (e.g., the control fields (cid:101) Ω ( t ) ),we have (cid:101) X = u ( t ) X ¯ u ( t ). We can also define the co-variant derivative for the rotating frame (cid:101) ∂ t , through therelation (cid:101) ∂ t X ( t ) = ∂ t X ( t ) − Ω ( t ) ∧ X ( t ); (59)likewise, for the noise kernel we have the covariant inte-gral operator (cid:90) τ d t (cid:48) (cid:101) N ij ( t, t (cid:48) ) X j ( t (cid:48) ) = (cid:90) τ d t (cid:48) N ij ( t, t (cid:48) ) × u ( t, t (cid:48) ) X j ( t (cid:48) ) ¯ u ( t, t (cid:48) ) (60)where u ( t, t (cid:48) ) ≡ u ( t ) ¯ u ( t (cid:48) ). In terms of the quantitiescarrying tildes, the Euler-Lagrange equations for S c canbe written in a generally covariant form, valid for arbi-trary time-dependent rotating frame Ω ( t ): (cid:101) ∂ t (cid:101) E i ( t ) = (cid:101) E i ( t ) ∧ (cid:101) Ω ( t ) , (cid:101) ∂ t (cid:101) Ω ( t ) = − λ (cid:101) E i ( t ) ∧ (cid:101) D i ( t ) . (61)A consequence of the fact that the form of S c expressedin terms of (cid:101) E i ( t ) is invariant under arbitrary Ω ( t ), isthat this general covariance carries over to all physicalequations. I.e., the relation between (cid:101) Ω ( t ) and (cid:101) E i ( t ) takesthe expected form (cid:101) Ω ( t ) = 12 (cid:15) ijk (cid:101) E i ( t ) (cid:104)(cid:101) E j ( t ) · (cid:101) ∂ t (cid:101) E k ( t ) (cid:105) . (62)We can take advantage of the freedom we have inchoosing Ω ( t ) to force the boundary conditions to takeon a simple form, (cid:101) E i (0) = (cid:101) E i ( τ ) = e i , (63)the only requirement being that in order to satisfy theboundary condition in the laboratory frame, we simply need (cid:0) u (0) , u ( τ ) (cid:1) = (cid:0) , u T (cid:1) , with u T being the tar-get operation. The solutions (cid:101) E i ( t ) then satisfy periodicboundary conditions, i.e., each member of the triad formsa closed loop on the unit sphere. The only other con-straint is that the triad rotates as a rigid body. Theloops are not constrained in any other way, i.e., they arefree to cross each other an arbitrary number of times.Without loss of generality, we choose Ω ( t ) to be aconstant, Ω D , so that all nontrivial dynamical behavioris displayed by the triad (cid:101) E i ( t ). This particular rotat-ing frame, with constant Ω D , corresponds to eliminat-ing the drift term whose effect is to take the triad E i ( t )through a free geodesic path on the unit sphere connect-ing the starting point at t = 0 with the target at t = τ .This procedure simply accounts for this trivial part ofthe evolution, allowing us to focus on corrections due tofluctuations in the environment. For this reason, it is ad-vantageous to solve the Euler-Lagrange equations in thisrotating frame. We take a further step, and work withthe difference δ (cid:101) Ω ( t ) ≡ (cid:101) Ω ( t ) − Ω D , and thus the set ofequations ∂ t δ (cid:101) Ω ( t ) = Ω D ∧ δ (cid:101) Ω ( t ) − λ (cid:101) E i ( t ) ∧ (cid:101) D i ( t ) , (64a) ∂ t (cid:101) E i ( t ) = (cid:101) E i ( t ) ∧ δ (cid:101) Ω ( t ) , (64b)satisfying boundary conditions given in Eq. (63). Thisset of variables holds the advantage that the trajecto-ries δ (cid:101) Ω ( t ) , (cid:101) E i ( t ), which describe deviations from the av-erage drift, are small in the limit where the noise matrix N ij ( t, t (cid:48) ) is small, and vanish in the limit of vanishingnoise, making them a natural set of variables to workwith.Before proceeding with examples, we note that the tra-jectories as determined from Eqs. (64a,64b) can belongto distinct topological sectors . This is easy to see in thecase where there is no noise, i.e. when Ω ( t ) = Ω D , sincethe set of drift vectors Ω ( n ) D = ( f + 2 πn ) ˆ r (for constant f and integer n ) give rise to trajectories for E i ( t ) thatsatisfy the same boundary conditions (i.e. correspond tothe same target), but differ in the number of times thetriad E i ( t ) winds around the axis determined by ˆ r . These winding numbers , n , then describe different topologicalsectors. The same situation arises in the case of nonzeronoise; here one also obtains sets of trajectories, differingin winding number, but satisfying the same boundaryconditions. When seeking solutions to the equations ofmotion, one can thus also specify the topological sectorof interest, which can without loss of generality be ac-counted for by the drift term Ω ( n ) D (where there is no riskof confusion we omit the label n for brevity).Finally, we mention that the laboratory frame compo-nents of the control field trajectories, the quantities weare ultimately interested in, can be determined entirely interms of the rotating frame quantities ω i ( t ) = ω ( t ) · e i = (cid:16) Ω D + δ (cid:101) Ω ( t ) (cid:17) · (cid:101) E i ( t ).1 D. Examples
A physically interesting situation arises when the noisehas a 1 /f character. In many systems, this accounts forthe material-dependent noise source that leads to deco-herence in quantum devices [16]. It is therefore of consid-erate importance for applications to mitigate this noiseeffectively. 1 /f noise can be understood as arising fromthe collective effect of multiple sources of telegraph noise that are coupled to the quantum system of interest, withthe weight of each source scaling as the inverse charac-teristic decay-rate of that source [16, 20].In what follows, we focus on the case of mitigatingnoise along a fixed axis, bearing in mind that general-izations are straightforward. Without loss of generality,we take the noise to point along the x axis, i.e., the onlynonvanishing component of the noise matrix is N xx ( t, t (cid:48) ).For 1 /f noise, this is obtained via the expression N xx ( t, t (cid:48) ) = ξ (cid:90) γ γ d γγ e − γ | t − t (cid:48) | , (65)where ξ denotes the strength of the noise, and γ ( γ )is the lower (upper) decay-rate cutoff for the ensembleof telegraph processes that are coupled to the spin. Forfrequencies ν within the range γ (cid:28) ν (cid:28) γ , the noisecorrelator in the frequency domain N xx ( ν ) ∼ ν − , i.e. is1 /f in character.For the noise form under consideration, we seek solu-tions for Ω ( t ) , E i ( t ) that dependend parametrically onthe Lagrange multiplier λ . The case λ − = 0 corre-sponds to making energy output infinitely costly, andgives rise to solutions corresponding to the geodesic pathon a sphere, i.e., we have Ω ( t ) = Ω D . For increasing λ − , the energy cost decreases, so that there is nontrivialcompetition between keeping energy costs low and com-pensating for the noise. In the limiting case of λ − → ∞ ,the energy costs become vanishingly small meaning thatthere are no constraints on the set of extremal controlfields we get to choose from. In this case, we can sim-ply take the control Ω ( t ) to be an infinitely sharp pulseacting over a vanishingly small transit time τ , and thisguarantees maximal fidelity. In actuality, as long as weensure that the spectral content of the noise and the con-trol do not overlap, which we are always free to do inthe case where there are no constraints on the control,we guarantee maximal fidelity at leading order in (cid:15) ; seeRefs. [6, 7].As we have mentioned, one advantage of this approachis that we can naturally find families of solutions corre-sponding to a given Ω D . Raising λ − = 0, we find a setof solutions that correspond to continuous deformationsof the geodesic on a sphere. As we raise λ − and loosenthe constraints on the energy output, we find solutionswhich improve the fidelity more and more.In order to illustrate our approach, we take the drift Ω D to be (2 π, , π ) /τ so that we have a nontrivialwinding-number (note that Ω D = ( √ − π, , π ) /τ would give us the same target rotation), and the target operation has both a nonvanishing component parallel tothe noise and a nonvanishing component orthogonal to it.In this case, we expect to obtain nontrivial time depen-dence in both the amplitude of the control field, | Ω ( t ) | ,and the direction (cid:98) Ω ( t ). It is convenient to scale all pa-rameters with respect to the transit time τ , i.e., we take ξ = 8 /τ (the strength of the noise relative to the controlis then ∼ (cid:15)ξ ), γ = 0 . /τ , and γ = 20 /τ . The valueschosen for the cutoffs γ and γ give us a wide range offrequencies for which the noise N xx is well approximatedby 1 /f noise.Our interest lies in the case where λ − is not too large,so that there is nontrivial competition between minimiz-ing energy output and maximizing the fidelity. The equa-tions of motion (61) can be solved straightforwardly, nu-merically. The quantities we are ultimately interested inare the control fields in the laboratory frame ω ( t ), whosecomponents are given by ω i ( t ) = Ω ( t ) · E i ( t )= ( Ω D + δ (cid:101) Ω ( t )) · (cid:101) E i ( t ) , (66)where Ω D = (2 π, , π ) /τ is the drift and δ (cid:101) Ω ( t ) ≡ (cid:101) Ω ( t ) − Ω D , with (cid:101) Ω ( t ) and (cid:101) E i ( t ) the control fields andthe triad as given in the rotating frame; which we findby solving the rotating-frame Euler-Lagrange equationsin the presence of the drift term Ω D (see Eqs. (64a,64b)and the accompanying discussion). The optimal controlfield history ω ( t ) in this nontrivial regime (with param-eters given in the caption) is shown in Fig. 1 (set of lightpurple arrows), where we have set λ − = 50 /τ . The driftfield Ω D ((2 π, , π ) /τ ) is shown (bold black arrow) forcomparison.To get a better understanding of the results obtainedin this example, let us take a closer look at how eachcomponent of ω ( t ) behaves. In Fig. 2, we plot the re-sults for δ ω ( t ) ≡ ω ( t ) − Ω D . The curves shown therecorrespond to several values of the Lagrange multiplier: λ − = (0 , , , , , /τ ; in each plot the solidcurve corresponds to λ − = 100 /τ , and the dashed curvescorrespond to a sequence of smaller values.Recalling the fact that in this example the noise liesalong the x axis, the interpretation of the plots becomesclear: δω x ( t ) is negative definite , meaning that the over-all amplitude of the control field is reduced relative tothe drift Ω D for the entire process . This is reasonablebecause a driving field along the x axis does nothing tocompensate for noise along the x direction: the drivingand noise terms would commute in this case. To com-pensate for the reduced drive along the x axis, the drivealong the z axis on average is increased. As far as thedrive along the y direction is concerned, although its timeaverage is zero, it takes on a nontrivial time-dependence.The behavior of the fields vary in concert in orderto satisfy the boundary conditions whilst reducing theweight of the control field ω ( t ) along the direction of thenoise (in this case the x direction) as much as possible.As λ − is increased, more and more of the weight lies2
0 1 2 3 4 5 6 7 8 9 -4 -3 -2 -1 0 1 2 3 0 1 2 3 4 5 6 7 8 9"inteq/gen_telegraph_gamma8_0_0_zeta8_0_0_W0_2pi_0_2pi_BETA_50_N_200_iter_40.txt" every 2:::i::i using (0.0):(0.0):(0.0):($3+$6):($4+$7):($5+$8):($2)"" every 2:::0::0 using (0.0):(0.0):(0.0):($3+$6):($4+$7):($5+$8) 0 0.2 0.4 0.6 0.8 1
FIG. 1. (Color online) Plot of ω ( t ) (in units of τ − ) for λ − = 50 τ − (light purple arrows), for the case of 1 /f noise. The driftfield Ω D = (2 π, , π ) /τ (bold black arrow), which also corresponds to the optimal control for the case λ − = 0, is shown forcomparison. The other parameter values are γ = 20 /τ , γ = 0 . /τ , and ξ = 8 /τ , as described in the text. (cid:45) (cid:45) (cid:45) t ∆ Ω x (cid:72) t (cid:76) (cid:45) t ∆ Ω y (cid:72) t (cid:76) (cid:45) (cid:45) t ∆ Ω z (cid:72) t (cid:76) FIG. 2. (Color online) Plot of the components of δ ω ( t ) (defined in text) vs. time, for λ − = 0 , , , , , τ − . The solidlines correspond to λ − = 100 τ − and the dashed lines correspond to intermediate values. As explained in the text, note thatthe x component of the control is reduced, relative to the noise-free case, at the expense of increasing the z component. Thisreduces the component of the control field along the noise (taken to be the x axis) in order to compensate more efficiently forthe noise; see the text for further discussion. along the y and z axes. As λ − is increased, and thus theenergy restrictions are lessened, we have at our disposallarger-amplitude control fields ω ( t ), and therefore largerfrequencies in the spectral content of E i ( t ). In light of thefact that 1 /f noise has higher weight at smaller frequen-cies, by increasing λ − we reduce the spectral overlapbetween the noise and the control, and hence increase the fidelity, which is what we set out to do. To recapitu-late, for the fidelity to be as large as possible, one shouldput as much of weight of the control field in a directionorthogonal to the noise for as long as possible. This allhas to be done in such a way as to satisfy the boundaryconditions (i.e., to obtain the sought after target oper-ation) and the energy constraints. This is why we canonly completely eliminate ω x ( t ) in favor of ω y,z ( t ) if we have access to arbitrarily large energy outputs.Recall that although the optimal control scheme (cid:16) Ω ( · ) , E i ( · ) (cid:17) is independent of the spin quantum num-ber s , the fidelity itself, however, is not, although thedependence on s is elementary in the limit of weak noise: F s [ E ( · )] = 12 s + 1 s (cid:88) j = − s e − ( j(cid:15) ) S [ E ( · )]+ O ( (cid:15) ) . (67)Thus, in the small- (cid:15) limit considered here, one does bestby reducing S = (cid:82) τ d t (cid:82) τ d t (cid:48) N ij ( t, t (cid:48) ) E i ( t ) · E j ( t (cid:48) ) asmuch at the constraint in energy output will allow, ob-taining an optimal control trajectory which is indepen-dent of (cid:15) . For a fixed value of (cid:15) (which we take to be 0 .
50 100 150 200 2500.350.400.450.500.55 1 Λ S FIG. 3. (Color online) Plot of the action S as a function of λ − , which determines the constraint on energy output forthe control fields Ω ( t ). Larger values of λ − correspond toless stringent constraints, i.e., larger energy outputs. As S approaches zero, the fidelity approaches unity (see Eq. (67) ).There is no limit to how small we can make S as long as weare willing to increase λ − . we plot S as a function of λ − in Fig. 3. Note that, inprinciple, there is no limit to how small S can be as longas we are willling to keep increasing λ − . For the case of λ − = 250 /τ , the maximum value present in Fig. 3, thefidelity for the s = 1 / . dominant noise contributions, whilechoosing the amplitudes | ω ( t ) | such that the triad E i ( t )has as little spectral overlap with the noise as possible,in other words, as much as is allowed by the constraintsimposed upon the control fields. IV. CONCLUSIONS
We have considered the problem of error mitigationin the control of quantum systems subject to time-dependent sources of noise, which in general includes en-vironmental noise and noise inherent to the experimen-tal control apparatus. We do this in terms of a fidelitymetric, which measures how faithfully the time evolutionmatrix (determined by the controls, internal system dy-namics, and the noise) reproduces a predetermined ‘tar-get’ unitary transformation at the end of a prescribedtransit time τ . We have tackled the problem througha modification of Schwinger-Keldysh path-integral tech-niques, which enables us to account for the effects of gen-eral sources of noise in a unified manner. By analogywith the Martin-Siggia-Rose scheme we ‘integrate out’noise sources, thus arriving at an effective deterministicformulation, which we use to represent the fidelity.Our methods yield the conditions obeyed by the con- trol field history such that the effects of noise are opti-mally mitigated and hence, on average, the fidelity cor-responding to the desired unitary transformation is aslarge as possible. These conditions are determined bysolving equations of motion that are found by extremiz-ing an effective action functional with the control fieldsplaying the role of the degrees of freedom. Our methodhas the advantage that it admits a wide range of con-straints through the use of Lagrange multipliers. Theseconstraints may correspond to those naturally found inexperiments, such as optimization of fidelity in the pres-ence of fixed energy input, which is the main type ofconstraint we consider in the article.As an application of our methods, we consider a systemcomposed of a single spin degree of freedom ˆS of arbitraryspin quantum number s coupled to a noise source that isof the 1 /f type, which is known to be a common sourceof quantum decoherence in many systems of interest. Weaddress the limit of weak noise, and study the problemin the case where the total energy output is constrainedat a fixed value. We use our methods to find the optimalcontrol field histories subject to applied constraints, andinterpret the results. In the case of weak noise, we findthat the optimal control field histories are independent ofthe spin quantum number s , and that the fidelity dependson s in an elementary manner.Although we have studied the case of a single spin asa specific example, it is straightforward to generalize ourmethods, e.g., by applying them to chains of coupledspins, atomic systems, as well as general noise distribu-tions. An intriguing avenue that merits exploration, andthat we have not pursued in this Paper, is the questionof many body effects on the control task, where interac-tions may have nontrivial repercussions. This is a ques-tion that can be addressed within the formalism we haveconstructed here. ACKNOWLEDGMENTS
We thank Kenneth R. Brown, Chingiz Kabytayev andJ. True Merrill for valuable discussions. This work wassupported by NSF grant DMR 12-07026 and by the Geor-gia Institute of Technology. Part of it was performed atthe Aspen Center for Physics, which is supported by NSFgrant PHY 10-66293.
Appendix A: Quaternion algebra
Here we briefly review quaternion algebra and explainthe notation used in our Paper. A quaternion q can bewritten in terms of its components as follows (using thesummation convention over repeated indices µ ) q = q µ e µ = q e + q e + q e + q e , (A1)4where the components q µ are real numbers, and thetetrad e µ are the basis quaternions . The tetrad e µ sat-isfy the quaternion multiplication table ( i , j , and k arerestricted to take nonzero values): e e i = e i e = e i e i e j = − δ ij + (cid:15) ijk e k . (A2)We can without loss of generality take e = 1; the quater-nion algebra can then be understood entirely in terms ofthe equation on the second line of Eq. (A2), i.e. we onlyneed to consider the triad e i , where i = 1 , ,
3. We referto the zeroth component q as the scalar part (in the lit-erature, it is often referred to as the real part in analogywith complex numbers), and we refer to q ≡ q i e i as the vector part (it is also, in analogy with complex numbers,the imaginary part ). In terms of its scalar and vectorparts, we can write an arbitrary quaternion q as q = q + q . (A3)Taking the analogy with complex numbers another step,we also have the quaternion conjugate ¯ q , where the con-jugation operation is defined through its action on thetriad e i . We have ¯ e i = − e i , (A4)and in terms of an arbitrary quaternion written in termsof its scalar and vector parts, we then have¯ q = q − q . (A5)In addition to the g eometric product between twoquaternions pq = r , we can also define the quaterniondot product p · q ≡
12 ( p ¯ q + q ¯ p ) , (A6)and the quaternion wedge product p ∧ q ≡
12 ( pq − qp ) . (A7)The magnitude of a quaternion is given by its modulus | q | , defined through the relation | q | = √ q ¯ q = (cid:112) ( q ) + q · q . (A8)Pure imaginary quaternions q , referred to as purequaternions , can naturally be interpreted as a geomet-ric vector in R . For pure quaternions, the quaterniondot and wedge products correspond to the usual vectordot and cross products respectively, p · q = 12 p i q j ( e i ¯ e j + e j ¯ e i )= p i q j δ ij p ∧ q = 12 p i q j ( e i e j − e j e j )= p i q j (cid:15) kij e k , (A9) where we use the quaternion multiplication table inEq. (A2) (repeated indices are to be summed over). Thegeometric quaternion product between two pure quater-nions pq subsumes both the dot and wedge products pq = − p · q + p ∧ q . (A10) Unit quaternions , defined as quaternions for which | u | = 1, can always be written in the form u = e p , (A11)where p is a pure quaternion. Unit quaternions are usedin the quaternion language to describe rotations. Givena unit quaternion v = e θ and any pure quaternion p wecan write down the rotated quaternion p (cid:48) as p (cid:48) = v p ¯ v = e θ p e − θ = p cos θ + (cid:98) θ ∧ p sin θ + (cid:0) p − ( (cid:98) θ · p ) (cid:98) θ (cid:1) (1 − cos θ ) , (A12)where p (cid:48) , p , and θ are all pure quaternions, and where wehave θ ≡ | θ | and (cid:98) θ ≡ θ / | θ | . The third line of Eq. (A12)is arrived at by the rules of quaternion algebra.The geometric interpretation of this relation is quite in-tuitive: θ describes both the angle of rotation (throughits modulus θ ), and the axis of rotation (through (cid:98) θ ). Notethat the quaternions ± v corresponds to the same rotation- this is analogous to the 2 to 1 correspondence between SU (2) and SO (3). Indeed, there is an isomorphism be-tween the description of rotations via unit quaternionsand via SU (2). An advantage of the quaternion descrip-tion of rotations is that it does not run into issues of‘gimbal lock’ that afflict more conventional descriptions,such as Euler angles. This robustness makes the quater-nion description very attractive not just in a theoretical,but a practical point of view as well.Finally, we note that composite rotations are alsoconveniently represented in terms of quaternions. It isstraightforward to show that the quaternion product oftwo unit quaternions is also a unit quaternion, and socan also be used to represent a rotation. The quaternioncorresponding to a rotation u followed by a rotation u is simply the quaternion product u = u u , which canbe easily seen from the following p (cid:48) = u p ¯ u = u u p u u = u ( u p ¯ u )¯ u , (A13)where in going from the second to third lines, we usedthe following relation for the conjugate of the product oftwo quaternions uv = ¯ v ¯ u . Appendix B: Relation between n ( t ) and m (cid:15) ( t ) for T e (cid:15) (cid:82) t d t (cid:48) n ( t (cid:48) ) = e (cid:15) m (cid:15) ( t ) Our goal is to find the relation between the purequaternions n ( t ) and m (cid:15) ( t ) (they can equivalently be5considered members of su (2) due to the isomorphism be-tween them) - we put a subscript (cid:15) in m (cid:15) ( t ) as a re-minder that it is generally a function of (cid:15) . In order tofind the relation between n ( t ) and m (cid:15) ( t ), we make use ofthe defining relation T e (cid:15) (cid:82) t d t (cid:48) n ( t (cid:48) ) = e (cid:15) m (cid:15) ( t ) , (B1)and the time derivative ddt T e (cid:15) (cid:82) t d t (cid:48) n ( t (cid:48) ) = (cid:15) n ( t ) T e (cid:15) (cid:82) t d t (cid:48) n ( t (cid:48) ) , (B2) in order to write the following expression (cid:15) n ( t ) = (cid:20) ddt e (cid:15) m (cid:15) ( t ) (cid:21) e − (cid:15) m (cid:15) ( t ) . (B3)To find the derivative of the exponential, we first rewriteit as an infinite product and get the result ddt e (cid:15) m (cid:15) ( t ) = ddt lim N →∞ δx → Nδx → N (cid:89) i =0 e (cid:15) δx m (cid:15) ( t ) = (cid:15) N →∞ δx → Nδx → N (cid:88) k =0 δx (cid:34) N − k (cid:89) i =0 e (cid:15) δx m (cid:15) ( t ) (cid:35) d m (cid:15) ( t ) dt N (cid:89) j = k e (cid:15) δx m (cid:15) ( t ) = (cid:15) (cid:90) d x e (cid:15) x m (cid:15) ( t ) d m (cid:15) ( t ) dt e (cid:15) (1 − x ) m (cid:15) ( t ) . (B4)Using this expression, we can rewrite Eq. (B3) as n ( t ) = (cid:90) d x e (cid:15) x m (cid:15) ( t ) d m (cid:15) ( t ) dt e − (cid:15) x m (cid:15) ( t ) . (B5) The geometric meaning of the integrand is simple: it cor-responds to a rotation of d m (cid:15) ( t ) /dt by an angle (cid:15)xm (cid:15) ( t )(where m (cid:15) ( t ) ≡ | m (cid:15) ( t ) | ) about the unit axis (cid:98) m (cid:15) ( t ) ≡ m (cid:15) ( t ) /m (cid:15) ( t ). Using quaternion algebra (or equivalentlythe su (2) algebra), we finde (cid:15) x m (cid:15) ( t ) d m (cid:15) ( t ) dt e − (cid:15) x m (cid:15) ( t ) = d m (cid:15) ( t ) dt + (cid:98) m (cid:15) ( t ) ∧ d m (cid:15) ( t ) dt sin x(cid:15)m (cid:15) ( t ) + (cid:98) m (cid:15) ( t ) ∧ (cid:18) (cid:98) m (cid:15) ( t ) ∧ d m (cid:15) ( t ) dt (cid:19) (1 − cos x(cid:15)m (cid:15) ( t )) . (B6)With this, it is now trivial to carry out the integral in x in Eq. (B5). In order to simplify the expression further,we make use of the relation d m (cid:15) ( t ) dt = ˙ (cid:98) m (cid:15) ( t ) m (cid:15) ( t ) + (cid:98) m (cid:15) ( t ) ˙ m (cid:15) ( t ) , (B7)where the overhead dot denotes a time derivative, alongwith the fact that (cid:98) m (cid:15) ( t ) and ˙ (cid:98) m (cid:15) ( t ) are orthogonal since (cid:98) m (cid:15) ( t ) is a unit pure quaternion. We obtain the expression n ( t ) = ˙ m (cid:15) ( t ) (cid:98) m (cid:15) ( t ) + ˙ (cid:98) m (cid:15) ( t ) sin (cid:15)m (cid:15) ( t ) (cid:15) + (cid:98) m (cid:15) ( t ) ∧ ˙ (cid:98) m (cid:15) ( t ) 1 − cos (cid:15)m (cid:15) ( t ) (cid:15) (B8) This equation is very useful, as it gives us an exact expres-sion for n ( t ) in terms of m (cid:15) ( t ) and its time derivatives.Fortunately it is possible to invert Eq. (B8) and unam-biguously solve for both ˙ m (cid:15) ( t ) and ˙ (cid:98) m (cid:15) ( t ) in terms of n ( t ), m (cid:15) ( t ), and (cid:98) m (cid:15) ( t ), where we find after some algebra˙ m (cid:15) ( t ) = n ( t ) · (cid:98) m (cid:15) ( t )˙ (cid:98) m (cid:15) ( t ) = − (cid:15) (cid:98) m (cid:15) ( t ) ∧ n ( t ) − (cid:18) (cid:15) (cid:15)m (cid:15) ( t )2 (cid:19) (cid:98) m (cid:15) ( t ) ∧ ( (cid:98) m (cid:15) ( t ) ∧ n ( t )) . (B9)Using Eq. (B7), we can combine both equations inEq. (B9) to find a single differential equation unambigu-ously relating m (cid:15) ( t ) and n ( t ). After a little algebra wefind6 d m (cid:15) ( t ) dt = n ( t ) − (cid:15) m (cid:15) ( t ) ∧ n ( t ) + (cid:18) − (cid:15)m (cid:15) ( t )2 cot (cid:15)m (cid:15) ( t )2 (cid:19) (cid:98) m (cid:15) ( t ) ∧ ( (cid:98) m (cid:15) ( t ) ∧ n ( t )) . (B10)Note that Eq. (B10) is an exact relation between m (cid:15) ( t )and n ( t ). Solving it is equivalent to summing up all termsin the Magnus expansion for the case where n ( t ) is anarbitrary time dependent linear combination of elementsof su (2), or equivalently, any pure quaternion since thetwo are isomorphic. For cases where (cid:15) is large, Eq. (B10)can be used to give us the exact solution — it can alsobe used to generate a diagrammatic expansion in (cid:15) . Bysumming up certain classes of diagrams, we can obtainuseful results even for moderately large (cid:15) .For small (cid:15) , we can solve Eq. (B10) perturbatively in afairly straightforward to arbitrary order. This is useful ifone is interested in finding a series expansion for higherorder contributions for the fidelity functional. We seek asolution of the form m (cid:15) ( t ) = ∞ (cid:88) j =0 (cid:15) j m ( j ) ( t ) , (B11)where m ( j ) ( t ) is the j th order term in the perturba-tion theory (we drop the subscript (cid:15) for the perturbationterms m ( j ) ( t ) since the (cid:15) dependence is accounted for inthe prefactor (cid:15) j ). Let us rewrite Eq. (B10) in a way thatis more amenable to this perturbative treatment. Wemake use of the following series expansion [21]1 − x x ∞ (cid:88) j =1 ( − j +1 B j x j (2 j )! , (B12)where B j are the first Bernoulli numbers. The firstBernoulli numbers can be obtained from the Bernoullipolynomials, defined through the generating function x e tx e x − ∞ (cid:88) j =0 B j ( t ) x j j ! , (B13)where we have B j ≡ B j (0), and are given explicitly by B j = j (cid:88) k =0 k (cid:88) (cid:96) =0 ( − (cid:96) k ! (cid:96) j (cid:96) !( k − (cid:96) )!( k + 1) . (B14)Using Eq. (B12), we then see that the third term inEq. (B10) only contains positive even powers of (cid:15)m (cid:15) ( t ),where in particular the leading order term is quadraticin (cid:15) . In a series expansion in (cid:15) , in the right hand side ofEq. (B10), we get terms of the form( − j +1 m j(cid:15) (cid:98) m (cid:15) ( t ) ∧ ( (cid:98) m (cid:15) ( t ) ∧ n ( t )) . These terms can be rewritten entirely in terms of m (cid:15) ( t ) only , simplifying the perturbative expansion since we do not have to consider the amplitude m (cid:15) ( t ) and the unitaxis (cid:98) m (cid:15) ( t ) separately. To do this, we introduce the nested wedge product, which can be defined recursively throughthe relation ( a ∧ ) j b ≡ a ∧ (cid:2) ( a ∧ ) j − b (cid:3) , (B15)where j is an integer greater than or equal to zero, andwhere we use the convention ( a ∧ ) b = b . Using thisdefinition for the nested wedge product, along with somequaternion algebra, we can show that( − j +1 m j(cid:15) (cid:98) m (cid:15) ( t ) ∧ ( (cid:98) m (cid:15) ( t ) ∧ n ( t )) = ( m (cid:15) ( t ) ∧ ) j n ( t ) . (B16)Next, we take advantage of the fact that the oddBernoulli numbers, B j +1 , vanish for all integers j > B = 1 and B = − / d m (cid:15) ( t ) dt = ∞ (cid:88) j =0 (cid:15) j B j j ! ( m (cid:15) ( t ) ∧ ) j n ( t ) . (B17)We now define the nested wedge product for N gener-ally distinguishable factors, which can be, for example,factors m ( j ) ( t ) , m ( j ) ( t ) , · · · , m ( j N ) ( t ), appearing in theperturbative expansion. It is defined recursively as m { j j ··· j N } ( t ) ∧ n ( t ) ≡ m ( j ) ( t ) ∧ (cid:104) m { j ··· j N } ( t ) ∧ n ( t ) (cid:105) . (B18)For the special case where the number of factors N van-ishes, i.e. { j · · · j N } = {∅} , we define m {∅} ( t ) ∧ n ( t ) ≡ n ( t ) — with this convention all other cases are uniquelydefined through Eq. (B18).Using Eqs. (B11,B17,B18), it is straightforward to findthe expression for arbitrary m ( j ) ( t ) in the perturbativeexpansion. The zeroth order j = 0 term is given by theexpression m (0) ( t ) = (cid:90) t d t (cid:48) n ( t (cid:48) ) , (B19)while for all other values j > m ( j ) ( t ) = j (cid:88) k =1 B k k ! (cid:88) (cid:80) k(cid:96) =1 i (cid:96) = j − k (cid:90) t d t (cid:48) m { i i ··· i N } ( t (cid:48) ) ∧ n ( t (cid:48) ) , (B20)where the second sum over the set i , i , · · · , i k inEq. (B20) satisfies the constraint that (cid:80) k(cid:96) =1 i (cid:96) = j − k .One can easily check that with this constraint, only m ( k ) ( t ) for 0 ≤ k ≤ j − j th order perturbation term entirelyin terms of lower order perturbation terms m ( k ≤ j − ( t )and n ( t ). Eqs. (B19,B20) reproduce what is expectedfrom the Magnus expansion [22], which is not surpris-ing since the same approximation scheme has been usedhere. It can be easily shown that our perturbative re-sult, Eqs. (B19,B20), readily generalizes to all Lie al-gebras, provided one replaces the nested wedge prod- ucts with nested commutators (Lie brackets) - in partic-ular the quaternion wedge product corresponds exactlyto the commutator for su (2). For the case of quater-nions (also su (2) due to the isomorphism), we have usedquaternion methods to derive an exact differential equa-tion Eq. (B10) in a straightforward manner, which whensolved is equivalent to summing up all terms in the Mag-nus expansion. The result given in Eqs. (B19,B20), alongwith Eq. (B11), gives us a perturbative solution to thedifferential equation in powers of (cid:15) — this reproduces theMagnus expansion term by term.Using Eqs. (B19,B20) we find explicit expressions forthe first few terms for in the perturbative expansion,given here m (0) ( t ) = (cid:90) t d t n ( t ) m (1) ( t ) = 12 (cid:90) t d t (cid:90) t d t n ( t ) ∧ n ( t ) m (2) ( t ) = 16 (cid:90) t d t (cid:90) t d t (cid:90) t d t { n ( t ) ∧ [ n ( t ) ∧ n ( t )] + n ( t ) ∧ [ n ( t ) ∧ n ( t )] } etc ... (B21)The form of Eq. (B10) also suggests a different way ofapproximating the solution which is better than the Mag-nus expansion in the sense that we include contributionsfrom all orders of (cid:15) at each iteration (with the exceptionof zeroth order which coincides with the expression ob-tained from the Magnus expansion). Because of this, wethen obtain a much better approximation which workswell even in the case were (cid:15) is large. We denote the [ j ]thorder approximation with a superscript [ n ] using brackets instead of parentheses to distinguish this from the Mag-nus expansion (note that we now include the subscript (cid:15) since each term now does depend explicitly on (cid:15) ). Forthe leading order we have m [0] (cid:15) ( t ) = (cid:90) t d t (cid:48) n ( t (cid:48) ) , (B22)and all higher order approximations j > m [ j ] (cid:15) ( t ) = (cid:90) t d t (cid:48) (cid:40) n ( t (cid:48) ) − (cid:15) m [ j − (cid:15) ( t (cid:48) ) ∧ n ( t (cid:48) ) + (cid:32) − (cid:15)m [ j − (cid:15) ( t (cid:48) )2 cot (cid:15)m [ j − (cid:15) ( t (cid:48) )2 (cid:33) (cid:98) m [ j − (cid:15) ( t (cid:48) ) ∧ ( (cid:98) m [ j − (cid:15) ( t (cid:48) ) ∧ n ( t (cid:48) )) (cid:41) . (B23)The approximate n th order expression is simply given bythe n th iterate of this procedure m (cid:15) ( t ) (cid:39) m [ n ] (cid:15) ( t ) . (B24)This procedure is equivalent to solving the exact equa-tion Eq. (B10) by iteration — when it is carried out to infinite order, it gives us the exact solution (as long asthe solution obtained this way is unique , then it is un-questionably the solution), i.e. we have m (cid:15) ( t ) = lim j →∞ m [ j ] (cid:15) ( t ) . (B25)8In practice we find that even for fairly large values of (cid:15) ,this procedure converges very quickly to a unique answer,i.e. there is some finite order N upon which m [ N ] (cid:15) ( t ) ispractically indistinguishable from the exact solution. Forextremely large values of (cid:15) this procedure may not con-verge — i.e. the iterates may jump back and forth be-tween two or more different values — in which case onemust modify the approximation to obtain a unique an-swer. One can always check whether this unique answeris the solution by plugging it back into Eq. (B10). Appendix C: Evaluation of the path integralexpression for the fidelity amplitude
Here we show how to evaluate the path integral expres-sion for the fidelity amplitude (given in Eq. (19) in themain Paper). We rewrite it here for convenience (in thisAppendix, unlike the main Paper, we write explicitly thenormalization prefactor, (2 s + 1) − , corresponding to thedimension of the Hilbert space) A s = 12 s + 1 Tr (cid:90) D α f D α b × | α b (0) (cid:105)(cid:104) α f (0) |× e − ( | α b ( τ ) | + | α f (0) | )+ α (cid:63)b ( τ ) · α f ( τ ) × e i (cid:82) τ d t [ α (cid:63)f · ( i∂ t −H (cid:15) ) · α f − α (cid:63)b · ( i∂ t −H c ) · α b ] , (C1)where | α f,b (0) (cid:105) are two-component coherent states cor-responding to the two-mode operator ˆa † ≡ (ˆ a † , ˆ a † ). Thetrace is taken over the complete set of states | ψ (cid:105) satis-fying the constraint (cid:104) ψ | ˆa † ˆa | ψ (cid:105) = s , fixing the quantumspin number s . The integral measure D α i for i = f, b isgiven by the following expressionD α i ≡ lim N →∞ N (cid:89) n =0 d α i ( t n ) , (C2)where t n = nτ /N and τ denotes the total transit time.The local (in time) measure d α i ( t n ) is given byd α i ( t n ) ≡ d α (1) i ( t n ) d α (2) i ( t n ) , (C3)where the superscripts (1) and (2) denote the componentsof the two-component field α i , and we haved α ( j ) i ( t n ) ≡ d α ( j ) i ( t n ) d α (cid:63) ( j ) i ( t n )2 πi = d (cid:60) α ( j ) i ( t n ) d (cid:61) α ( j ) i ( t n ) π , (C4)where (cid:60) α ( j ) i ( t n ) and (cid:61) α ( j ) i ( t n ) denote the real and imag-inary parts of α ( j ) i ( t n ) respectively. The first and secondlines of Eq. (C4) are equivalent, and it is a matter ofconvenience as to which representation we use. To evaluate Eq. (C1), we introduce the generatingfunctional G s [ J ] G s [ J ] = 12 s + 1 Tr (cid:90) D α f D α b × | α b (0) (cid:105)(cid:104) α f (0) |× e − ( | α b ( τ ) | + | α f (0) | )+ α (cid:63)b ( τ ) · α f ( τ ) × e i (cid:82) τ d t [ α (cid:63)f · ( i∂ t −H c ) · α f − α (cid:63)b · ( i∂ t −H c ) · α b ] × e (cid:82) τ d t [ J (cid:63) · α f + α (cid:63)f · J ] , (C5)where J and J (cid:63) are two-component source fields whichcouple only to the forward branch of the path integral(due to the fact that the noise field only couples to thisbranch — see discussion in main Paper). We can eval-uate A s (which includes the noise contribution) via thefollowing expression A s = exp (cid:26) − i(cid:15) (cid:90) τ d t δδ J ( t ) H n ( t ) δδ J (cid:63) ( t ) (cid:27) G s [ J ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) J ( t )=0 J (cid:63) ( t )=0 . (C6)Note that G s [ J ] depends only on the noise-free term inthe Hamiltonian (which shows up symmetrically in bothforward and backward branches). For this reason, itis a simpler task to calculate G s [ J ] first, and then useEq. (C6) to obtain A s , as opposed to calculating A s di-rectly. Taking the generating functional route also holdsthe advantage that the physics is more transparent.Our first goal is to evaluate G s [ J ] exactly, which weshow in the following subsection. Once we have thisquantity we will use it to obtain an exact expression forthe fidelity amplitude A s .
1. Evaluation of the generating functional G s [ J ] In what follows, we shall evaluate the path integral ex-pression for G s [ J ] (defined in Eq. (C5)) exactly . We startwith a change of variables (a Keldysh rotation ) definedby ψ ( t ) = α f ( t ) + α b ( t )2 η ( t ) = α f ( t ) − α b ( t ) (C7)It is easy to check that the Jacobian associated with thischange of variables is exactly unity. The description ofthe path integral in terms of these fields has a nice physi-cal interpretation. The symmetric combination ψ is usu-ally referred to as the ‘classical’ component in the lit-erature (it is the only component that survives in theclassical limit, giving rise to a unique classical trajectorycorresponding to the saddle point of G s [ J ]). The anti-symmetric combination η is referred to as the ‘quantum’component since it accounts for deviations from the clas-sical trajectory [12]. The evaluation of the path integralin terms of this set of variables simplifies greatly as weshall see below.9Let us rewrite the expression for G s [ J ] in terms of thisnew set of variables. After some algebra (and integrationby parts in order to move time derivatives from η ’s to ψ ’s) we obtain the following expression G s [ J ] = (cid:90) D ψ D η W ( ψ , ψ (cid:63) , η , η (cid:63) ) e − | η τ | × e (cid:82) τ d t [ i η (cid:63) · ( i∂ t ψ −H c · ψ − i J / ψ (cid:63) · J ] × e (cid:82) τ d t [( − i∂ t ψ (cid:63) − ψ (cid:63) ·H c − i J (cid:63) / · i η + J (cid:63) · ψ ] , (C8)where we use the shorthand notation ψ ≡ ψ (0), η ≡ η (0), and η τ ≡ η ( τ ). The function W ( ψ , ψ (cid:63) , η , η (cid:63) ) inthe integrand of Eq. (C8) is given by the expression W ( ψ , ψ (cid:63) , η , η (cid:63) ) = e ( ψ (cid:63) · η − η (cid:63) · ψ ) × s + 1 Tr | ψ − η (cid:105)(cid:104) ψ + η | , (C9)where the states | ψ ± η (cid:105) are coherent states, and werecall that the trace in Eq. (C9) is constrained to beover states for which the spin quantum number s is fixed ,i.e. the constrained subspace contains the two-mode Fockstates | n , n (cid:105) for which n + n = 2 s .Before evaluating the path integral expression for G s [ J ]in Eq. (C8), we take a moment to write down explicitlythe form taken by the integral measures in the new setof variables, ψ and η . We haveD ψ ≡ lim N →∞ N (cid:89) n =0 d ψ ( t n ) , D η ≡ lim N →∞ N (cid:89) n =0 d η ( t n ) , (C10)with the local measures d ψ ( t n ) and d η ( t n ) taking theexpected formd ψ ( t n ) ≡ d ψ (1) ( t n ) d ψ (2) ( t n )d η ( t n ) ≡ d η (1) ( t n ) d η (2) ( t n ) , (C11)where as usual, the superscripts denote the componentsof the fields. Importantly, the expression for the mea-sures d ψ ( j ) ( t n ) and d η ( j ) ( t n ) slightly differ. Suppress-ing time arguments for brevity, we haved ψ ( j ) ≡ d ψ ( j ) d ψ (cid:63) ( j ) πi = 2 d (cid:60) ψ ( j ) d (cid:61) ψ ( j ) π d η ( j ) ≡ d η ( j ) d η (cid:63) ( j ) πi = d (cid:60) η ( j ) d (cid:61) η ( j ) π , (C12)where we see that the measure for ψ is a factor of fourlarger than the measure for η . Note that the expressionsin the second and third columns are entirely equivalentrepresentations, and one can simply choose whicheverrepresentation is most convenient when performing cal-culations.Going back to the expression for G s [ J ] in Eq. (C8),we can immediately evaluate the integrals over η , η (cid:63) , η τ and η (cid:63)τ . The only contribution to the integrand for thefields η , η (cid:63) that survives the time-continuum limit of the path integral comes from the term W ( ψ , ψ (cid:63) , η , η (cid:63) )given in Eq. (C9), giving us the result W ( ψ , ψ (cid:63) ) ≡ (cid:90) d η W ( ψ , ψ (cid:63) , η , η (cid:63) )= ( − s s + 1 L (1)2 s (4 | ψ | )e − | ψ | , (C13)where L (1)2 s ( x ) is an associated Laguerre polynomial,which we express here in terms of its Rodrigues repre-sentation L ( k ) n ( x ) = e x x − k n ! d n dx n (e − x x k + n )= n (cid:88) j =0 ( k + n )!( n − j )!( k + j )! ( − x ) j j ! . (C14)In the process of evaluating Eq. (C13), we made use ofthe following expression giving the overlap between thetwo-component coherent states | ψ ± η (cid:105) and the two-component Fock states | n, s − n (cid:105) : (cid:104) n, s − n | ψ ± η (cid:105) = e − | ψ ± η | (cid:112) n !(2 s − n )! (cid:18) ψ (1)0 ± η (1)0 (cid:19) n × (cid:18) ψ (2)0 ± η (2)0 (cid:19) s − n , (C15)where the superscripts (1) and (2) on the right hand sideof Eq. (C15) denote the components of the respectivefields. The function W ( ψ , ψ (cid:63) ) given in Eq. (C13), re-ferred to as the Wigner function in the context of quan-tum dynamics, can be interpreted as a distribution overthe fields ψ , ψ (cid:63) . Note that this distribution function,as described by W ( ψ , ψ (cid:63) ), is generally not positive def-inite, and in our case it shows oscillatory behavior. Asfar as the integral over η τ , η (cid:63)τ is concerned, the only con-tribution that survives the time-continuum limit is givenby (cid:90) d η τ e − | η τ | = 1 . (C16)We now proceed to evaluate the integrals over the fields η ( t ) , η (cid:63) ( t ) in the bulk (i.e. for all t > Dirac delta functions lim N →∞ N − (cid:89) n =0 (cid:34) δ (cid:16) ψ ( t n +1 ) + iδt [ H c ( t n ) · ψ ( t n ) + i J ( t n ) / (cid:17) × δ (cid:16) ψ (cid:63) ( t n +1 ) − iδt [ ψ (cid:63) ( t n ) · H c ( t n ) + i J (cid:63) ( t n ) / (cid:17) (2 π ) (cid:35) , (C17)where δt ≡ τ /N . This result makes the evaluation of theintegrals over ψ ( t ) and ψ (cid:63) ( t ) (for t >
0) simple as well.0The delta functions constrain the fields ψ ( t ) , ψ † ( t ) (for t >
0) to obey the following equations of motion i∂ t ψ = H c · ψ + i J i∂ t ψ (cid:63) = − ψ (cid:63) · H c − i J (cid:63) , (C18)where we have suppressed time arguments for brevity.Assuming general initial conditions ψ (0) = ψ we can write down the exact solution for all t > ψ ( t ) = U c ( t, · ψ + 12 (cid:90) t d t (cid:48) U c ( t, t (cid:48) ) · J ( t (cid:48) ) ψ (cid:63) ( t ) = ψ (cid:63) · U c (0 , t ) − (cid:90) t d t (cid:48) J (cid:63) ( t (cid:48) ) · U c ( t (cid:48) , t ) , (C19)where we have U c ( t, t (cid:48) ) = T exp (cid:20) − i (cid:90) tt (cid:48) H c ( t (cid:48)(cid:48) )d t (cid:48)(cid:48) (cid:21) . (C20)Note that in general we have ψ (cid:63) ( t ) (cid:54) = [ ψ ( t )] (cid:63) , as one cansee from Eq. (C19), and these fields have to be treatedas being independent of one another. This also appliesto the fields ψ (cid:63) and ψ .Applying the results obtained in Eqs. (C13), (C16),(C17), and (C19) to Eq. (C8), we get the following ex-pression for G s [ J ]: G s [ J ] = e (cid:82) τ d t (cid:82) t d t (cid:48) [ J (cid:63) ( t ) ·U c ( t,t (cid:48) ) · J ( t (cid:48) ) − J (cid:63) ( t (cid:48) ) ·U c ( t (cid:48) ,t ) · J ( t )] × (cid:90) d ψ W ( ψ , ψ (cid:63) ) e (cid:82) τ d t [ J (cid:63) ( t ) ·U c ( t, · ψ + ψ (cid:63) ·U c (0 ,t ) · J ( t )] . (C21)All that remains to be done in order to obtain the finalresult for G s [ J ] is the evaluation of ordinary integrals overthe fields ψ and ψ (cid:63) . This is a Gaussian integral witha polynomial prefactor in the integrand (see Eq. (C13)),and it can therefore be evaluated exactly. Before doing sowe rewrite the expression for W ( ψ , ψ (cid:63) ) by making useof the following integral representation for the associatedLaguerre polynomial L ( k ) n ( x ) = (cid:73) d z πi ( − k e xz z (1 + z ) k +1 z n +1 , (C22)where the contour runs counterclockwise and encloses thepole at z = 0, but not the pole at z = 1. We can use theexpression in Eq. (C22) to rewrite W ( ψ , ψ (cid:63) ) as follows W ( ψ , ψ (cid:63) ) = 12 s + 1 (cid:73) d z πi e − ( − z z ) | ψ | (1 + z ) z s +1 (C23)This greatly simplifies the evaluation of the integrals over ψ and ψ (cid:63) in Eq. (C21), and we obtain the followingresult: (cid:90) d ψ · · · = (cid:73) d z πi e ( z − z ) (cid:82) τ (cid:82) τ d t d t (cid:48) J (cid:63) ( t ) ·U c ( t,t (cid:48) ) · J ( t (cid:48) ) (2 s + 1)(1 − z ) z s +1 . (C24)It is a simple matter now to combine the expression inEq. (C24) with the prefactor in Eq. (C21) to obtain theexpression for G s [ J ]. After some algebra, we finally get G s [ J ] = (cid:73) d z πi e (cid:82) τ (cid:82) τ d t d t (cid:48) J (cid:63) ( t ) ·G c ( z,t,t (cid:48) ) · J ( t (cid:48) ) (2 s + 1)(1 − z ) z s +1 , (C25) where G c ( z, t, t (cid:48) ) (which plays a role similar to that of a Green’s function ) is given by the expression G c ( z, t, t (cid:48) ) = (cid:18) z − z + Θ( t − t (cid:48) ) (cid:19) U c ( t, t (cid:48) ) . (C26)The quantity Θ( t − t (cid:48) ) denotes the Heaviside (unit step)function, defined asΘ( t − t (cid:48) ) = (cid:26) t > t (cid:48) t < t (cid:48) . (C27)It is straightforward to evaluate the integral over z in Eq. (C25) if we wish, but the expression for G s [ J ] ismuch more useful for doing calculations as is, and for thisreason this is the expression used in the main Paper (seeEq. (22)). The (continuous) variable z can be thought ofas a conjugate variable to the (discrete) variable s , andthe right hand side of Eq. (C25) can be interpreted as an integral transform between the z -representation and the s -representation, i.e. G s [ J ] = (cid:73) d z πi (cid:101) G z [ J ](2 s + 1)(1 − z ) z s +1 (C28)where we have (cid:101) G z [ J ] ≡ e (cid:82) τ (cid:82) τ d t d t (cid:48) J (cid:63) ( t ) ·G c ( z,t,t (cid:48) ) · J ( t (cid:48) ) . (C29)In practice, it is easier to compute quantities using thecontinuous z -representation generating functional ˜ G z [ J ](where the expression is a simple Gaussian) and take thetransform back into the s -representation as a final step.In the next subsection of this Appendix, we use G s [ J ] tocalculate the fidelity amplitude A s .1
2. Using G s [ J ] to calculate the fidelity amplitude A s With the expression for G s [ J ] (see Eq. (C25)), wecan make use of Eq. (C6) to calculate A s by takingfunctional derivatives of G s [ J ]. Because G s [ J ] is givenby Gaussian, the right hand side of Eq. (C6) can berewritten in an entirely equivalent representation by wayof introducing fictitious two component quantum fields φ ( t ) † ≡ ( φ (cid:63) ( t ) , φ (cid:63) ( t )) that are completely described interms of the expectation value (cid:104) φ ( t ) (cid:105) q = (cid:104) φ (cid:63) ( t ) (cid:105) q = (cid:104) φ ( t ) φ † ( t ) (cid:105) q = G c ( z, t, t (cid:48) ) , (C30)where the Green’s function G c ( z, t, t (cid:48) ) is the same onegiven in Eq. (C26), and (cid:104)·(cid:105) q denotes a quantum expec-tation value over the fields φ ( t ) with all higher orderexpectation values are entirely determined through theuse of Wick’s theorem. With the aid of Eq. (C30), it isstraightforward to show that the expression (suppressingtime arguments for brevity) A s = (cid:73) d z πi (cid:68) e − i(cid:15) (cid:82) τ d t φ † H n φ (cid:69) q (2 s + 1)(1 − z ) z s +1 (C31)is equivalent to the defining relation for A s given inEq. (C6). This formulation, given in terms of the dy-namics of the fictitious fields φ , has the advantage ofbeing physically more intuitive.We shall now evaluate the quantum expectation valueby making use of the following cumulant expansion (cid:68) e − i(cid:15) (cid:82) τ d t φ † H n φ (cid:69) q = exp (cid:40) ∞ (cid:88) m =1 X m (cid:41) , (C32)where the quantities X m are defined by the relation X m ≡ m ! (cid:28)(cid:28)(cid:18) − i(cid:15) (cid:90) τ d t φ † H n φ (cid:19) m (cid:29)(cid:29) q . (C33)The double brackets (cid:104)(cid:104)·(cid:105)(cid:105) q on the right hand side ofEq. (C33) denote cumulant averages. It is a well knownfact that in a diagrammatic expansion only connected di-agrams contribute to cumulant averages. In what follows,we shall use the diagrammatic expansion to exactly eval-uate the right hand side of Eq. (C32), i.e. we will performan exact resummation of all connected diagrams.Given the form of Eq. (C32), we see that each diagramcontributing to X m contains exactly m vertices and m propagators. Considering this together with the fact thatonly connected diagrams contribute (since we are tak-ing cumulant averages), places a severe restriction on theform allowed for the diagrams: all contributing diagramshave the topology of a single non-selfcrossing closed loop .For convenience, we assign a definite orientation to thisloop by placing arrows on the propagators (but note thatdifferent orientations are not distinct and should not becounted as such). FIG. 4. a ) Diagram corresponding to entire contribution for X . b ) Diagram corresponding to entire contribution to X . c )The 2! diagrams corresponding to the entire contribution for X . Both diagrams give the same contribution, since all vari-ables are internal, so in practice one only need evaluate a sin-gle representative diagram (see discussion in text). Likewise,for a given order m , all ( m − a ), b ), and c ) are given in Eqs. (C34a,C34b,C34c) re-spectively (see text for Feynman rules). For all diagrams, theinteger labels i = 1 , , · · · etc. next to the points refer to thetime labels t i , which are all dummy variables to be integratedover. Though we have chosen a counterclockwise orientationfor the diagrams, one is free to choose whichever orientationone wishes (i.e. diagrams with different orientations are theexact same diagram). The Feynman rules are easily determined , and we statethem here. In a given diagram, each vertex (with itstime label t i ) corresponds to a factor H n ( t i ), and eachpropagator with starting point t i and ending point t j (asdetermined from the direction of the arrow) correspondsto the Green’s function G c ( z, t i , t j ). To obtain the valueof an m th order diagram (i.e. one contributing to X m ),start at an arbitrary vertex and write down all of the fac-tors corresponding to each vertex and propagator in theorder determined by the direction of the arrows (takingcare not to count the starting point twice). Then simplyintegrate over all time variables, take the matrix trace,and multiply by an overall prefactor ( − i(cid:15) ) m /m . Thatthis prefactor is ( m − m there are ( m − m − X m , it suffices to evaluate a single repre-sentative diagram, so, i.e., in Fig. (4c) we only need toevaluate either the left or right diagram, but not both.The diagrams corresponding to X m for m = 1 , , i = 1 , , · · · etc.) correspond the time labels t i . We have arbitrarily chosen to use a counterclockwiseorientation when drawing the loop diagrams, though as2we noted earlier different orientations correspond the thesame diagram. As we have stated earlier, both diagramsshown in Fig. (4c) give the same contribution, since alllabels correspond to internal variables. As an example,we evaluate the diagrams shown in Fig. (4), giving ex-pressions corresponding to X , X , and X . We find X = ( − i(cid:15) ) (cid:90) τ d t Tr H n ( t ) G c ( z, t , t ) (C34a) X = ( − i(cid:15) ) (cid:90) τ d t (cid:90) τ d t Tr (cid:104) H n ( t ) G c ( z, t , t ) ×H n ( t ) G c ( z, t , t ) (cid:105) (C34b) X = ( − i(cid:15) ) (cid:90) τ d t (cid:90) τ d t (cid:90) τ d t Tr (cid:104) H n ( t ) ×G c ( z, t , t ) H n ( t ) G c ( z, t , t ) H n ( t ) G c ( z, t , t ) (cid:105) , (C34c) and it is straightforward to see to how the expressiongeneralizes for arbitrary X m .Now that we know the expression for all X m , all thatremains is to evaluate the sum (cid:80) ∞ m =1 X m . Let us firstintroduce the shorthand notation ζ ≡ z − z Θ ij ≡ Θ( t i − t j ) , (C35)and rewrite the expression for X m as follows: X m = ( − i(cid:15) ) m m (cid:90) τ d t (cid:90) τ d t · · · (cid:90) τ d t m Tr [ H n ( t ) G c ( z, t , t ) H n ( t ) G c ( z, t , t ) · · · H n ( t m ) G c ( z, t m , t )]= ( − i(cid:15) ) m m (cid:90) τ d t (cid:90) τ d t · · · (cid:90) τ d t m Tr [ H n ( t ) U c ( t , t ) H n ( t ) U c ( t , t ) · · · H n ( t m ) U c ( t m , t )] × ( ζ + Θ )( ζ + Θ ) · · · ( ζ + Θ m )= ( − i(cid:15) ) m m (cid:90) τ d t (cid:90) τ d t · · · (cid:90) τ d t m Tr (cid:2) ( U † c ( t , H n ( t ) U c ( t , · · · ( U † c ( t m , H n ( t m ) U c ( t m , (cid:3) × ( ζ + Θ )( ζ + Θ ) · · · ( ζ + Θ m )= ( − i(cid:15) ) m m (cid:90) τ d t · · · (cid:90) τ d t m Tr (cid:104) (cid:101) H n ( t ) · · · (cid:101) H n ( t m ) (cid:105) ( ζ + Θ ) · · · ( ζ + Θ m ) , (C36)where in the second line of Eq (C36) we simply usethe definition of G c ( z, t, t (cid:48) ) (see Eq. (C26)), in the thirdline we use the property U c ( t, t (cid:48) ) = U c ( t, U c (0 , t (cid:48) ) = U c ( t, U † c ( t (cid:48) ,
0) along with the cyclic property of tracesTr AB · · · Y Z = Tr
ZAB · · · Y , and in the fourth line thequantity (cid:101) H n ( t ) is defined as (cid:101) H n ( t ) = U c ( t, † H n ( t ) U c ( t, . (C37)Note that (cid:101) H n ( t ) is simply the noise Hamiltonian in therotating frame, or equivalently, in the interaction picture(with respect to the control Hamiltonian H c ( t, (cid:80) m X m cannotbe explicitly carried out since the expression for X m , asgiven in Eq. (C36), does not factorize (the factors of Θ ij make it impossible to factorize the time integrals). Thissituation can be remedied by rearranging (cid:80) m X m into apower series in ζ ∞ (cid:88) m =1 X m = ∞ (cid:88) m =1 Y m ζ m . (C38)Unlike X m , the coefficients Y m do factorize, making this rearrangement advantageous. They take the form Y m = Tr ∆ m m , (C39)where ∆ is a 2 × ∞ (cid:88) m =1 X m = Tr log(1 l − ζ ∆) − , (C40)where 1 l is the 2 × infinite set of X q contribute to Y m , since all X q with q ≥ m containterms proportional to ζ m . Starting with the first term, Y (i.e. the term in the sum (cid:80) m X m proportional to ζ ), wefind that it exactly vanishes. This is due to the fact thatthe contribution coming from each X m is proportional toΘ Θ · · · Θ m , and in order for this to be nonvanishingwe need t > t · · · > t m − > t m > t , an impossibility.Next, we seek the expression for Y by collecting allterms linear in ζ from (cid:80) m X m . It is easy to see from3Eq. (C36) that a given X m contributes exactly m terms,furthermore all of these terms give the exact same con-tribution since they only differ in the labeling of internalvariables. It then suffices to take a single representativeand multiply the result my m . Let us take as the repre-sentative the term proportional to Θ Θ · · · Θ m − ,m .The effect of this factor is to simply cut off the lim-its in the time integrals so that t > t · · · > t m ,and we are left with a time ordered sequence of factors (cid:101) H n ( t ) (cid:101) H n ( t ) · · · (cid:101) H n ( t m ) which can be easily factorizedvia the help of the time ordering operator T . The (linearin ζ ) contribution from X m is given explicitly by X m → ζ ( − i(cid:15) ) m Tr (cid:90) τ d t (cid:90) t d t · · · (cid:90) t m − d t m (cid:101) H n ( t ) × (cid:101) H n ( t ) · · · (cid:101) H n ( t m )= ζ Tr 1 m ! T (cid:20) − i(cid:15) (cid:90) τ d t (cid:101) H ( t ) (cid:21) m . (C41)Collecting all terms in (cid:80) m X m proportional to ζ , we ob- tain Y = Tr T (cid:40) ∞ (cid:88) m =1 m ! (cid:20) − i(cid:15) (cid:90) τ d t (cid:101) H n ( t ) (cid:21) m (cid:41) = Tr (cid:104) T e − i(cid:15) (cid:82) τ d t (cid:101) H ( t ) − l (cid:105) ≡ Tr ∆ , (C42)giving us the sought for expression for the matrix ∆.Now we proceed to find Y , showing that the generalresult given in Eq. (C39) holds. Looking at Eq. (C36), wesee that all X m for m ≥ ζ and therefore contribute to Y . In order to more easilyunderstand the pattern that emerges, let us consider thecontributions from the first few X m . The lowest orderterm to contribute to Y is X , from which we get theexpression X → ζ ( − i(cid:15) ) (cid:90) τ d t (cid:90) τ d t (cid:101) H n ( t ) (cid:101) H n ( t )= ζ (cid:20) T (cid:18) − i(cid:15) (cid:90) τ d t (cid:101) H n ( t ) (cid:19) × T (cid:18) − i(cid:15) (cid:90) τ d t (cid:101) H n ( t ) (cid:19)(cid:21) , (C43)where in the second line of Eq (C43) we have just rewrit-ten the expression in a way that suggests what the pat-tern will be for higher order terms.In order to more clearly see the pattern that emerges,let us work out the ζ order contribution coming from X . We have X → ζ ( − i(cid:15) ) (cid:20)(cid:90) τ d t (cid:90) τ d t (cid:90) τ d t (cid:101) H n ( t ) (cid:101) H n ( t ) (cid:101) H n ( t ) (Θ + Θ + Θ ) (cid:21) = ζ ( − i(cid:15) ) Tr (cid:20)(cid:90) τ d t (cid:90) τ d t (cid:90) τ d t (cid:101) H n ( t ) (cid:101) H n ( t ) (cid:101) H n ( t ) Θ (cid:21) = ζ ( − i(cid:15) ) Tr (cid:20)(cid:18)(cid:90) τ d t (cid:90) t d t (cid:101) H n ( t ) (cid:101) H n ( t ) (cid:19) (cid:18)(cid:90) τ d t (cid:101) H n ( t ) (cid:19)(cid:21) = ζ (cid:34) T (cid:18) − i(cid:15) (cid:90) τ d t (cid:101) H n ( t ) (cid:19) T (cid:18) − i(cid:15) (cid:90) τ d t (cid:101) H n ( t ) (cid:19) + 11! T (cid:18) − i(cid:15) (cid:90) τ d t (cid:101) H n ( t ) (cid:19) T (cid:18) − i(cid:15) (cid:90) τ d t (cid:101) H n ( t ) (cid:19) (cid:35) = ζ (cid:88) j =1 j ! T (cid:18) − i(cid:15) (cid:90) τ d t (cid:101) H n ( t ) (cid:19) j − j )! T (cid:18) − i(cid:15) (cid:90) τ d t (cid:101) H n ( t ) (cid:19) − j . (C44)In the second line we have used the fact that the termsproportional to Θ and Θ are identical to the onepropotional to Θ . This can be seen by simply rela-beling variables along with taking cyclic permutations of (cid:101) H n ( t i ), since the trace operation remains invariant un-der this. In the third line, we simply apply Θ to cutoff the integral over t , and note that the expression fac- torizes as the parentheses suggest. In the third line werewrite the expression as a sum of two terms, via the useof time-ordering operators T , and by once again takingadvantage of the invariance of the trace operation un-der cyclic permutations of matrices. In the final line, werewrite the sum is a way that is suggestive of how higherorder X m contribute to Y .4As one can easily guess from the expression inEq. (C44) (we do not show the proof explicitly here,though it is easy to prove by induction) the contribu-tion to Y coming from X m for general m is given by theexpression X m → ζ (cid:104) m − (cid:88) j =1 j ! T (cid:18) − i(cid:15) (cid:90) τ d t (cid:101) H n ( t ) (cid:19) j × m − j )! T (cid:18) − i(cid:15) (cid:90) τ d t (cid:101) H n ( t ) (cid:19) m − j (cid:105) . (C45)We are now ready to write down the entire contributionto Y , coming from all X m , where we find Y = 12 ∞ (cid:88) m =2 Tr (cid:104) m − (cid:88) j =1 j ! T (cid:18) − i(cid:15) (cid:90) τ d t (cid:101) H n ( t ) (cid:19) j × m − j )! T (cid:18) − i(cid:15) (cid:90) τ d t (cid:101) H n ( t ) (cid:19) m − j (cid:105) = Tr (cid:20) (cid:16) T e − i(cid:15) (cid:82) τ d t (cid:101) H n ( t ) − l (cid:17) (cid:21) = Tr ∆ , (C46)where we see that in the last line of Eq. (C46), the matrix∆ is the same matrix appearing in Eq. (C42).Continuing with the same procedure used above to de-termine Y and Y , it is simple to show (though we donot explicitly show the proof here) that for general order m we get the expression Y m = Tr ∆ m m , (C47)reproducing the expression given in Eq. (C39), which iswhat we intended to show. Using this expression, alongwith Eqs. (C38) and (C40), we obtain the exact resultfor the quantum average in Eq. (C32). We have (cid:68) e − i(cid:15) (cid:82) τ d t φ † H n φ (cid:69) q = exp (cid:40) ∞ (cid:88) m =1 Y m ζ m (cid:41) = exp (cid:8) Tr log(1 l − ζ ∆) − (cid:9) = exp (cid:8) log Det(1 l − ζ ∆) − (cid:9) = 1Det(1 l − ζ ∆) . (C48)With this expression, all that remains is to evaluate theintegral over z in Eq. (C31) in order to obtain the ex-pression for A s .Before doing so, let us first define the matrix δδ ≡ l + ∆= T e − i(cid:15) (cid:82) τ d t (cid:101) H n ( t ) , (C49)in terms of which the analysis to follow simplifies. Re-calling that ζ ≡ z/ (1 − z ), we see that1Det(1 l − ζ ∆) = (1 − z ) Det(1 l − zδ ) , (C50) where we used the fact that for any 2 × A andc-number c we have Det( cA ) = c Det A . This is a veryconvenient representation since the factor of (1 − z ) inEq. (C50) cancels a similar factor appearing in Eq. (C31)which simplifies the evaluation of the integral over z . Italso turns out to be very convenient to reexpress the de-terminant on the right hand side of Eq. (C50) as (cid:104) Det(1 l − zδ ) (cid:105) − = (cid:104) − z Tr δ + z Det δ (cid:105) − = (cid:104) − z Tr δ + z (cid:105) − , (C51)where the first line is an exact relation for any × δ , and in the second line we used the fact that δ is a SU (2) matrix (see Eq. (C49)) with unit determinant.As a final step before evaluating the integral over z in Eq. (C31) to find A s , we take advantage of the factthat the right hand side of Eq. (C51) takes the form ofa generating function for the Chebyshev polynomials ofthe second kind [21], which play an important role in thedevelopment of spherical harmonics in four dimensions.We have 11 − z Tr δ + z = ∞ (cid:88) j =0 V j (cid:16)
12 Tr δ (cid:17) z j , (C52)where V j ( x ) denotes the j th order Chebyshev polynomialof the second kind. Eq. (C52) is valid as long as theconditions | z | < | Tr δ | ≤ (cid:68) e − i(cid:15) (cid:82) τ d t φ † H n φ (cid:69) q = (1 − z ) ∞ (cid:88) j =0 V j (cid:16)
12 Tr δ (cid:17) z j . (C53)We are now ready to evaluate Eq. (C31) to find A s .Using Eq. (C53), we find A s = (cid:73) d z πi (cid:68) e − i(cid:15) (cid:82) τ d t φ † H n φ (cid:69) q (2 s + 1)(1 − z ) z s +1 = 12 s + 1 ∞ (cid:88) j =0 (cid:73) d z πi V j (cid:16) Tr δ (cid:17) z s +1 − j = 12 s + 1 V s (cid:16)
12 Tr δ (cid:17) , (C54)where in going from the second to the third line, we useCauchy’s residue theorem. In order to arrive at the ex-pression for A s shown in Eq. (19) in the main Paper, wemake use of the following relation V j (cos θ ) = sin ( j + 1) θ sin θ . (C55)5We then have A s = 12 s + 1 sin (cid:104) (2 s + 1) cos − (cid:16) Tr δ (cid:17)(cid:105) sin (cid:104) cos − (cid:16) Tr δ (cid:17)(cid:105) = 12 s + 1 s (cid:88) j = − s e − ji cos − (cid:16) Tr δ (cid:17) , (C56)where the expression given in the second line turns outto be more convenient for calculations. By carrying outthe sum in the second line, one arrives at the expressiongiven in the right hand side of the first line, showing thatthe two expressions are entirely equivalent. Taking thesimplest case, s = 1 /
2, we get the expression A / = 12 (cid:104) e i cos − (cid:16) Tr δ (cid:17) + e − i cos − (cid:16) Tr δ (cid:17)(cid:105) = 12 Tr δ = 12 T e − i(cid:15) (cid:82) τ d t (cid:101) H n ( t ) , (C57) where in the third line we made use of Eq. (C49). Makinguse of Eq. (C57), we finally arrive at the expression givenin Eq. (19) in the main Paper: A s = 12 s + 1 s (cid:88) j = − s e − ji cos − A / . (C58)This relation is remarkable in the fact that it essentiallyshows us we can understand the behavior of the spin s system entirely in terms of quantities associated withthe spin 1 / s system isentirely contained in the dynamics of a spin-half system. [1] M. Nielsen and I. Chuang, Quantum Computation andQuantum Communication (Cambridge University Press,Cambridge, 2000).[2] D. d’Alessandro,
Introduction to quantum control and dy-namics (CRC press, 2007).[3] H. M. Wiseman and G. J. Milburn,
Quantum measure-ment and control (Cambridge University Press, 2010).[4] C. H. Bennett and D. P. DiVincenzo, Nature , 247(2000).[5] L. Viola, E. Knill, and S. Lloyd, Physical Review Letters , 2417 (1998).[6] A. G. Kofman and G. Kurizki, Physical Review Letters , 270405 (2001).[7] A. G. Kofman and G. Kurizki, Physical Review Letters , 130406 (2004).[8] T. J. Green, J. Sastrawan, H. Uys, and M. J. Biercuk,arXiv:1211.1163 (2012).[9] C. Kabytayev, T. J. Green, K. Khodjasteh, M. J. Bier-cuk, L. Viola, and K. R. Brown, Physical Review A ,012316 (2014).[10] J. Schwinger, Journal of Mathematical Physics , 407(1960).[11] L. V. Keldysh, Soviet Physics JETP , 1018 (1965).[12] A. Kamenev, Field theory of non-equilibrium systems (Cambridge University Press, Cambridge, 2011).[13] A. Polkovnikov, Annals of Physics , 1790 (2010).[14] P. C. Martin, E. D. Siggia, and H. A. Rose, PhysicalReview A , 423 (1973).[15] A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalat-tore, Reviews of Modern Physics , 863 (2012).[16] E. Paladino, Y. M. Galperin, G. Falci, and B. L. Alt-shuler, Reviews of Modern Physics , 361 (2014).[17] U. Weiss, Quantum Dissipative Systems (World Scien-tific, 1999).[18] J. Klauder and B. Skagerstam,
Coherent States: Appli-cations in Physics and Mathematical Physics (World Sci-entific, 1985).[19] Strictly speaking, one should use the lab frame quantity | ω ( t ) | , but in our system due to rotational invariance | Ω ( t ) | = | ω ( t ) | for any and all realizations of the con-trol field.[20] J. Bergli, Y. M. Galperin, and B. L. Altshuler, NewJournal of Physics , 025002 (2009).[21] E. T. Whittaker and G. N. Watson, A course in mod-ern analysis , 4th ed. (Cambridge University Press, Cam-bridge, 1990).[22] S. Blanes, F. Casas, J. A. Oteo, and J. Ros, PhysicsReports470