Quantum control with a multi-dimensional Gaussianquantum invariantSelwyn Simsek and Florian Mintert

Physics Department, Blackett Laboratory, Imperial College London, Prince Consort Road, SW7 2BW, United KingdomMarch 8, 2021

The framework of quantum invariants is anelegant generalization of adiabatic quantumcontrol to control fields that do not need tochange slowly. Due to the unavailability of in-variants for systems with more than one spatialdimension, the benefits of this framework havenot yet been exploited in multi-dimensionalsystems. We construct a multi-dimensionalGaussian quantum invariant that permits thedesign of time-dependent potentials that letthe ground state of an initial potential evolvetowards the ground state of a final potential.The scope of this framework is demonstratedwith the task of shuttling an ion around a cor-ner which is a paradigmatic control problem inachieving scalability of trapped ion quantuminformation technology.

1 IntroductionThe development of hardware for quantum informa-tion processing has reached a stage in which tens ofqubits can be accurately controlled [1, 2]. In order tocontrol such a large register of qubits, it is advisableto arrange them into several smaller units that arecoherently interconnected. Such interconnections canbe realised in a multitude of different ways [3, 4]. Inthe case of trapped ions, one of the most prominentenvisioned architectures is based on segmented traps,in which ions can be moved, i.e. shuttled, betweendifferent segments [4, 5] of a segmented trap.

Since trapped ion quantum logic usually requiresthe motional states of ions to be close to their quan-tum mechanical ground state [6], it is desirable thatany such shuttling process should transfer ions to theirmotional ground state with high fidelity. Perform-ing all operations of a quantum algorithm within thetime-scale imposed by the system’s coherence times,requires fast shuttling [5] implying that the transientmotional states during shuttling are far away from themotional ground state. Any shuttling process thusneeds to end with a stage of deceleration in which anoriginally rapidly moving ion is being transferred toits quantum mechanical motional ground state.

In practice, shuttling is realised in terms of time-dependent trapping potentials resulting from volt-

ages applied to trap electrodes [7, 8]. Describingthe quantum dynamics of a particle through sucha time-dependent potential landscape is generally aformidable numerical challenge, but in the presentcontext the problem can be substantially simplified.The initial ground state is a Gaussian wavepacketcharacterised by the expectation values of positionand momentum and the corresponding covariancesof those operators. For realistic trapping potentials,the potential can be approximated to be harmonicin the spatial domain occupied by such a wavepacket[9]. Within this approximation, an initial Gaussianwavepacket remains Gaussian under time evolution,and thus the state is characterised in terms of ex-pected position in phase space and covariances, i.e. asmall number of parameters [10–14].

Identifying time-dependent gate voltages that re-alise a shuttling protocol that transfers one or severalions from their ground state in the initial potentialto the ground state of their final potential, typicallyrequires testing a large number of different potentialsolutions. As such, the reduction of numerical effortresulting from the Gaussian approximation is invalu-able.

An equally important reduction in effort can beachieved with the framework of quantum invariants[15], since those allow one to directly construct shut-tling protocols that end with the ions in their mo-tional ground state [16]. The framework of quantuminvariants and its use for control of wave-packets viainvariant-based inverse engineering is well-establishedfor one-dimensional problems [16–20].

Invariant-based inverse engineering has been alsosuggested to realise the transport of cold atoms inoptical lattices [21] or optical tweezers [22] as well asatom clouds with a corresponding experimental im-plementation [23]. Invariant-based inverse engineer-ing has also motivated the study of fast transportof spin-orbit-coupled Bose-Einstein condensates [24],and quantum invariants have been used as a theo-retical tool to compute topological phases in planarwaveguides [25] and minispace quantum cosmologies[26]. They may also be constructed for light beampropagation in nonlinear inhomogeneous media [27].

Processes in which an ion is being shuttled across ajunction in a more complicated trap geometry, how-ever, cannot yet be approached with invariant-based

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inverse engineering, because the development of in-variants for systems with more than one translationaldegree of freedom [28]is still in its infancy. Here, wedevelop a framework of quantum invariants that issuitable for optimal control of Gaussian wave-packetsin any number of spatial dimensions via invariant-based inverse engineering.

In Sec. 2, invariant-based inverse engineering andGaussian states are summarised. Sec. 3 introducesthe invariant that is at the core of this paper anddiscusses how this invariant can be used to designtime-dependent potentials that result in ground-state-to-ground-state shuttling. This section is followedby explicit examples of such protocols discussed inSec. 4, while the actual derivation of the invariant isin Sec. 5, and Sec. 6 contains a discussion of how well-established one-dimensional invariants are containedas special case in the present framework. A readerwho is more interested in the underlying frameworkthan in its application to a specific control problem,can skip Sec. 4 or read Secs. 5 and 6 in any order.

2 Invariant-based controlIn this section several elements of Gaussian quantumstates and quantum invariants upon which the presentwork builds will be reviewed.

In order to ease notation, from hereon in the mo-mentum variables pi will include a scaling factor de-pending on the mass of each particle, such that the ki-netic energy reads

∑i p

2i /(2m) instead of the usual ex-

pression∑

i p2i /(2mi) that explicitly reflects the pos-

sibility of having different ions with different masses.A corresponding scaling of the position variables xi

ensures that xi and pi, as well as their classical coun-terparts, are conjugate to each other.

2.1 Gaussian dynamicsThe framework uses the Gaussian approximation ofwavepackets that simplifies the description in termsof general quantum states to a description in terms ofclassical trajectories and covariances. If X denotesthe 2d-dimensional vector comprised of the phasespace operators of a d-dimensional system, i.e. d po-sition operators and d momentum operators

X =(r1, r2, ..., rn, p1, p2, ..., pn

), (1)

then any Gaussian wavepacket is completely charac-terised in terms of the expectation value of X, denotedin the following by the vector X, and by the 2d×2d −dimensional covariance matrix Σ with the elements

Σij = 12 〈XiXj + XjXi〉 − 〈Xi〉〈Xj〉 , (2)

where the symbol 〈 〉 denotes the expectation valuewith respect to the Gaussian quantum state.

In a potential that is not always harmonic, an ini-tially Gaussian quantum state will lose its Gaussiancharacter, but if the wavepacket is sufficiently well lo-calised in real-space, then the potential can be approx-imated by its second-order Taylor expansion aroundthe center of the wavepacket, and the quantum stateremains Gaussian [11].

The center of the wavepacket, X = 〈X〉, followsthe classical trajectory, i.e. it satisfies the equation ofmotion

X = S ~V , (3)

where~V = ∇Hc (4)

is the vector containing the first derivatives of theclassical Hamiltonian Hc with respect to the phasespace variables, evaluated at the centre of thewavepacket X, and the symplectic matrix

S =[

O 1

−1 O

], (5)

is defined in terms of the d× d−dimensional identitymatrix 1 and zero-matrix O.

The covariance matrix satisfies the equation of mo-tion [29]

Σ = ΣΩS − SΩΣ (6)

in terms of the matrix Ω containing the second deriva-tives

Ωij = ∂2Hc

∂xi∂xj(7)

of the classical Hamiltonian Hc with respect to thephase space variables evaluated at X.

From hereon in it is assumed that the anharmoniccomponent of H, which is to say the Taylor expan-sion of H beyond the second order, is negligible, inwhich case the approximation holds and the systemHamiltonian can be expressed as

H = 12X

T ΩX + ~V X . (8)

2.2 Quantum invariantsThe problem at hand lies in finding Ω and ~V such thatX(T ) matches the desired phase space position at thefinal instance in time T of the shuttling protocol, andsuch that Σ(T ) corresponds to the covariances of thequantum mechanical ground state of H(T ). Quan-tum invariants allow one to derive straightforwardlyexpressions for Ω and ~V that ensure that the bound-ary conditions at t = 0 and t = T are satisfied [20].

A quantum invariant is any operator I(t) satisfyingthe equation of motion [30]

∂I(t)∂t

= i[I(t), H(t)] . (9)

Crucially, the instantaneous eigenstates of an in-variant, are solutions of the Schrodinger equation

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with the time-dependent Hamiltonian H(t) [15]. Aquantum invariant with non-degenerate spectrumthat commutes with the Hamiltonian at the begin-ning and the end of a time interval of interest, i.e.[I(0), H(0)] = 0 and [I(T ), H(T )] = 0, can thus beused to identify shuttling protocols that ensure thatthe system ends up in an eigenstate of the final Hamil-tonianH(T ), if it is initialised in an eigenstate of H(0)[16]. If the system is initialised in the ground state ofH(0), then it will be transferred to the ground stateof H(T ) since the only eigenstate of the quadraticoperator H(T ) that is Gaussian is the ground state.This is similar to the idea of adiabatic transitions,but in contrast, the invariant based method is exactand holds also under fast changes in the Hamiltonian,and the system state remains an eigenstate of the in-variant I(t) rather than an eigenstate of the systemHamiltonian H(t).

2.3 Control with quadratic invariantsIn the case of quadratic, time-dependent, Hamilto-nians, quantum invariants can be constructed fromwhich one can deduce quadratic Hamiltonians H(t)that correspond to those invariants. This, how-ever, is not very useful in practice since, the result-ing Hamiltonians contain in general bilinear terms ofphase space operators; in particular, the part corre-sponding to the kinetic energy will be of the form∑

ij hij(t)pipj , which is not necessarily equal to the

desired term∑

i p2i /2m.

Since, in practice, shuttling must be realised interms of modulation of only the trapping potential, itis crucial to ensure that Hamiltonians resulting fromthe invariant-based framework are of the form

H(t) =∑

i

p2i

2m + 12m

∑ij

Mij xixj −∑

i

~Fixi , (10)

with tuneable parameters ~F and M , and no Lorentz-type terms of the form xipj + xj pi, or terms linear inthe momenta pi.

In one-dimensional systems, there does exist a classof quantum invariants [16] that ensures the Hamilto-nians are of this form and gives a way to deduce Mand ~F from the invariant. Despite the utility and ap-plicability of these one-dimensional invariants, gen-eralisations to higher dimensional systems have notbeen found to date.

3 Optimal control with high-dimensional quadratic invariantsAs the present work deals with quadratic Hamiltoni-ans, it is most useful to work with an invariant thatcontains the phase space operators X only up to sec-

ond order. Any such invariant can be written as

I(t) = 12X

T ΓX + ~W · X + θ , (11)

parametrised in terms of a time-dependent, Hermitianmatrix Γ(t), a real vector ~W (t), and a real scalar θ(t).I(t) is an invariant if and only if Eq. (9) holds,

which is the case if the equations

dΓdt

= ΩSΓ− ΓSΩ, (12)

d ~W

dt= ΩS ~W − ΓS ~V , (13)

dθ

dt= ~V TS ~W, (14)

with Ω and ~V as defined in Eqs.(3) and (7) are sat-

isfied. For any choice of Γ and ~W , one may deducechoices of Ω, ~V and θ which satisfy Eqs. (12), (13)and (14).

This approach would permit us to find a quadraticHamiltonian for an invariant specified by functionsΓ(t) and ~W (t) that one would be free to choose. Formost choices of such invariants, however, any compat-ible Hamiltonian will not necessarily be of the formgiven in Eq. (10), i.e. it would not be realizable inpractice. It is thus essential to find instances of Γ and~W , such that Ω is of the form

Ω =[mM O

O1m1

], (15)

and such that ~V is of the form

~V =[−~F0

]. (16)

3.1 A suitable invariantWhile a general invariant of the form specified inEq. (11) does not correspond to a Hamiltonian ofdesired form, the following explicit parametrizationforces the Hamiltonian to be of the form of Eq. (10),as shown in Sec.5.

The quadratic part Γ of the invariant is given by

Γ = m(R2 + <

([R, R]A −RA2R

)) J − R, R2

−J − R, R2

R2

m

where the symbol [x, y]z = xzy − yzx denotes a gen-eralized commutator and x, y = xy+yx is the anti-commutator.

Γ is parametrised in terms of a d× d real, positiveand semi-definite square matrix R satisfying R(0) =0, and the matrices A and J are determined uniquelyfrom the relations

A = iR−2 + 12[R−1, R] + 1

2R−1JR−1 ,

J , R−2 = [R, R−1] + [R,R−2]R . (17)

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The linear part ~W of the invariant I is given by

~W = −Γ[

~L

m~L

], (18)

parametrized by a d-dimensional real vector ~L,

and the scalar part is given by

θ(t) = 12~WT Γ−1 ~W . (19)

The Hamiltonian corresponding to an invariant isdetermined by the equations of motion Eqs. (12)and (13). As shown in the following, the quanti-

ties M and ~F (defined in Eq. (15) and (16)) usedto parametrize and accessible Hamiltonians are de-termined by Eqs. (20) and (21).

The quadratic component M is determined by

R2,M = 2[R, R]A − R, R − 2RA2R (20)

in terms of R following Eq. (50). This is still notan explicit solution for M , but it can be obtained interms of an expansion into the eigenstates of R asstated in Eq. (53).

Once M is obtained, the linear component ~F is ob-tained through the relation

~F = m(~L+M~L

)(21)

following Eq. (61).R and ~L may be chosen freely subject to a set of

boundary conditions discussed in the next section.

3.2 Boundary conditionsAs stated in Sec. 2.2, a prerequisite for invariant-basedcontrol is that the invariant and Hamiltonian com-mute at initial and final times, i.e.

[I(0), H(0)] = [I(T ), H(T )] = 0 . (22)

Since the initial and final Hamiltonians H(0) andH(T ) are determined by the choice of problem athand, this commutativity must be ensured only byimposing boundary conditions on the invariant I(t),and not H(t). Since the invariant I(t) is parametrised

in terms of ~L and R via Eqs. (11), (17) and (18), this

implies boundary conditions for ~L and R.

The choices

R(t) = M(t)− 14 , R(t) = 0 ,

~L(t) = (mM(t))−1 ~F (t) , and ~L(t) = 0(23)

directly result in commutativity of I(t) and H(t).Imposing the conditions of Eq. (23) for t = 0 andt = T thus imposes the desired boundary conditionsEq. (22).

An additional set of conditions arise from the re-quirement that I be an invariant. The equations

R, R+ R2,M = 2[R, R]A − 2RA2R

~L+M~L =~F

m, (24)

which are derived in Sec. 5 as Eqs. (50) and (61), mustbe satisfied at all times. Substituting Eqs. (23) intothese equations gives an additional set of boundaryconditions

~L(t) = 0 , and R(t) = 0. (25)

For any time other than t = 0 and t = T , the condi-tions of Eqs.(23) and (25) do not need to be satisfied,

and any time-dependent form of ~L and R that satisfiesthe boundary conditions may be employed.

In practice, however, some choices of ~L(t) and R(t)can result in trajectories X(t) or covariances Σ(t) ofthe quantum state that are conflicting with practicalrequirements, such as a trapped ion colliding with atrap electrode. It is thus desirable to relate the vari-ables ~L(t) and R(t) characterizing the invariant I tothe variables X(t) and Σ(t) characterizing the quan-tum state.

Since the invariant I(t) is quadratic, this is mostelegantly done via the fact that the time-dependentquantum state is the instantaneous ground state ofI(t), which is characterized by covariance matrix

Σg = 12Γ−1 (26)

and phase-space coordinate

Xg = −Γ−1 ~W . (27)

Since Γ (given in Eq. (17)) is symplectic, its inversecan be expressed as Γ−1 = −SΓS, resulting in theexplicit form

Σg =

12

R2

m

R, R − J2

R, R+ J2 m

(R2 + <([R, R]A −RA2R)

)

for the covariance matrix.With the form of ~W given in Eq. (18), the explicit

solution (Eq. (27)) for Xg reads

Xg =[

~L

m~L

]. (28)

That is, Xg is determined straightforwardly in terms

of ~L, which means that the trajectory can be easilychosen at will.

It is of course desirable to attribute an equally clearphysical interpretation to the quantity R as found for~L in the sense of the trajectory. This, however, seems

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feasible only in the adiabatic limit, in which the time-derivative R becomes negligible. J in Eq. (17) thusalso becomes negligible, and Eq. (20) reduces to

R = M−14 . (29)

R thus directly determines the curvature of the trap-ping potential. Apart from this limiting case, how-ever, the relation between R and the potential inthe system Hamiltonian is substantially more compli-cated, consistent with the fact that diabatic dynamicsis generally more complicated than adiabatic dynam-ics.

4 Shuttling protocolWith the general framework for optimal control basedon high-dimensional invariants established, its prac-tical use can be demonstrated with the problem ofshuttling an ion through a two-dimensional potentiallandscape, such as the transfer from an initial trap-ping potential

Vi = 12m

(ω2

t x2 + ω2

r(y − r)2) (30)

centered around the initial position [0, r] to a finalpotential

Vf = 12m

(ω2

r(x− r)2 + ω2t y

2) (31)

centered around the final position [r, 0]. In additionto the change in position, in particular, also the shapeof the potential will change during the shuttling pro-cess with the (typically strong) radial confinement(characterized by the frequency ωr) initially in the y-direction and finally in the x-direction, and the (typ-ically weak) axial confinement (characterized by thefrequency ωt) initially in the x-direction and finally inthe y-direction.

As discussed above in the context of Eq. (28), thetrajectory of the ion can be chosen at will, as long asit satisfies the required boundary conditions. We arethus free to chose an arc

~L(t) = r

sin(π

2 p(τ))

cos(π

2 p(τ)) (32)

with τ = t/T , radius r, and a function p(τ) satisfyingthe boundary conditions p(0) = 0 and p(1) = 1. Mo-tivated by its successful use in one-dimensional prob-lems [17, 31, 32] we will choose

p(τ) = 10τ3 − 15τ4 + 6τ5 , (33)

which ensures that the boundary conditions

~L(0) = ~L(T ) = ~L(0) = ~L(T ) = 0 (34)

are satisfied.For the quadratic component R(t) of the invariant,

the boundary conditions read

R(0) =[ √

ωr 00 √

ωt

]=: Ri , (35)

and

R(T ) =[ √

ωt 00 √

ωr

]=: Rf , (36)

as well as R = 0 and R = 0 at t = 0 and t = T .A suitable time-dependent matrix R(t) that satis-

fies the boundary conditions can be defined with theansatz

R(t) = (1− p(τ))Ri + p(τ)Rf + τ3 (1− τ)3Rc , (37)

The matrix Rc that will determine how the trappingpotential rotates along the trajectory of the ion canstill be chosen at will, and the following examples arebased on the choice

Rc = (ωtωr) 14

[1 11 1

]. (38)

With the given specific choices for R(t) and ~L(t)one can solve Eqs.(12) and (13) for Ω and ~V , which

yields the desired objects M and ~F via Eqs.(15) and(16).

The explicit time-dependent solution for M de-pends on the system parameters ωr and ωt and onthe duration T of the shuttling protocol. Due to scale-invariance, the shuttling protocol depends only on twoindependent parameters, and in the following, we willconsider the variation of ωr

ωtand ωtT .

The explicit time-dependent solution for ~F dependsadditionally also on the mass m of the ion and theradius r of the trajectory. Discussing the solution ofthe time-dependent trapping potential in terms of itscenter ~C = 1

mM−1 ~F instead of the force ~F yields a

solution that is independent of m, and that simplyscales linearly with r.

As such, one can exemplify shuttling protocols forall values of ωr, ωt, m, r and T in terms of the twoindependent parameters ωr

ωtand ωtT .

Fig. 1 depicts some explicit solutions for time-dependent trapping potentials resulting in ground-state to ground-state shuttling protocols. Insets (a)and (d) correspond to a very fast shuttling protocolwith ωtT = 3, whereas insets (b) and (e) depict in-stances of a slower shuttling protocol with ωtT = 5,and insets (c) and (f) corresponds to a shuttling pro-tocol with ωtT = 10 that is close to adiabatic. Theinsets (a)-(c) correspond to a highly anisotropic trapwith ωr = 10ωt, whereas insets (d)-(f) correspond toa less anisotropic trap with ωr = 2ωt.

The arc-shaped trajectory ~L(t) of the ion is de-picted in purple, and the center of the trapping po-tential is depicted in blue. In the close-to-adiabatic

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0

r

0 r

y

x

(a)

ωr

ωt= 10 , ωtT = 3

0

r

0 r

y

x

(b)

ωr

ωt= 10 , ωtT = 5

0

r

0 r

y

x

(c)

ωr

ωt= 10 , ωtT = 10

0

r

0 r

y

x

(d)

ωr

ωt= 2 , ωtT = 3

0

r

0 r

y

x

(e)

ωr

ωt= 2 , ωtT = 5

0

r

0 ry

x

(f)

ωr

ωt= 2 , ωtT = 10

Figure 1: Trajectories for center of trapping potential (blue) and trap frequencies (yellow ellipses) for shuttling protocols withincreasing duration from the left insets to the right insets. The purple line denotes the trajectory of the ion, which traces outthe same path in each instance. Fast protocols (left) result in substantial deviations between the trajectories of trap centerand ion, and this effect is even more pronounced for the more isotropic trapping potentials.

cases (c) and (f) the trajectories of ion and center oftrapping potential nearly perfectly coincide, which isconsistent with the fact that the ion remains in theinstantaneous ground state of the trapping potentialin the adiabatic limit. Deviations between the twotrajectories are clearly discernible for the faster shut-tling protocols. In insets (b) and (e) the ion trajec-tory remains close to the trajectory of the trappingpotential, indicating that this protocol is close to theregime of validity of the adiabatic approximation. Ininsets (a) and (d), however, the trajectory of the cen-ter of trapping potential shares no similarity with thearc-shaped trajectory of the ion. The ion will thusbe substantially displaced from the center of trappingpotential, but the carefully chosen dynamics of thetrap center ensures that the ion comes to rest at thefinal instance of the protocol.

Comparison between the instances of the stronglyanisotropic trapping potentials featured in (a)-(c)with the instances of less anisotropic trapping poten-tials (d)-(f) indicate that the isotropy has only a mi-

nor impact on the dynamics of the trapping potentialin the slow protocols, but that anisotropy tends toalter dramatically the trajectory of the center of thetrapping potential.

The dynamics of the quadratic component of thetrapping potential is represented by yellow ellipses in-dicating equipotential curves, and these ellipses aredepicted for t = jT/8 with j = 0, . . . , 8. Even inthe fast shuttling protocols, the shape of the trap-ping potential rotates without discernible rapid ornon-monotonic dynamics, which is in quite some con-trast to the seemingly wild trajectories of the centerof trapping potential.

In order to gauge the experimental resources re-quired for the present shuttling protocol, it is in-structive to specify the maximum of the electric fieldmaxt∈[0,T ]‖~F (t)/e‖ that is required to keep the ion onits trajectory. For an Ytterbium-171 ion, with turningradius r = 30 µm a trap frequency ωt = 2π × 1 MHz

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this results in an electric field strength

‖ ~E‖ = 2.10× 105Vm−1 (39)

for the cases depicted in insets (a) to (c) of Fig. 1,and an electric field strength

‖ ~E‖ = 8.36× 103Vm−1 (40)

for the cases depicted in insets (d) to (f). The electricfield strength required in these examples thus dependonly on the isotropy of the trapping potential, butnot on the duration of the shuttling protocol. Thisseems surprising, but also encouraging since it sug-gests that increasing the speed of shuttling does notrequire increasingly strong control fields.

5 Construction of the invariantSince the derivation of the invariant introduced inSec. 3.1 is rather involved, it is deferred to this Sec-tion, with some technical steps discussed in Appen-dices A and B.

The goal is not to find an invariant for a givenHamiltonian; instead, the time-dependent invariantis the starting point, and the quest is for a Hamilto-nian such that Eqs. (12), (13) and (14) are satisfiedfor the given invariant I(t). This can be done in sub-sequent steps, first determining the quadratic compo-nent of the invariant, parametrized in terms of Γ, andsubsequently determining the linear component of theinvariant, parametrized in terms of ~W in Eq. (11).

5.1 A suitable invariant – the quadratic com-ponentAs stated in Eq. (15), Ω must take the form

Ω =[mM O

O1m1

], (41)

where M is a real symmetric matrix. The lower rightsub block 1

m1 is associated with the kinetic energywhile the term M is associated with the potential en-ergy, and is the unknown to be deduced from anyproposed invariant.

For a general matrix Γ, the resultant quadratic partΩ of the system Hamiltonian will not be of the desiredform given in Eq. (15), but the parametrization

Γ = <[mP †P −P †P−P †P 1

mP†P

], (42)

with four sub-blocks defined in terms of a d × d-dimensional, complex matrix P will result in the de-sired form. This can be seen by expressing the equa-tion of motion for Γ (Eq. (12)) in terms of P , resultingin [

m<(P †D +D†P ) −<(D†P )−<(P †D) 0

]= 0 (43)

where D = P + PM . Any choice of P satisfying thedifferential equation

P + PM = 0 (44)

thus results in a quadratic term Γ for the invariant Ithat is consistent with the initial ansatz in Eq. (15).

With the current parametrisation of Γ in terms ofP , one could construct M as

M = −P−1P . (45)

Usually, however, this does not result in the identi-fication of a suitable matrix M , because M definedin this fashion would not necessarily be symmetric orreal. It is thus necessary to find additional restrictionson P that guarantee the symmetry and reality of M .

To this end, it is helpful to express P in terms ofits polar decomposition

P = UR , (46)

where U is unitary and R is Hermitian and positivesemi-definite. It is possible to leave the factor R as afree parameter and determine U through the require-ment that M be real-symmetric.

In practice, this task can be formulated in terms ofthe anti-Hermitian operator

A = U†U . (47)

Since A satisfies the differential equation U = UA,the unitary factor U is uniquely determined in termsof A(t) and the initial condition U(0) = 1.

As derived explicitly in Sec.A, requiring M to beHermitian requires A to be of the form

A = iR−2 + 12[R−1, R] + 1

2R−1JR−1 , (48)

with

J =∫ t

0dτ [M(τ), R(τ)2] . (49)

With this specification of A in terms of R, R and M ,one obtains the differential equation

R, R+ R2,M = 2[R, R]A − 2RA2R (50)

for R.So far, it is only imposed thatM ought to be Hermi-

tian, and not that it should be real as well. As shownin Sec. B, however, M is indeed real-symmetric for alltimes, if R is chosen such that it is real and positiveat all times and that

R(0) = 0 . (51)

Even though Eq.(50), together with Eqs.(48) and(49) may be used to determine M(t), they are un-wieldy in practice, since they relate M(t) to the inte-gral J that depends on the entire history of M . SinceM is real-symmetric, however, one can – as discussed

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in detail in Appendix A arriving at Eq. (76) – specifyJ explicitly as

J =∑

ij

(λi − λj)(λi + λj)2

λ2i + λ2

j

〈Φi|R|Φj〉 |Φi〉 〈Φj |

(52)in terms of the eigenvalues λi and eigenvectors |Φi〉 ofR. With this explicit form of J , in terms of R and R,but without dependence on M , one can finally solveEq. (50) for M , which yields

M =∑

ij

〈Φi|B|Φj〉λ2

i + λ2j

|Φi〉 〈Φj | , (53)

with

B = 2[R, R]A − 2RA2R− R, R . (54)

Any real positive matrix R that satisfies the ini-tial condition R(0) = 0 (Eq. (51)) thus yields thequadratic term Γ of the invariant via Eq. (42) thatis consistent with the desired form for Ω specified inEq. (15) with a real-symmetric potential matrix M .

Substituting for P = UR and A = U†U in theAnsatz for Γ (Eq. (42)), results in the form of Γ thatis quoted in Eq. (17) in Sec. 3.1.

5.2 A suitable invariant – the linear componentThe goal of this section is to show that the ansatz

~W = −Γ[

~L

m~L

], (55)

of Eq.(18) results in a linear part ~V of the Hamiltonianthat is of the form

~V =[−~F0

], (56)

specified in Eq. (16).~V can be expressed in terms of ~W as

~V = S−1Γ−1(

ΩS ~W − ~W)

(57)

via the equation of motion for ~W given in Eq. (13).Rather than expressing the right-hand-side in termsof ~W , it will turn out helpful to express it in terms of~Z defined via the relation ~W = −Γ~Z. Together withthe equation of motion for Γ given in Eq. (12), thisyields

~V = S−1 ~Z − Ω~Z. (58)

With the explicit form of the quadratic term Ω ofthe Hamiltonian specified in Eq. (15) this results inthe explicit form

~V =[−mM ~Zx − ~Zp

~Zx − 1m~Zp

](59)

for ~V , where the vector ~Z is decomposed into spatialand momentum parts

~Z =[~Zx

~Zp

]. (60)

Finally, choosing ~Zp = m~Zx ensures that ~V is of theform given in Eq. (16) as desired, and the identifica-

tion of ~L with ~Zx results in the sought-after form of~W (Eq. (18)).

The resulting equation of motion

~L+M~L =~F

m, (61)

permits the determination of ~F in terms of ~L and R(through M).

5.3 A suitable invariant – the scalar compo-nentThe scalar part θ(t) of the invariant I is determinedby Eq. (14). Strictly speaking, it does not need tobe constructed, since the eigenstates of the invariantthat are the relevant objects are independent of thisphase term. For the sake of completeness, however,the solution

θ(t) = 12~ZT Γ~Z = −1

2~ZT ~W = 1

2~WT Γ−1 ~W , (62)

which satisfies Eq. (14) is stated.

6 The one-dimensional caseThe invariant derived in Sec. 5 does not seem to sharemany similarities with the invariant that are routinelyused in the one-dimensional case [33]. The regu-lar one-dimensional framework is, however, naturallycontained as special case in the present framework, aswe will show in the following.

Since all matrices such asM andR reduce to scalarsMs and Rs in the one-dimensional case, the commuta-tors and generalized commutators vanish in the one-dimensional case. In particular, the object J definedin terms of a commutator in Eq. (49) vanishes, so thatEq. (48) simplifies to

As = iR−2s . (63)

Substituting Eq. (63) into Eq. (50) and exploiting thefact that all objects commute, gives that

RsRs +MsR2s = R−2

s . (64)

The quadratic part Γ defined in Eq. (17) simplifiesto

Γs =[m(R2

s +R−2s

)−RsRs

−RsRs1mR

2s

], (65)

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in the case of commutativity. From Eqs. (18), (19)and (65), it is possible to write the invariant I outexplicitly in terms of r and p,

Is = 12m

(Rs(p−mL)−mRs(r − L)

)2

+ 12m

(r − LRs

)2, (66)

which is the Ermakov-Lewis invariant [30] with addi-tional terms linear in position and momentum [33].Dividing both sides of Eq. (64) by Rs gives

Rs +RsMs = R−3s , (67)

which is the Ermakov equation that appears in theoriginal treatment [30], where the symbol ρ is usedinstead of Rs, and the symbols ω2 and α are usedinstead of Ms and ~L.

7 OutlookThe extension of the framework of quantum invariantsto systems with quadratic potentials of more than onetranslational degree of freedom opens up a wide per-spective for the control of quantum systems. Whilethe present work is motivated by control of trappedions, none of the features of the present frameworkare specific to this system, and the only preconditionfor the use of the presented invariant is the validity ofthe Gaussian approximation. Since also shuttling ofcold neutral atoms has been experimentally realisedin one-dimensional optical lattice potentials [21] withtime-dependent potential designed with the Ermakov-Lewis invariant, the presently derived framework shallopen up possibilities to extend such tasks to higher-dimensional optical lattices.

The ability to systematically devise time-dependentpotentials that realise ground-state to ground-statetransfer permits to identify those potentials that areoptimized with respect to additional desirable proper-ties. Typical examples might include shuttling proto-cols with particularly weak forcing or potentials thatsuit best the geometry of the electrodes that gener-ate the potentials. As such, the present frameworkpromises to substantially advance the shuttling oftrapped ions, which is a central step towards achievingscalability of quantum information technology withtrapped ions.

The specific choice of invariant discussed here, doesnot necessarily need to be seen as the unique gen-eralization of one-dimensional invariants, but ratheras evidence that such generalizations are possible. Itthus seems conceivable that different generalizationcan be found, and that different restrictions of theachievable Hamiltonians can be found. One may, forexample, envision the Lorentz force caused by a homo-geneous magnetic field and aim at finding an invariant

that ensures that no inhomogeneities in the magneticfield arise.

In addition to the extension of experimental taskssuch as noise-resilient shuttling [18, 32, 34] to higherdimensional settings, the present work can also initi-ate new steps in the conceptual developments of quan-tum invariants.

AcknowledgementsWe are indebted to stimulating discussions withAdam Callison, Alexander Paige, Pedro Taylor-Burdett, Sebastian Weidt and Winnie Hensinger.This work was supported through a studentship inthe Centre for Doctoral Training on Controlled Quan-tum Dynamics at Imperial College London funded byEPSRC(EP/L016524/1).

A Derivation of the invariant equa-tionsThis section contains the explicit derivation ofEqs. (48),(49) and (50).

The equation of motion P + PM = 0 (Eq. (44))with P expressed in terms of its polar decompositionP = UR (Eq. (46)) results in

K := AR+A2R+ 2AR+ R+RM = 0 . (68)

Since M is Hermitian, and the relation RK = K†Rholds because K vanishes, this is equivalent to

d

dtRAR =1

2([R, R] + [M,R2]

). (69)

The right hand side of Eq. (69) can be rewritten interms of a total derivative to obtain

d

dtRAR = 1

2d

dt

([R, R] + J

), (70)

with

J =∫ t

0dτ [M(τ), R(τ)2] . (71)

This can be directly integrated, resulting in the ex-plicit solution

A = R−1CR−1 + 12[R−1, R] + 1

2R−1JR−1, (72)

for A.The matrix C is the constant of integration that can

be chosen freely. With the choice C = i1 motivatedby the Ermakov equation in the one-dimensional case[30, 35], this results in Eq. (48).

Using again the fact that K vanishes, we can formthe relation RK + K†R = 0 and rearrange to deriveEq. (50).

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B Reality of M

This section contains the proof that M(t) determinedby Eq.(50) is indeed real and symmetric for all times,provided that the initial condition R(0) = 0 is satis-fied.

Eq. (50) reads explicitly

R, R+ R2,M = 2[R, R]A − 2RA2R , (73)

and there is indeed no evident reason why M deter-mined from this relation together with Eqs. (48) and(49) should be real and symmetric.

In the following, explicit, closed-form expressionsfor J , A and M are derived, that hold under the as-sumption that M is real and symmetric. It is thendemonstrated that such closed-form expressions formindeed a solution of Eqs. (48), (49) and (50), whichnot only completes the proof that M is real symmet-ric, but gives a direct way to calculate M .

B.1 Necessary conditions for the reality of M

Under the assumption that M is real and symmetric,the imaginary part of the equation of motion for R

(Eq. (50)) reads

J , R−2 = [R, R−1] + [R,R−2]R . (74)

This determines J in terms of R and R, and, in con-trast to the defining relation (Eq. (49)) for J it istime-local.

In order to solve Eq. (74) for J , it is helpful toconsider the eigen-decomposition

R =∑

j

λj |Φj〉〈Φj | (75)

of the real and positive semi-definite matrix R.In terms of matrix elements with respect to the eigen-states |Φj〉 of R, Eq. (74) can directly be solved, yield-ing

Jjk = (λj − λk)(λj + λk)2

λ2j + λ2

k

Rjk , (76)

whereOjk = 〈Φj |O|Φk〉 (77)

is a short hand notation for the matrix elements ofoperator O, and

Ojk = 〈Φj |O|Φk〉 (78)

does generally not coincide with ∂∂tOjk because of the

time-dependence of the eigenstates |Φj〉.Similarly to Eq.(76), also Eq. (50) can be solved for

Mjk, resulting in

Mjk = 1λ2

j + λ2k

(2∑

l

(λkRjlAlk − λjAjlRlk − λjλkAjlAlk

)− Rjk (λj + λk)

). (79)

Eq. (48) expressed similarly in terms of matrix elements reads

Ajk = 12λjλk

(2iδjk − (λj − λk)Rjk + Jjk

). (80)

Substituting Eq. (76) into Eq. (80) yields

Ajk = i

λjλkδjk + λj − λk

λ2j + λ2

k

Rjk. (81)

Substituting Eq. (81) into Eq. (79) yields

Mjk = δjk

λ4j

− (λj + λk)λ2

j + λ2k

Rjk + 2∑

l

RjlRlkp(λj , λk, λl) , (82)

with

p(λj , λk, λl) =λ3

l (λj + λk)− λ2l

(λ2

j + λ2k − λjλk

)− λ2

jλ2k

(λ2j + λ2

k)(λ2j + λ2

l )(λ2k + λ2

l ) . (83)

The closed-form expressions for J , A and ulti-mately M , in terms of their matrix elements, givenin Eqs, (76), (81) and (82), hold if M is real symmet-ric.

Conversely, if Eqs. (76), (81) and (82) hold, thenM is guaranteed to be real symmetric since Eq. (82)is manifestly symmetric and defined solely using realquantities. As a consequence, M is real symmetric if

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and only if these expressions hold.

B.2 Direct proof of the reality of M

What remains is thus to verify directly that takingEqs. (76), (81) and (82) as defining relations for J ,A and M , implies that Eqs. (48), (49) and (50) aresatisfied. Doing this completes the proof that M isreal symmetric.

Henceforth it is assumed that Eqs. (76), (81) and(82) hold. As Eq. (81) is derived using Eq. (80) whichis Eq. (48) expressed in terms of the matrix elementsof the eigenstates of R, Eq. (48) holds by construction.Similarly, Eq. (82) is derived from Eq. (79), which it-self is Eq. (50) expressed in terms of matrix elements,so Eq. (50) additionally holds.

It remains to show that Eq. (49), or its equivalentdifferential form

J (0) = 0 , and (84)J = R2M −MR2 . (85)

is satisfied Since R(0) = 0, Eq. (84) follows directlyfrom inspection of Eq. (76).

Eq.(85) reads in terms of matrix elements

Jjk = (λ2j − λ2

k)Mjk. (86)

In order to complete the proof, it is sufficient to showthat Eq. (86) is satisfied. As J is defined in terms ofits matrix elements Jjk in Eq. (76), it is necessary torelate these to the ˙Jjk that appear in Eq. (86).

The time-derivative of Jjk reads

d

dtJjk =

∑l

WjlJlk −∑

l

JjlWlk + Jjk (87)

withWjk = 〈Φj |Φk〉. (88)

Substituting this expression into Eq. (86) gives

d

dtJjk =

∑l

WjlJlk −∑

l

JjlWlk + (λ2j − λ2

k)Mjk,

(89)which is equivalent to Eq. (86).

The next step is to differentiate Eq. (76) and usethe result to show that Eq. (89) is satisfied identically.

Defining, motivated by Eq. (89),

Ξ = d

dtJjk−

∑l

WjlJlk +∑

l

JjlWlk +(λ2k−λ2

j )Mjk,

(90)the remaining task is to show that Ξ = 0.

Substituting for Jjk and Mjk using Eqs. (76) and(82) in Eq. (90) gives

Ξ = d

dt

((λ2

j − λ2k)(λj + λk)

λ2j + λ2

k

)Rjk +

(λ2j − λ2

k)(λj + λk)λ2

j + λ2k

d

dt

(Rjk

)−∑

l

(λ2l − λ2

k)(λl + λk)λ2

l + λ2k

WjlRlk +∑

l

(λ2j − λ2

l )(λj + λl)λ2

j + λ2l

RjlWlk

+(λj + λk)(λ2

k − λ2j )

λ2j + λ2

k

Rjk − 2(λ2

k − λ2j

)p(λj , λk, λl)RjlRlk . (91)

Given that the states |Φi〉 are the eigenvectors of R, the explicit expressions for Rij , Rij and Rij read

Rjk = λjδjk , (92)Rjk = (λj − λk)Wjk + λjδjk , (93)

Rjk =∑

l

(λj + λk − 2λl)WjlWlk + (λj − λk)Wjk + 2(λj − λk)Wjk + λjδjk . (94)

Substituting Eq. (93) and Eq. (94) into Eq. (91) gives

Ξ = Wjkr(λj , λk, λj , λk) +∑

l

WjlWlks(λj , λk, λl), (95)

where

r(λj , λk, λj , λk) = (λj − λk) ddt

((λ2

j − λ2k)(λj + λk)

λ2j + λ2

k

)−

(λ2j − λ2

k)(λj + λk)(λj − λk)λ2

j + λ2k

− 2(λj − λk)(λ2k − λ2

j )(λjp(λj , λk, λj) + λkp(λj , λk, λk)

), (96)

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and

s(λj , λk, λl) = − (λ2l − λ2

k)2

λ2l + λ2

k

+(λ2

j − λ2l )2

λ2j + λ2

l

+(λj + λk)(λ2

k − λ2j )(λj + λk − 2λl)

λ2j + λ2

k

− 2(λj − λl)(λk − λl)(λ2j − λ2

k)p(λj , λk, λl) . (97)

Both r and s are rational functions of their arguments, and they are indeed identically vanishing. ThereforeΞ = 0 which completes the proof.

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