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Minimizing irreversible losses in quantum systems bylocal counterdiabatic drivingDries Selsa,b,1 and Anatoli Polkovnikova

aDepartment of Physics, Boston University, Boston, MA 02215; and bTheory of Quantum and Complex Systems, Universiteit Antwerpen, B-2610 Antwerpen,Belgium

Edited by Steven M. Girvin, Yale University, New Haven, CT, and approved March 27, 2017 (received for review December 2, 2016)

Counterdiabatic driving protocols have been proposed [DemirplakM, Rice SA (2003) J Chem Phys A 107:9937–9945; Berry M (2009) JPhys A Math Theor 42:365303] as a means to make fast changesin the Hamiltonian without exciting transitions. Such driving inprinciple allows one to realize arbitrarily fast annealing protocolsor implement fast dissipationless driving, circumventing standardadiabatic limitations requiring infinitesimally slow rates. Theseideas were tested and used both experimentally and theoreti-cally in small systems, but in larger chaotic systems, it is knownthat exact counterdiabatic protocols do not exist. In this work,we develop a simple variational approach allowing one to findthe best possible counterdiabatic protocols given physical con-straints, like locality. These protocols are easy to derive and imple-ment both experimentally and numerically. We show that, usingthese approximate protocols, one can drastically suppress heatingand increase fidelity of quantum annealing protocols in complexmany-particle systems. In the fast limit, these protocols provide aneffective dual description of adiabatic dynamics, where the cou-pling constant plays the role of time and the counterdiabatic termplays the role of the Hamiltonian.

counterdiabatic driving | adiabatic gauge | transitionless driving |variational principle | complex systems

Despite the time-reversal symmetry of the microscopicdynamics of isolated systems, losses are ubiquitous in any

process that tries to manipulate them. Whether it is the heat pro-duced in a car engine or the decoherence of a qubit, all lossesarise from our lack of control on the microscopic degrees of free-dom of the system. Since the early days of thermodynamics andactually, even before, the adiabatic process has emerged as a uni-versal way to minimize losses, leading to the concept of Carnotefficiency—the cornerstone of modern thermodynamics. Despiteits conceptual importance, practical implications of the Carnotefficiency are limited, because the maximal efficiency goes handin hand with zero power. Nonetheless, by sacrificing some of theefficiency, one can run the same Carnot cycle at finite power (1).Heat engines might seem to be a problem of the past, but theunderstanding of finite-time thermodynamics in small (quantum)systems has become increasingly important because of develop-ments in quantum information and nanoengineering.

Developing and understanding methods to induce quasiadia-batic dynamics at finite times are paramount to the advancementof quantum information technologies. In general, one coulddistinguish between two (complementary) approaches. On onehand, one can, for a fixed setup, try to develop optimal drivingprotocols that result in minimal loss under certain constraints.Such protocols were recently suggested as appropriate geodesicpaths in the parameter space in both the context of thermody-namics (2) and the context of adiabatic-state preparation (3, 4).The optimal protocols were also analyzed numerically using var-ious optimum control ideas (5–9). On the other hand, one cantry to engineer fast nonadiabatic protocols that lead to the sameresult as the fully adiabatic protocol. In particular, transition-less driving protocols were recently proposed and explored insmall single-particle systems (10–15) with numerical extensions

to larger interacting systems (16). In this approach, one intro-duces an auxiliary counterdiabatic (CD) Hamiltonian drive ontop of a target Hamiltonian to suppress all transitions betweeneigenstates. A general problem with this approach is that, in com-plex chaotic systems, the exact CD Hamiltonian is nonlocal andexponentially sensitive to any tiny perturbations. The goal of thiswork is to overcome these difficulties by providing a route to find-ing approximate optimal CD driving protocols by restricting to aclass of physical operators (e.g., those accessible in experiments).In this work, we focus specifically on local CD protocols.

The ideas of CD driving are certainly not new and are used onan everyday basis in nature. Let us illustrate these ideas using anexample of a waiter bringing a tray with a glass of water from thebar to a table (Fig. 1). As we will show in this work, this simpleexample contains very important insights, which will become rel-evant to the core of this paper. The goal of the waiter is to deliverthe water to the table with a high fidelity (i.e., without spilling orsplashing it). Of course, the glass should be vertical in the begin-ning of the process (i.e., when the waiter is leaving the bar) andat the end of the process (i.e., when the waiter reaches the table).The simplest protocol that can be adopted by the waiter is adia-batic, where he slowly moves along the shortest path (geodesic)connecting the bar and the table, keeping the tray vertically atall times. This protocol will work but will require a lot of time,and thus, the efficiency of such an “adiabatic waiter” will be verylow. An efficient waiter has to serve more customers by goingfaster, and doing so requires a different tactic. When accelerat-ing to reach a finite speed, a pseudoforce will act on the drinks,which will cause the drinks to spill or even tip over. This effectcan be avoided by acting on the drinks with an equal and oppositeforce, which is exactly what waiters do. Moreover, the same tiltcan counter a drag force caused by the air if the waiter runs very

Significance

Losses are ubiquitous in manipulating complex systems. Theyarise from our lack of control on the microscopic degrees offreedom of the system. A universal way to minimize lossesis to consider adiabatic processes. These processes are, how-ever, very slow, which significantly limits their power. In thiswork, we show how to speed up these protocols for generalcomplex (quantum) systems. Although dissipation cannot beavoided, we show how it can be reduced significantly withonly local access to the system. Applications range from quan-tum information technologies to preparing experiments andeven controlling complicated classical systems, such as thosefound in nature.

Author contributions: D.S. and A.P. designed research; D.S. and A.P. performed research;D.S. and A.P. contributed new reagents/analytic tools; D.S. analyzed data; and D.S. andA.P. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.1To whom correspondence should be addressed. Email: dsels@bu.edu.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1619826114/-/DCSupplemental.

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Static Moving Counter diabaticA B C

Fig. 1. The CD waiter. (A) A waiter’s goal is to deliver a tray with a glass ofwater from the bar to a customer without spilling. In the beginning and theend of the task, the system should look like the situation in A. An adiabaticwaiter can always be in the situation in A, but with a desire to be moreefficient and speed up the protocol, a naive waiter will find himself in theundesirable situation in B somewhere during the task. (C) By tilting the tray,an example of CD driving, the situation in B can be avoided, and the desiredtasks can be achieved much faster.

fast. In fact, by tilting the tray while moving, the waiter induces aCD force. Despite the fact that the system of the tray, the glass,and water is complex and chaotic, it is clear from our everydayexperience that this CD protocol can be extremely efficient. Letus highlight several important points, which we can learn fromthis intuitive example. We will come back to these points laterwhen we discuss various physical examples.

To implement an efficient CD protocol, one has to introducenew degrees of freedom (like a tilt), which do not show up inthe initial and final states as well as the adiabatic path.

The system does not generally follow an instantaneous groundstate: at intermediate times, the waiter tilts the tray and movesit fast, which correspond to a highly excited state of the systemin the laboratory frame.

The CD protocol corresponds to adding local terms to theHamiltonian of the system, like the gravitational field. Thisprotocol is only sensitive to the velocity and acceleration ofthe waiter.

As we will show, these observations underlie crucial ideasbehind engineering CD protocols in complex systems, such aslocality and gauge equivalence. Using these ideas as a guidingprinciple, we develop a simple variational approach allowing oneto find local and robust approximate counteradiabatic Hamilto-nians. These counterterms allow one to achieve truly spectacu-lar results in suppressing heating or targeting ground states ofgapped or gapless many-particle systems with a very high fidelityat very fast speeds. An important advantage of the variationalmethod is that it allows one to find efficient CD protocols with-out the need of diagonalizing the Hamiltonian, in particular, inthe thermodynamic limit. Moreover, one can check the accuracyof the variational ansatz by analyzing the stability of the protocolwith respect to adding additional terms.

Local CD DrivingCD Driving in Quantum and Classical Systems. Let us have a closerlook at how transitions between eigenstates actually arise andhow one can suppress them. Consider a state |ψ〉 evolving underthe Hamiltonian H0 (λ(t)), which is time-dependent through theparameter λ(t). In general, λ can be a multicomponent vectorparameter (for example, in the case of a waiter, λ can standfor his x and y coordinates), but in this work, we will focus onthe single-component case to avoid extra complications. If theparameter changes in time, then for a moving observer in theinstantaneous eigenbasis of H0, the laws of physics are modified.This phenomenon is, of course, very well known for the case ofan accelerated or rotating frame, but in fact, it applies to all types

of motion. Specifically, the Hamiltonian picks up an extra contri-bution and becomes

H eff0 = H0 − λAλ; [1]

Here, Aλ is the adiabatic gauge potential in the moving frame.It is geometric in origin and related to the infinitesimal trans-formations of the instantaneous basis states in the quantumcase and the infinitesimal canonical transformations of conjugatevariables (like coordinates and momenta) in the classical case(details are in Methods and ref. 17).

In the moving frame, the Hamiltonian H0 is diagonal (station-ary), and therefore, all nonadiabatic effects must be caused bythe second term. The idea of the CD driving is to evolve the sys-tem with the Hamiltonian

HCD(t) = H0 + λAλ,

such that, in the moving frame, H effCD(t) = H0 is stationary and no

transitions occur. Note that, by construction in the zero velocitylimit |λ|→ 0, the CD Hamiltonian HCD(t) reduces to the orig-inal Hamiltonian H0(t) as expected. It is easy to show that thegauge potential Aλ satisfies the following equation (details arein Methods and ref. 17):

[i~∂λH0 − [Aλ,H0] ,H0] = 0. [2]

This equation immediately extends to classical systems by replac-ing the commutator with the Poisson brackets: [. . . ]→ i~ {. . . }.Although in this work, we focus on quantum systems, all generalresults and methodology equally apply to classical systems.

One can immediately see that Eq. 2 reduces to familiarGalilean transformations in the case of translations. Let usassume that

H0 =p2

2m+ V (q − λ(t)), [3]

that is, λ(t) is the position of the center of the potential. Inthis case, the gauge potential is just the momentum operatorAλ = p. Indeed, we have i~∂λH0≡−i~∂qV =−i~[∂q ,H0] =[p,H0], such that Eq. 2 is automatically satisfied. Thus, the exactCD Hamiltonian is

HCD = H0 + λp. [4]

If the waiter implements the protocol (Eq. 4), then he would beable to move a tray with glasses fast without exciting it. Noticethat this is not what the waiter actually does. The reason is that itis very hard to realize the term linear in momentum. Moreover,from Eq. 2, we see that, in systems satisfying time-reversal sym-metry, where the Hamiltonian H0 is real, the gauge potential andhence, the corresponding CD terms are always strictly imaginary.Therefore, the CD driving always breaks the time-reversal sym-metry. The situation is actually much better than it seems. TheCD term in this setup plays the role of the vector potential inelectromagnetism:

HCD =p2

2m+ V (q − λ) + λp

=(p + mλ)2

2m+ V (q − λ)− mλ2

2. [5]

It is very well-known that, if the vector potential is curl-free,which is always the case in one dimension, it can be removed(gauged away) at the expense of introducing a scalar potential viap→ p + ∂q f and H →H + ∂t f for an arbitrary function f (q , t).Choosing f (q , t) =−mλq , we see that

HCD ∼p2

2m+ V (q − λ)−mλq . [6]

Thus, as expected from the Galilean invariance, the CD drivingamounts to adding an extra gravitational field proportional to theacceleration. Here, we used the tilde sign instead of an equal sign

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to highlight that the right-hand side is gauge-equivalent ratherthan equal to the Hamiltonian (Eq. 4). There is an importantphysical difference between the two CD protocols. Although fol-lowing the imaginary CD protocol (Eq. 4) amounts to instanta-neously following eigenstates of H0, following the real CD pro-tocol (6) amounts to instantaneously following eigenstates of agauge-equivalent Hamiltonian. Only when the velocity λ is zerodo the two Hamiltonians coincide, such that CD driving leadsto identical results. This subtlety is precisely the reason why thewaiter, following the real CD protocol, does not keep the glass inthe instantaneous ground state, except when the velocity and theacceleration are zero.

Although Eq. 2 is linear and looks very simple, this simplic-ity is actually deceptive. In fact, one can show that, in genericchaotic systems, it has no solution. In quantum chaotic systems,the exact analytic expression for Aλ suffers from the problemof small denominators (17), and in classical chaotic systems, itcan be expressed through a formally divergent integral (18). Thephysical reason behind is very intuitive. By trying to find theexact gauge potential, we are requiring too much. We essen-tially want to find a transformation that keeps the system inexact many-body eigenstates without any excitations. However,achieving this is clearly impossible or at least exponentially hard[e.g., in chaotic many-particle systems, eigenstates are essentiallyrandom vectors sensitive to exponentially small perturbations ofthe Hamiltonian (19), and therefore, the exact gauge potentialshould have the same exponential sensitivity to the details of themany-body spectra and access to all microscopic degrees of free-dom. However, finding such gauge potential is hardly our goaleither. We are generally interested in either suppressing dissipa-tion (i.e., suppressing transitions between levels resulting in sub-stantial energy changes) or following very special states like theground states, which are also robust to small perturbations. Thus,our goal should be finding approximate gauge potentials, whichsatisfy requirements of robustness and locality and strongly sup-press physical diabatic effects rather than completely eliminatethem, which is precisely what we are going to discuss next.

Variational Principle and Local CD Protocols. Our goal is to set upa variational procedure allowing one to determine the best pos-sible Aλ under some constraints, like locality, robustness, or justexperimental accessibility. Unconstraint minimization should, ofcourse, result in the exact gauge potential (Eq. 2). It is easy toshow that solving Eq. 2 is equivalent to minimizing the Hilbert–Schmidt norm of the operator

Gλ(Aλ) ≡ ∂λH0 +i

~[Aλ,H0]

with respect to Aλ. Indeed, finding the minimum of this norm isequivalent to the Euler–Lagrange equation,

δS(Aλ)

δAλ= 0, [7]

of the action (Methods)

S (Aλ) = Tr[G2λ(Aλ)

]. [8]

For classical systems, trace should be replaced by an integral overthe phase space. Instead of minimizing the action over the wholeHilbert space of operators, one can now restrict to a subspace ofphysical operators. Let us introduce A∗λ as an approximate adi-abatic gauge potential; then, one can simply calculate the bestconstrained approximation by minimizing over all allowed A∗λ.To do so, one simply has to be able to evaluate the action (Eq.8) for the allowed operators A∗λ. Evaluating the trace of localoperators or their products is very straightforward and can beusually done with minimal efforts. Physically, the action (Eq. 8)defines the average transition rate (over all possible states) whenthe classical parameter λ(t) is a weak random white noise pro-

cess. Therefore, minimizing the action is equivalent to suppress-ing processes, such as heating and energy diffusion, under a whitenoise drive.

Quite often, one is interested in suppressing transitions froma low-temperature manifold of states, in particular, the groundstate. Then, targeting the gauge potential, which suppresses tran-sitions everywhere in the spectrum, is overdemanding. Instead,one can define the action through a finite temperature norm:

S (A∗λ, β) =⟨G2λ(A∗λ)

⟩− 〈Gλ(A∗λ)〉2 , [9]

where the angular brackets denote usual average with respect tothe thermal density matrix ρ(β) = 1/Z exp[−βH0]. The Hilbert–Schmidt norm (Eq. 8) is recovered as the infinite temperaturelimit (β→ 0) of this norm [note that subtracting the second termin Eq. 9 is not affecting the result, because 〈Gλ(A∗λ)〉 is inde-pendent ofA∗λ]. In the zero temperature limit β→∞, the action(Eq. 9) reduces to the variance of Gλ in the ground state. Theexact gauge potential minimizes the action (Eq. 8) for any tem-perature. However, details of the variational solution can dependon β. In this work, we are focusing on finding CD protocols min-imizing the infinite temperature action (Eq. 8), leaving the anal-ysis of the finite/zero temperature action for future work.

ApplicationsThis section is primarily concerned with applications in quantummechanics; for convenience, we set ~= 1. At the end of the sec-tion, we show how to apply the current variational method toclassical systems.

1D Lattice Fermions. Let us investigate the performance of vari-ational CD protocols using an example of noninteracting latticespinless fermions in a time-dependent potential. Despite the sim-plicity of the setup, the problems that we will be addressing arehighly nontrivial, because we will consider inserting and movingobstacles, breaking translational symmetry. In turn, this pertur-bation generally leads to strong mixing of single-particle orbitalsand strong nonadiabatic effects originating from scattering offermions from the obstacle. Consider a single-band tight bindingmodel in an external potential

H0 = −JL−1∑j=1

(c†j cj+1 + c†j+1cj

)+

L∑j=1

Vj (λ)c†j cj , [10]

where c†j and cj are fermionic creation and annihilation oper-ators, respectively, and Vj (λ) is the external potential. In prin-ciple, the dependence of V on λ can be arbitrary. Here, wewill focus on a particular example of inserting the potential:Vj (λ) =λvj . In SI Text, we additionally analyze a moving poten-tial: Vj (λ) =V (j − λ). Because the Hamiltonian is real, as wediscussed earlier, the adiabatic gauge potential should be purelyimaginary [which is also clear from the form of the action (Eq.8)]. The most local Hermitian imaginary operator that one canfind for a free system is the current; therefore, we will look forsolutions of the form

A∗λ = i

L−1∑j=1

αj

(c†j+1cj − c†j cj+1

).

Substituting this potential into the action (Eq. 8) and extremizingwith respect to the coefficients αj , we find the following equation(SI Text):

−3J 2∂2j αj + (∂jVj (λ))2αj = −J∂j∂λVj (λ), [11]

where ∂jαj ≡αj+1 − αj , ∂jVj ≡Vj+1 −Vj , and ∂2j αj ≡αj+1 −

2αj + αj−1 are discrete lattice derivatives. If the potential issmooth on the scale of the lattice spacing, then the discretederivatives can be substituted by continuous derivatives. In gen-eral expression Eq. 11 is a set of linear equations, which can

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always be solved numerically. However, there are several caseswhere one can find approximate or exact analytic solutions.

The imaginary hopping terms in A∗λ explicitly break time-reversal symmetry, which might create some difficulties in imple-menting them in practice. However, restricting the variationalansatz only to the nearest neighbors allows one to once againperform a simple gauge transformation similar to the vectorpotential shift, also known as the Peierls substitution. Namely,cj → cj e

−ifj, such that the new Hamiltonian becomes real (moredetails are in SI Text):

HCD ∼ −∑

j

Jj

(c†j+1cj + h.c.

)+∑

j

Uj c†j cj [12]

with

Jj = J

√1 + (λαj/J )2, [13]

Uj = Vj −j∑

i=1

J

J 2 + (λαj )2

(λαj + (λ)2∂λαj

). [14]

As earlier, the tilde sign indicates that the right-hand side isgauge-equivalent to the CD Hamiltonian, and the two coincideonly at λ= 0. This equivalence is again very similar to the tilt-ing of the tray in the waiter example. It is easy to see thatthe CD Hamiltonian has two distinct limits. At small velocities|dVj/dt |= |λvj |� J 2, the renormalization of hopping in neg-ligible, and the CD term is a correction to the potential pro-portional to the acceleration λ exactly like in the waiter exam-ple. Conversely, in the high-velocity limit, renormalization of thepotential is negligible, and the CD Hamiltonian contains therenormalized hopping, which scales linearly with the velocity.This local hopping renormalization plays a role similar to therefractive index by locally changing the group velocity of parti-cles in a way that essentially traps scattered particles.

Because the CD Hamiltonian depends only on the velocity andacceleration λ and λ, respectively, it is convenient (although notnecessary) to deal with λ(t), which has vanishing first and secondderivatives at the beginning and the end of the protocol, such thatHCD =H0 at these points. An example of such a protocol that wewill use in this work is

λ(t) =λ0 + (λf −λ0) sin2

(π

2sin2

(πt

2τ

)), t ∈ (0, τ). [15]

This protocol ramps λ(t) from the initial value λ0 to the finalvalue λf during the time τ .Uniform linear potential. In the case of a general time-dependentforce Vj (λ) =λj , with λ playing the role of an effective electricfield, the solution of Eq. 11 is very simple: αj =−J/λ2. It is easyto check that this solution is, in fact, exact up to the boundaryterms (details are in SI Text, Fig. S1) (i.e., A∗λ =Aλ). Remark-ably, because α is constant, the effective potential Uj remainslinear. Additionally, the effective hopping is constant across thelattice, which allows us to absorb the hopping renormalizationinto the timescale, because one can always rescale the Hamilto-nian by an arbitrary factor without exciting the system.

As a result of these transformations, the real CD Hamiltonianis structurally the same as the naive Hamiltonian. One simplyhas to switch on the electric field in a different way to avoidending in an excited state; that is, for each protocol λ, the CDprotocol is

λCD =λ√

1 + µ2

(1− µµ

1 + µ2

), where µ = 1/λ. [16]

This protocol is illustrated in Fig. 2 for λ0 = 0.1J and λf = J(blue line in Fig. 2) for a particular choice τ = 5/J . To allow forenough transport of particles, there is an initial pulse in the fieldwith an amplitude that is significantly larger than the final one.

0 0.2 0.4 0.6 0.8 1-0.5

0

0.5

1

1.5

2

2.5

3

t/τ

λ(t)

Fig. 2. CD protocol. Example of a naive adiabatic protocol (Eq. 15) to switchon a linear potential from λ0 = 0.1J to a final value of λf = J without excit-ing the system (dashed red line) and a CD protocol that does exactly that ina time τ = 5/J (blue line).

After this pulse, the field is much flatter and first goes oppositeto the target field before reversing to the right direction to reachthe final desired field λf . The naive and the CD protocols comecloser to each other as one increases the duration τ , althoughthey always significantly differ at small fields.

Let us also comment on the opposite limit of instantaneousprotocol, τ→ 0, where the bare coupling turns on as a step-like function: λ(t)→λ0 + (λf − λ0)θ(t). In this case, theGalilean term dominates the CD Hamiltonian. It is then con-venient to formally use the chain rule and parametrize time interms of the coupling i∂tψ= i λ∂λψ, such that the Schrodingerequation reads

i∂λ|ψ〉 = A∗λ|ψ〉. [17]

If the gauge potential is exact, then this equation describes thefastest route to perform adiabatic evolution. In particular, a slowturning on of the uniform field is equivalent to dynamics gov-erned by the Hamiltonian

A∗λ = − iJ

λ2

∑j

(c†j+1cj − c†j cj+1

). [18]

Because the coupling λ effectively plays the role of time, we seethat the total time required to load the system into the groundstate according to Eq. 17 diverges as the initial or final electricfield approaches zero. As we will discuss elsewhere, this diver-gence is fundamentally related to the divergence of the quantumspeed limit. Let us point out that, at τ→ 0, there is no smoothreal CD protocol, because it contains the acceleration terms thatbecome singular.Fighting Anderson orthogonality: Inserting a potential. Anothersomewhat more involved problem is the adiabatic insertion of ascattering potential into the Fermi gas. This problem is harderthan it might seem. The difficulty can be understood from theperspective of Anderson’s orthogonality catastrophe (20), whichstates that the ground state of the homogeneous Fermi gas andthe gas with a single impurity are orthogonal in the thermody-namic limit. In addition, the system is gapless; as a consequence,standard arguments exploited in the adiabatic quantum comput-ing literature (21, 22) suggest that, to load the potential adiabat-ically, one has to scale the ramp velocity to zero with the inversesystem size. We checked that this is indeed the case for the naive

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loading protocol. The situation changes dramatically with the CDdriving.

To obtain the gauge potential, we numerically solve Eq. 11 ina box of size L with vanishing boundary conditions αj=−L/2 =αj=L/2 = 0. In Fig. 3, we show time dependence of the squaredfidelity of the wave function and the instantaneous ground state:F 2(t) = |〈ψ(t)|ψGS (t)〉|2 for different protocols. The total sys-tem size is L= 512 at half-filling (256 particles). The system isinitially prepared in the ground state at zero potential, and then,we are turning on a repulsive Eckart potential of the form

Vj (λ) =λ

cosh2 j/ξ, j ∈ [−L/2,L/2].

We choose ξ= 8 and turn on λ(t) according to the protocol (Eq.15) with λ0 = 0 and λf = 2J . The total duration of the protocolis τ = 10/J . The naive protocol indeed fails completely, givingthe very small final fidelity F 2(τ)≈ 2 · 10−19 as expected. Thisresult is only marginally better than the fidelity of the initial stateand the final state, which is 5 · 10−20. The CD protocol, however,gives fidelity of the order of 1/2, gaining more than 18 orders ofmagnitude. This value implies a 50% chance of preparing the sys-tem in the exact many-body ground state. Notice that, althoughthe imaginary CD protocol (purple line in Fig. 3) keeps instan-taneous fidelity high at all times, for the real protocol, exactlylike in the waiter case, the instantaneous fidelity at intermediatetimes drops to a very small value. High fidelity is only recoveredat the end of the protocol, where the velocity λ becomes close tozero. For additional details on the performance see Figs. S2–S6.

1D Spin Chain. In the previous examples, we focused on free parti-cle 1D systems, where an exact solution forAλ always exists, andthe variational approach merely helps one to find a simpler andeasier way to implement local approximate A∗λ. Many-particlesystems are intrinsically chaotic, and as we already discussed,the exact gauge potential simply does not exist in the form ofa local operator. In such cases, approximate methods for find-ing A∗λ are simply required to find CD protocols. To illustratethe power of the variational approach, let us consider an Isingspin chain in the presence of a transverse and longitudinal field.

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fide

lity

F2 (t)

t/τ

Fig. 3. Inserting local potential. The probability to be in the adiabaticground state when inserting a scattering potential quickly decays to a smallvalue for the naive protocol (dashed red line). By CD driving with a localcomplex gauge, the state stays much closer to the ground state, and a finalfidelity of about 1/2 is reached (purple line). A gauge-equivalent real Hamil-tonian, with renormalized hopping and potential, results in the same finalfidelity but is almost orthogonal to the ground state at intermediate times(blue line).

This system is one of the simplest nonintegrable models withvery rich phase diagram (23, 24). The Hamiltonian of this sys-tem reads

H0 =∑

j

(Jσz

j σzj+1 + Zjσ

zj + Xjσ

xj

), [19]

where σzj and σx

j are the Pauli matrices. We allow all couplingsto depend on some tuning parameter λ, which in turn, dependson time. The simplest gauge potential, which is purely imaginary,is the magnetic field along the y direction:

A∗λ =∑

j

αjσyj . [20]

Recall that Gλ ≡ ∂λH + i [A∗λ,H ]; hence,

Gλ =∑

j

((X ′j − 2Zjαj )σ

xj + 2αJ (σx

j σzj+1 + σz

j σxj+1)

)+∑

j

((Z ′j + 2Xjαj )σ

zj + J ′σz

j σzj+1

), [21]

where prime stands for the derivative with respect to λ. BecausePauli matrices are traceless, computing the Hilbert–Schmidtnorm of this operator is trivial and amounts to adding up squaresof coefficients in front of independent spin terms:

2−LTr(G2λ) =

∑j

((X ′j − 2Zjαj )

2 + 8α2j J

2)+∑

j

((Z ′j + 2Xjαj )

2 + (J ′)2). [22]

Minimizing the quadratic form with respect to αj immediatelyyields the optimal variational solution:

αj =1

2

ZjX′j −XjZ

′j

Z 2j + X 2

j + 2J 2. [23]

Note that the y magnetic field is strictly local, because it onlydepends on the local values of the x and z magnetic fields. ForJ = 0, this gauge potential is exact, because it is simply a gen-erator of local spin rotations in the x − z plane. Note that A∗λvanishes if either hx = 0 or hz = 0, implying that the leading con-tribution to Aλ in this case actually comes from two-spin terms.To second order, we can include two-spin terms into the varia-tional ansatz:

A∗λ =∑

j

[αjσ

yj + βj

(σy

j σzj+1 + σz

j σyj+1

)]+∑

j

γj

(σy

j σxj+1 + σx

j σyj+1

). [24]

The coefficients αj , βj , γj can directly be found by minimizingthe quadratic action (Eq. 8), resulting in a linear set of coupledequations, which can be easily solved numerically. This varia-tional solution dramatically enhances performance of the anneal-ing protocol, loading spins from an initial product state to theground state of the Hamiltonian (Eq. 19) (SI Text; specific pro-tocols are analyzed in Figs. S7–S9).

Following the orthogonality catastrophe example, let us con-sider a CD protocol for turning on a local magnetic field. Specif-ically, we consider turning on an additional x magnetic field λfrom 0 to the final value λf =−10J in a periodic chain describedby the Hamiltonian H0 + λσx

0 , where H0 is given by Eq. 19with J = 1, Zj = 2, and Xj = 0.8. We compare three protocols:bare protocol with no CD driving, CD protocol with only localCD driving (Eq. 20), and the optimal two-spin CD protocol(Eq. 24). We verified that the variational coefficients αj , βj , γj

rapidly (exponentially) decay with j away from the site j = 0,and therefore, the CD protocol effectively remains local. Notethat we can find the CD protocol in the thermodynamic limit;

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however, we have to apply it to a finite chain to verify its perfor-mance. The probability of recovering the system in the groundstate for a chain of 15 spins is shown in Fig. 4. For fast proto-cols, there is a significant reduction in excess energy and a corre-sponding increase in fidelity for the CD protocols. In particular,in the instantaneous quench limit τ→ 0, the CD driving givesalmost a factor of 10 gain in fidelity and a similar reduction inthe heating. Interestingly, in this limit, dynamics is entirely gov-erned byA∗λ, which is exponentially localized near j = 0. Loadinginto a strong magnetic field can be also viewed as changing theboundary conditions in the system; effectively, it cuts the chainin two pieces. Moreover, because of time-reversal symmetry, thefidelity of joining the chain back together is identical. There-fore, in this case,A∗λ can be interpreted as an effective boundaryHamiltonian generating adiabatic boundary transformations onthe ground-state wave function.

Next, let us use the same example to analyze another appli-cation of the CD driving, namely suppression of dissipation in anoisy system. Let us now assume that the site j = 0 is subject toa small white noise in the magnetic field in the x direction. Phys-ically, this noise can stem from coupling of this site to a nearbyimpurity or a quantum dot. A convenient measure of dissipationin the system is the rate of spread of energy fluctuations dtδ

2E . If

we implement exact CD protocol, then the system follows instan-taneous eigenstates, and the energy fluctuations remain constantin time, so that the rate is zero. At finite temperatures, dtδ

2E is

related to the usual heating rate dE/dt by the fluctuation dissi-pation relation (19). However, the spread of energy fluctuationshas a well-defined limit even for the infinite temperature states,where the heating rate vanishes. Within Fermi’s golden rule andunder the assumption that the spectral density of the fluctuat-ing field is white, the rate at which the fluctuations will increaseif we initialize the system in a pure state |n〉 takes the follow-ing form:

∂tδ2E = Sλλ 〈n| ([H0,Gλ])2 |n〉, [25]

where Sλλ is the power spectral density of the noise (i.e., theinverse bandwidth of the noise). Fig. 5 shows the normalized pro-duction rate ∂tδ

2E/Sλλ for every eigenstate of an ergodic chain

of 15 spins. We clearly see a reduction in fluctuations across theentire spectrum, comparable with the reduction in excess energyfor the ground-state protocol discussed above. Interestingly, CDdriving not only reduces dissipation but also, reduces its fluctua-tions between different eigenstates.

To understand better the performance of the CD protocol, wecan analyze the spectral decomposition of the dissipation as afunction of the absorption frequency. Specifically, we look at the

0

2

4

6

8

10

12

14

16

E-E 0

10 10 10 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

F2

10 10 10 10

A B

Fig. 4. Local spin flip. A single spin in a chain of 15 spins is aligned withthe x direction by switching a strong local magnetic field in the x direction(details are in the text). A shows the fidelity to recover the system in theground state after a protocol of duration τ . B shows the final energy abovethe ground state. The red squares are associated with the naive protocol,the green circles are associated with strictly local CD driving, and the bluecircles are associated with CD driving (Eq. 24).

lifetime of a state Γn within Fermi’s golden rule when we subjectthe system to a periodic drive with frequency ω:

Γn(ω) = Sλλ(ω)∑m

|〈m|Gλ |n〉|2 δ(Em −En −ω), [26]

where the δ function is broadened, such that it contains sev-eral eigenstates leading to a smooth dependence of Γn(ω). Theresult is shown in Fig. 6. All three protocols show a very nar-row peak in the transition rate around ω= 0 followed by a muchbroader distribution at large frequencies. The small frequencytransitions represent mixing between nearby eigenstates. Theylead to very small dissipation and are not affected by the CDdriving terms. Conversely, the high-frequency transitions leadingto large energy transfer from the noise to the system are stronglysuppressed by the CD driving.

Classical Spin Chain. Let us briefly show how the developed ideascan be applied to classical systems. Specifically, we will considera classical rotor model with the Hamiltonian

H0 = J∑

j

~Sj~Sj+1 + X0(λ)S x

0 + Z0(λ)S z0 , [27]

where ~Sj are 3D classical angular momenta. Because ~S2j =S2 is

conserved under the Hamiltonian dynamics, we will set S2 = 1.This Hamiltonian, with appropriate rescaling of the couplingconstants, can be obtained as a large spin limit of a quantumHeisenberg spin chain with an additional local magnetic field.This Hamiltonian is a direct analog of a quantum spin modelanalyzed earlier (Eq. 19), except that we use isotropic Heisen-berg coupling. Because the Heisenberg model is not integrable,there is no need to introduce extra static magnetic fields. As inthe quantum case in the leading order, we will seek the gaugepotential in the form

A∗λ = α0(λ)S y0 . [28]

Recall that the Poisson bracket between any two functions canbe expressed as{

A(~Sj

),B(~Sj

)}=∑

j

εabcSaj∂A

∂S bj

∂B

∂S cj

,

where a, b, c = {x , y , z}, and εabc is a fully antisymmetric tensor.Using this definition we find that

Gλ = ∂λH0 + {H0,A∗λ} = (X ′0 − αZ0)S x0 + (Z ′0 + αX0)S z

0

+Jα (S z0 (S x

1 + S x−1)− S x

0 (S z1 + S z

−1)). [29]

From this expression, we easily obtain ||Gλ||2 by integrating thesquare of Gλ over spin directions:

||Gλ||2 =1

3(X ′0 − αZ0)2 +

1

3(Z ′0 + αX0)2 +

4

9J 2α2. [30]

Minimizing this with respect to α, we find the optimum solution:

α =Z0X

′0 −X0Z

′0

X 20 + Z 2

0 + 43J 2. [31]

This expression is very similar to the quantum result (Eq. 23)with a slightly different prefactor in the J 2. (The overall factor1/2 in front is just because of the difference between spins andPauli matrices.) For the classical model with only z − z interac-tions, the only difference in α would be in a prefactor in front ofJ 2: 2/3 instead of 4/3. In a similar fashion, one can extend thevariational ansatz to higher orders, including various terms oddin powers of S y like S y

j S(z ,x)j+1 .

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US

PHYS

ICS

10 -4 10 -3 10 -2 10 -1 10 00

2

4

6

8

10

12

14

16In

duce

d Fl

uctu

atio

ns

Fig. 5. Spin chain fluctuations. Normalized energy fluctuations productionrate in a chain of L = 15 spins with X = 0.9, Z = 0.8, and J = 1 when a singlespin is subject to a random weak magnetic field in the x direction. The effec-tive inverse temperature of each eigenstate β is defined to match its energyin an equivalent thermal ensemble. The red, green, and blue dots representresults for no CD term, the best single-site CD term, and the best two-siteCD term, respectively.

MethodsConsider a state |ψ〉 evolving under a time-dependent Hamiltonian H0(λ(t)):

i~∂t |ψ〉 = H0(λ(t)) |ψ〉, [32]

where the full time dependence of the Hamiltonian is caused by an exter-nal drive λ(t). Let us go to the rotating frame where the Hamiltonianremains stationary (diagonal at all times). This diagonalization can bealways achieved by a λ-dependent unitary transformation U(λ(t)), whichalso expresses the state in the instantaneous eigenbasis of the Hamiltonian.It is easy to see that the wave function in the moving frame |ψ〉= U(λ)|ψ〉satisfies the effective Schrodinger equation (17):

i~∂t|ψ〉 =(

H0(λ(t))− λAλ

)|ψ〉, [33]

where H0 is the Hamiltonian in the instantaneous basis, and Aλ is the adia-batic gauge potential in the moving frame:

H0(λ(t)) = U†H0(λ(t))U =∑

n

εn(λ)|n〉〈n|,

Aλ = U†i∂λU. [34]

Differentiating the relation H0 = U†H0U with respect to λ and using thefact that ∂λH0 commutes with H0, one can check that the adiabatic gaugepotential satisfies the following equation (17):

i~ (∂λH0 + Fad) = [Aλ, H0], [35]

where Fad is the adiabatic or generalized force operator:

Fad = −∑

n

∂λεn(λ) |n(λ)〉 〈n(λ)|, [36]

and Aλ = UAλU† = i(∂λU)U† is the adiabatic gauge potential in the lab-oratory frame. Although we used the moving frame to derive Eq. 35, it isan operator equation, which is valid in any frame, including the laboratoryframe. Note that Eq. 2 trivially follows from Eq. 35, because Fad by construc-tion commutes with the Hamiltonian.

Now let us discuss in more detail how Eq. 35 can be reformulated asthe minimization problem leading to the variational ansatz. For this pur-pose, let us choose some trial gauge potential A∗λ and define an oper-ator Gλ:

Gλ = ∂λH0 +i

~[A∗λ, H0].

This operator also has a well-defined classical limit. It is clear from Eq. 35that, forA∗λ =Aλ, we have Gλ =−Fad. The diagonal elements of Gλ in the

basis of the Hamiltonian do not depend on A∗λ: 〈n|Gλ|n〉= ∂λεn(λ). Thus,different choices ofA∗λ only affect the off-diagonal elements of Gλ. For thetrue gauge potential Aλ, Gλ has no off-diagonal elements; it thus corre-sponds to the operator Gλ with minimal Hilbert–Schmidt norm. Formally,this equivalence can be seen from the distance between Gλ and −Fad:

D(A∗λ)= Tr

[(Gλ + Fad)

2]

= Tr

[(∂λH0 + Fad +

i

~[A∗λ, H0

])2]. [37]

Using cyclic properties of the trace, this distance becomes

D(A∗λ)= S

(A∗λ)− Tr

[F2

ad

], [38]

with the action S(A∗λ) given by Eq. 8. Minimizing the distance with respectto A∗λ results in Eq. 2. We would like to stress that an enormous gain hasbeen made by moving from the original Eq. 35 to Eq. 2, because the adia-batic force has been eliminated. In fact, neither the action (Eq. 8) nor theEuler–Lagrange Eq. 2 make any reference to the adiabatic force, which isgenerally hard to compute. Nonetheless, by exactly minimizing (Eq. 8), onerecovers the exact adiabatic gauge potential Aλ = i(∂λU)U†, which impliesthat the matrix elements of Aλ between instantaneous eigenstates satisfy〈m(λ)| Aλ |n(λ)〉= 〈m(λ)| i∂λ |n(λ)〉. Consequently, this CD drive reproducesthe full adiabatic evolution, including the Berry phase. Approximate gaugepotentials can, therefore, be used to extract an approximate Berry curvatureand phase.

Discussion and ConclusionBuilding on the concept of transitionless driving, we have devel-oped a variational principle that allows one to construct approx-imate variational CD protocols. Using this variational ansatz,we obtained best local CD protocols and showed that they canstrongly decrease heating of highly excited states and increasefidelity of the ground-state preparation by many orders of mag-nitude. Efficient CD protocols can find many different applica-tions from constructing fast and efficient annealing protocols forboth quantum computers and quantum simulators to engineer-ing thermodynamic engines operating close to the maximum effi-ciency at fast speeds. A key advantage of the variational methodis that it allows one to find such protocols without the needfor knowing any details about the spectrum of the Hamiltonianor the structure of its eigenstates, which are usually very hardto obtain.

-15 -10 -5 10 150

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

Tra

nsit

ion R

ate

0 5

Fig. 6. Spin chain transitions. Average normalized transition rate overstates with effective temperature β= 0.1. The parameters and the colorsare the same as in Fig. 5.

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We illustrated the ideas using various examples: (i) inserting(and moving) a local potential barrier into a Fermi sea, effec-tively fighting the Anderson orthogonality catastrophe, and (ii)locally flipping a spin on an ergodic quantum spin chain. In allcases, we showed that CD causes a dramatic decrease of heat-ing and improvements in the final fidelity even in the gaplessregimes, where standard arguments based on the gap indicatethat the high-fidelity state preparation is not possible at thesefast rates. It remains to be seen how efficient and robust the

CD protocol can be in more complicated situations, such ascrossing phase transitions or annealing systems with slow glassydynamics.

ACKNOWLEDGMENTS. We thank C. Ching and C. Jarzynski for useful dis-cussions. D.S. was supported by the Fonds Wetenschappelijk Onderzoek asa postdoctoral fellow of the Research Foundation–Flanders. A.P. was sup-ported by Air Force Office of Scientific Research Grant FA9550-16-1-0334,National Science Foundation Grant DMR-1506340, and Army Research OfficeGrant W911NF1410540.

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