# Quantum invariant-based control of interacting trapped ions

Selwyn Simsek  Blackett Laboratory, Imperial College London, South Kensington
Campus, London SW7 2AZ, UK  Florian Mintert  Blackett Laboratory, Imperial
College London, South Kensington Campus, London SW7 2AZ, UK

###### Abstract

Invariant-based inverse engineering is an elegant approach to quantum control
with corresponding experimental implementations that perform tasks with
applications in quantum information processing such as shuttling trapped ions.
We build on recent work to generalise invariant-based inverse engineering to
control two coupled harmonic oscillators in any number of spatial dimensions.
This may be used to perform experimentally relevant tasks such as separation
of trapped ions, which is demonstrated numerically, achieving transfer
fidelities of over 96% as well as low motional number excitations.

††preprint: APS/123-QED

##  I Introduction

Trapped ions are among the most developed platforms for quantum information
processing. Both single-qubit quantum gates and entangling gates can be
implemented with high fidelity [1, 2, 3], and there exists the prospect of
realising a scalable architecture with microfabricated, structured traps. The
resulting quantum charge-coupled device can trap a large number of ions in
various trapping zones, and shuttling ions between such zones results in a
high interconnectivity of all the trapped ions [4, 5, 6, 7].

While the development of logical gates for trapped ions is fairly advanced
with many schemes for fast and noise-resilient gate operations [8, 9, 3],
shuttling of trapped ions is a much less mature field. Because of limited
coherence times, it is important to separate an ion from one chain of
interacting ions and to shuttle it towards another chain as fast as possible
[5]. Since the realization of logical gates requires the motion of trapped
ions to be close to their ground state [3], it is equally important that
motional excitations of the ions at the end of any shuttling protocol are kept
to a minimum.

Ideally, a shuttling protocol would thus transfer ions that initially occupy
their motional ground state to the ground state of a final trapping potential.
Such a task fits exactly into the setting of adiabatic control, but the
necessarily slow dynamics conflicts greatly with the requirement to have fast
shuttling protocols. Invariant-based inverse engineering is well suited to
construct shuttling protocols that ensure ground state to ground state
transfer without the requirement of slow dynamics. In particular for harmonic
potentials, the framework of quantum invariants is rather well developed, and
the harmonic approximation is excellent for trapped ions. Invariant-based
inverse engineering has thus become a frequently used technique in the context
of trapped ions with many conceptual developments [10, 11, 12, 13, 14, 15, 16,
17, 18, 19, 20] and experimental implementations with atoms and trapped ions
[21, 22]. While most approaches to quantum dynamics would focus on solving the
dynamics induced by a given Hamiltonian, invariant-based approaches rely on an
Ansatz for the dynamics and aim at constructing a Hamiltonian that gives rise
to this desired dynamics. The challenge lies in ensuring that the Ansatz
results in a Hamiltonian that can be realized in practice. While an Ansatz
resulting in a Hamiltonian that is quadratic in position and momentum
operators may be found reasonably straightforwardly, only the trapping
potential of trapped ions can be modified in practice. The kinetic energy,
however, is determined by the mass of the ions and any Ansatz resulting in a
Hamiltonian with a different kinetic energy does not help to find a
practically realizable shuttling protocol.

There exists a class of invariants that result in Hamiltonians with a given
kinetic energy term that can be used to find time-dependent trapping
potentials that realise ground state to ground state shuttling protocols for
individual trapped ions [23]. Generalizing such approaches to interacting
trapped ions, however, brings a crucial further difficulty: while the actual
trapping potential of the trapped ions can be chosen at will, the Coulomb
interaction between trapped ions cannot be modified in any way. Any approach
that relies on the ability to choose the full potential term of the system
Hamiltonian would thus not result in practically realizable shuttling
protocols. Incorporating the constraints imposed by interactions is possible
if the dynamics of each ion can be restricted to one single motional degree of
freedom [24], but many operations required for quantum information procession,
such as swapping two ions [25, 26] do not admit a one-dimensional
approximation.

The main contribution of this paper is the derivation of a quantum invariant
that results in a Hamiltonian with given kinetic energy and pairwise
interaction between the ions without a restriction on the number of motional
degrees of freedom of any ion. While a rigorous basis for invariant-based
inverse engineering is given for invariants with a non-degenerate ground
state, the present invariant has a degenerate spectrum. Despite this caveat,
however, this invariant can be used to devise shuttling protocols that are
implemented in terms of time-dependent trapping potentials only, as
demonstrated with the example of separating two initially interacting ions.
Even though these degeneracies result in slight imperfections, the framework
results in accurate shuttling protocols well beyond the adiabatic
approximation.

##  II Quantum invariants

A quantum invariant ${\cal I}$ is a Hermitian operator that satisfies the
equation of motion

$\frac{\partial{\cal I}(t)}{\partial t}=i[{\cal I}(t),H(t)],$ (1)

with the generally time-dependent system Hamiltonian $H(t)$. Crucially, the
instantaneous eigenstates of an invariant may be chosen to be solutions of the
time-dependent Schrödinger equation with the Hamiltonian $H(t)$ [27].

If the invariant ${\cal I}$ has a non-degenerate ground state, and ${\cal I}$
commutes with the Hamiltonian $H(t)$ at initial and final times, which is to
say that $[{\cal I}(0),H(0)]=0$ and $[{\cal I}(T),H(T)]=0$, then the ground
state of ${\cal I}$ is also an eigenstate of $H(t)$ at $t=0$ and $t=T$. One
has thus found a solution of the time-dependent Schrödinger equation that
yields the evolution of an eigenstate of the initial Hamiltonian $H(0)$
towards an eigenstate of the final Hamiltonian $H(t)$.

With invariant-based inverse engineering, one would start out with an Ansatz
for the time-dependent invariant ${\cal I}(t)$, and then construct a
Hamiltonian such that the equation of motion Eq. (1) is satisfied. The crucial
difficulty lies in finding an invariant that gives rise to a Hamiltonian that
can be experimentally realised.

For the translational degrees of freedom of a single trapped ion, for example,
the trapping potential can be tuned in terms of voltages applied to the trap
electrodes, but the kinetic energy term dependent on momentum and mass of the
ion is an intrinsic system property that can not be modified in practice. It
is absolutely essential that the Hamiltonian resulting from an invariant has
exactly this kinetic energy term.

Going from a single trapped ion or several non-interacting trapped ions to the
problem of interacting trapped ions, yields the Coulomb interaction as another
component to the Hamiltonian that can not be modified.

It is thus essential that the Hamiltonian resulting from an invariant is of
the form with the intrinsic kinetic energy of the trapped ions and the Coulomb
interaction as given by the position of the ions. Only single-ion potential
terms may appear as quantities that need to be chosen such that Eq. (1) is
indeed satisfied.

Realistically, this is achievable only within some approximation, such as the
Gaussian approximation in which the Hamiltonian is approximated in a second
order Taylor expansion around the expected position of the ions. For the
Coulomb interaction, this implies the approximation

$\displaystyle\frac{1}{\lvert\hat{\vec{x}}_{1}-\hat{\vec{x}}_{2}\rvert}\simeq$
$\displaystyle\frac{1}{|\vec{r}|}-\frac{\vec{\delta}_{\hat{x}}\vec{r}}{|\vec{r}|^{3}}-\frac{\vec{\delta}_{\hat{x}}\vec{\delta}_{\hat{x}}}{2|\vec{r}|^{3}}+\frac{3}{2}\frac{(\vec{\delta}_{\hat{x}}\vec{r})^{2}}{|\vec{r}|^{5}}\
,$ (2)

with

$\vec{r}=\vec{x}_{1}-\vec{x}_{2}\hskip 14.22636pt\mbox{and}\hskip
14.22636pt\vec{\delta}_{\hat{x}}=\hat{\vec{x}}_{1}-\hat{\vec{x}}_{2}-\vec{r}$
(3)

defined in terms of the expectation values
$\vec{x}_{1}=\langle\hat{\vec{x}}_{1}\rangle$ and
$\vec{x}_{2}=\langle\hat{\vec{x}}_{2}\rangle$ the (vectorial) positions of the
two ions.

The full Hamiltonian for ${N}$ ions within the Gaussian approximation is thus
of the form

$\displaystyle H$
$\displaystyle=\sum_{i=1}^{{N}}\frac{\vec{\hat{p}}_{i}^{2}}{2m_{i}}+\frac{1}{2}m_{i}\vec{\hat{x}}_{i}^{T}\mathcal{M}_{i}\vec{\hat{x}}_{i}-{\vec{\mathcal{F}}}_{i}\cdot\vec{\hat{x}}_{i}$
$\displaystyle+\sum_{i<j}^{{N}}\vec{\hat{x}}_{i}^{T}D(\vec{x}_{i}-\vec{x}_{j})\
\vec{\hat{x}}_{j}\ .$ (4)

It contains an interaction matrix $D(\vec{x}_{i}-\vec{x}_{j})$ with

$D(\vec{r})=C\left(\frac{\mathbbm{1}}{|\vec{r}|^{3}}-\frac{3\vec{r}\vec{r}^{T}}{|\vec{r}|^{5}}\right)\
,$ (5)

that depends on the spatial separation of the Gaussian wave-packets of pairs
of ions. The real-symmetric matrices $\mathcal{M}_{i}$ and real vectors
${\vec{\mathcal{F}}}_{i}$ that characterize the harmonic potential experienced
by the individual ions, have contributions of both the actual trapping
potential and the Coulomb interaction with the other ions. Since any
contribution from the Coulomb interaction to the single-ion terms of the
Hamiltonian can be compensated by the trapping potential, the matrices
$\mathcal{M}_{i}$ and vectors ${\vec{\mathcal{F}}}_{i}$ are thus taken as
freely tuneable objects, whereas $D(\vec{r})$ can not be modified at all
through changes in the trapping potential.

The ground state of any such quadratic Hamiltonian is Gaussian. It is thus
characterized in terms of the expectation value $Z=\langle X\rangle$ of the
$2{N}{d}$-dimensional vector

$X=\begin{pmatrix}\hat{\vec{x}}_{1},\hat{\vec{x}}_{2},\ldots,\hat{x}_{{N}},\hat{\vec{p}}_{1},\hat{\vec{p}}_{2},\ldots,\hat{p}_{{N}}\end{pmatrix}.$
(6)

of phase space operators, and the corresponding covariance matrix $\Sigma$
with elements

$\Sigma_{ij}=\frac{1}{2}\langle X_{i}X_{j}+X_{j}X_{i}\rangle-\langle
X_{i}\rangle\langle X_{j}\rangle\ .$ (7)

If the Hamiltonian changes in time while remaining quadratic, any initially
Gaussian state remains Gaussian, but the phase space vector $Z$ and the
covariance matrix become time-dependent. This is independent of whether the
change in the Hamiltonian is adiabatic or not, and the vector $Z$ satisfies
the classial Hamiltonian equations of motion and can be understood as the
classical phase space trajectory of the ions.

##  III A quantum invariant for interacting particles

The present invariant

${\cal I}=\frac{1}{2}(X-Z)^{T}\Gamma(X-Z)$ (8)

is a regular quadratic function of the vector $X$ of phase space operators and
the phase-space trajectory $Z$ of the ions. With the form of a general
quadratic operator, ${\cal I}$ is an invariant (i.e. satisfies Eq. (1)) for a
general quadratic Hamiltonian

$H=\frac{1}{2}X^{T}\Omega X+V\cdot X$ (9)

if and only if the equations of motion

$\displaystyle\dot{\Gamma}$ $\displaystyle=\Omega{\cal S}\Gamma-\Gamma{\cal
S}\Omega,$ (10) $\displaystyle\dot{Z}$ $\displaystyle={\cal S}V+{\cal S}\Omega
Z$ (11)

for $\Gamma$ and $Z$ with the $2{N}{d}$-dimensional symplectic matrix

${\cal S}=\begin{pmatrix}\mathbbm{O}&\mathbbm{1}\\\
-\mathbbm{1}&\mathbbm{O}\end{pmatrix}$ (12)

are satisfied.

###  III.1 Quadratic part

The specific features that make ${\cal I}$ satisfy the desired properties are
encoded in the real-symmetric matrix

$\Gamma=\real\begin{pmatrix}G^{\dagger}G\end{pmatrix}\ ,$ (13)

defined in terms of the ${d}\times{N}{d}$ dimensional matrix

$\displaystyle G$
$\displaystyle=(S\otimes\mathbbm{1})\begin{pmatrix}{Y}_{1}&\ldots&Y_{N}&m_{1}\dot{Y}_{1}&\ldots&m_{N}\dot{Y}_{N}\end{pmatrix}$
(14)
$\displaystyle=\begin{pmatrix}m_{1}\dot{Y}_{1}&\ldots&m_{N}\dot{Y}_{N}&-{Y}_{1}&\ldots&-Y_{N}\end{pmatrix},$
(15)

comprised of complex, square matrices $Y_{i}$ each referring to one individual
ion.

In order to find the equations of motion for the matrices $Y_{i}$ that result
from Eq. (10), it is helpful to notice that, in the case of interacting ions
with the Hamiltonian given in Eq. (4), the matrix $\Omega$ has the explicit
form

$\Omega=\begin{pmatrix}\Omega_{xx}&\mathbbm{O}\\\
\mathbbm{O}&\Omega_{pp}\end{pmatrix}\ ,$ (16)

with

$\Omega_{pp}=\begin{pmatrix}\frac{\mathbbm{1}}{m_{1}}&0&0&0&\ldots&0\\\
0&\frac{\mathbbm{1}}{m_{2}}&0&0&\ldots&0\\\
0&0&\frac{\mathbbm{1}}{m_{3}}&0&\ldots&0\\\ \vdots&\vdots&\vdots&&&\vdots\\\
0&0&0&\ldots&0&\frac{\mathbbm{1}}{m_{N}}\end{pmatrix},$ (17)

for the kinetic energy, and

$\Omega_{xx}=\begin{pmatrix}m_{1}\mathcal{M}_{1}&D[1,2]&D[1,3]&\ldots&D[1,{N}]\\\
D[1,2]&m_{2}\mathcal{M}_{2}&D[2,3]&\ldots&D[2,{N}]\\\
D[1,3]&D[2,3]&m_{3}\mathcal{M}_{3}&\ldots&D[3,{N}]\\\
\vdots&\vdots&\vdots&&\vdots\\\
D[1,{N}]&D[2,{N}]&D[3,{N}]&\ldots&m_{N}\mathcal{M}_{N}\end{pmatrix},$ (18)

for the potential energy, where $D[i,j]$ is a short hand notation for
$D(\vec{x}_{i}-\vec{x}_{j})$.

With this explicit form of $\Omega$, the equation of motion for $G$ resulting
from Eq. (10) reads

$Re\left(G^{\dagger}\left(\dot{G}+G{\cal S}\Omega\right)+\left(\dot{G}+G{\cal
S}\Omega\right)^{\dagger}G\right)=0\ .$ (19)

With the specific choice of $G$ given in Eq. (15), one obtains the explicit
form

$\dot{G}+G{\cal
S}\Omega=\begin{pmatrix}Q_{1}&Q_{2}&\ldots&Q_{N}&\mathbbm{O}&\ldots&\mathbbm{O}\end{pmatrix}.$
(20)

with

$Q_{i}=m_{i}\left(Y_{i}\mathcal{M}_{i}+\ddot{Y}_{i}\right)+\sum_{j\neq
i}Y_{j}D[i,j]\ .$ (21)

The equation of motion (Eq. (19)) to be solved, is satisfied if all the
$Q_{i}$ vanish, which is the case if the differential equations

$\ddot{Y}_{i}+Y_{i}\mathcal{M}_{i}=-\sum_{j\neq i}\frac{Y_{j}D[i,j]}{m_{i}}\
,$ (22)

for the complex matrices $Y_{i}$ are satisfied.

###  III.2 Construction of the trapping potential

With the parametrization of the invariant ${\cal I}$ given in Eqs.(8), (13)
and (15) the ion dynamics is defined in terms of the phase space trajectory
$Z$ and the time-dependent matrices $Y_{i}$. Any choice for those quantities
completely determines the right-hand-side of Eq. (22). The matrices
$\mathcal{M}_{i}$ that characterize the quadratic component of the trapping
potential for the individual ions are thus determined by

$\mathcal{M}_{i}=-Y_{i}^{-1}\left(\ddot{Y}_{i}+\sum_{j\neq
i}\frac{Y_{j}D[i,j]}{m_{i}}\right)\ .$ (23)

This is the desired prescription of how to chose the trapping potential, given
in terms of the matrices $\mathcal{M}_{i}$, such that the equation of motion
Eq. (1) for the invariant is satisfied. Crucially, ensuring the validity of
the equation of motion is what results in the determination of the regular
trapping potential, and indeed no modification of the interaction is
necessary.

There is, however, an aspect that requires some further care. The matrices
$\mathcal{M}_{i}$ given by Eq. (23) are not necessarily symmetric and real as
they would need to be in order to specify a harmonic trapping potential. It is
thus essential to ensure that this property will be given while choosing the
matrices $Y_{i}$. Similar to the one-dimensional case [28], the problem can be
simplified sufficiently further for symmetric trapping potentials. As
discussed in more detail in Sec. IV this will indeed result in the
construction of real and symmetric matrices $\mathcal{M}_{i}$ that correspond
to actual trapping potentials.

##  IV Inverse engineering the invariant

In the case of two ions, there is only one interaction term
$D(\vec{x}_{1}-\vec{x}_{2})$. For a symmetric trap configuration with
$\mathcal{M}_{1}=\mathcal{M}_{2}=\mathcal{M}$, the two ($i=1,2$) equations of
motion in Eq. (22) are thus of exactly the same form. One can therefore work
with one single matrix $P=\sqrt{m_{1}}Y_{1}=\sqrt{m_{2}}Y_{2}$ to define an
invariant, as was considered recently [28] in a one-dimensional context.

The equation of motion (Eq. (22)) for $P$ then reduces to

$\ddot{P}+P\left(\mathcal{M}+\frac{D}{\sqrt{m_{1}m_{2}}}\right)=0\ ,$ (24)

where the argument $[i,j]$ of the interaction matrix $D$ is dropped. This is
exactly of the form of the special case of Eq. (22)) for non-interacting ions
with the terms $D(\vec{x}_{i}-\vec{x}_{j})$ vanishing.

The problem is thus reduced to the case of a single trapped ion with the only
difference that the dynamics depends on the modified quadratic component

$\tilde{\mathcal{M}}=\mathcal{M}+\frac{D}{\sqrt{m_{1}m_{2}}}$ (25)

of the potential, instead of the component $\mathcal{M}$ as given in the
system Hamiltonian in Eq. (4).

It is thus sufficient to resort to existing solutions [23] for single ions
that ensure that $\tilde{\mathcal{M}}$ and thus also $\mathcal{M}$ is real
symmetric.

In this context it is helpful to express $P$ in a polar decomposition $P=UR$
with a unitary $U$ and a positive semi-definite matrix $R$ that satisfies the
initial condition $\dot{R}(0)=0$. The matrix $\Gamma$ of the invariant ${\cal
I}$ can then be determined by the choice of $R(t)$ instead of the choice of
$P(t)$, and the quadratic component $\mathcal{M}$ of the trapping potential
resulting from a choice of $R(t)$ is determined by the relation

$\\{R^{2},\tilde{\mathcal{M}}\\}=2[\dot{R},R]_{A}-2RA^{2}R-\\{\ddot{R},R\\}\
,$ (26)

with the generalized commutator
$[\alpha,\beta]_{\gamma}=\alpha\gamma\beta-\beta\gamma\alpha$, and the
matrices $A$ and ${\cal J}$ satisfying

$A=iR^{-2}+\frac{1}{2}[R^{-1},\dot{R}]+\frac{1}{2}R^{-1}{{\cal J}}R^{-1}\ ,$
(27)

and

$\\{{{\cal J}},R^{-2}\\}=[\dot{R},R^{-1}]+[R,R^{-2}]_{\dot{R}}\ .\\\ $ (28)

The unitary $U$ in the polar decomposition is specified to satisfy
$A=U^{\dagger}\dot{U}$. Even though the matrix $\tilde{\mathcal{M}}$ enters
Eq. (26) in terms of an anti-commutator with $R^{2}$, Eqs. (26) to (28)
determine $\mathcal{M}$ uniquely, since the anti-commutator as linear map is
indeed invertible. The resulting matrix $\mathcal{M}$ is provably real and
symmetric [23].

In order to ensure that the time-dependent Hamiltonian realises ground state
to ground state transfer, it is important that ${\cal I}$ and $H$ commute at
the initial time $t=0$ and the final time $t=T$ of the protocol. Requiring
this commutativity is equivalent to requiring that the time-derivative
$\dot{\cal I}$ vanishes because of the equation of motion $\dot{\cal
I}=i[{\cal I},H]$ of quantum invariants (Eq.(1)).

Since the invariant ${\cal I}$ is parametrized in terms of the phase-space
trajectory $Z(t)$ of the ions and the positive semi-definite matrix $R(t)$,
this implies that the time derivatives $\dot{Z}$ and $\dot{R}$ need to vanish
at $t=0$ and $t=T$.

(a) $T=3\omega_{t}^{-1}$

(b) $T=5\omega_{t}^{-1}$

(c) $T=10\omega_{t}^{-1}$

Figure 1: Separation of two ions into a T-junction. Inset (a) depicts a fast,
ostensibly diabatic protocol, whereas inset (c) depicts a close-to-adiabatic
protocol; inset (b) depicts an intermediate protocol. The dynamics of the trap
centers are depicted by blue lines, and the dynamics of the ions are depicted
by purple lines. The dynamics of trap centers and ions deviate increasingly
with increasing diabaticity, but the shuttling protocols ensure that the ion
trajectories end up at the classical equilibrium positions. Due to the finite
Coulomb interaction, the initial equilibrium positions of the ions does not
coincide with the trap center as highlighted in the insets.

The condition that $\dot{Z}$ vanishes asserts that only phase space
trajectories may be chosen such that the velocities of the ions and the
classical forces vanish, which is exactly the classical boundary conditions.
The condition that $\dot{R}$ vanishes requires a bit more care. Since $\Gamma$
is parameterized in terms of $R$ and $\dot{R}$, the time-derivative
$\dot{\Gamma}$ is a function of $R$ and its first two derivatives. It is thus
natural to require that $\dot{R}$ and $\ddot{R}$ vanish at $t=0$ and $t=T$. At
instances at which $\dot{R}$ and $\ddot{R}$ vanish, Eqs. (26), (27) and (28)
reduce to

$\\{R^{2},\tilde{\mathcal{M}}\\}=-2RA^{2}R\ ,\ A=iR^{-2}\ \mbox{and}\ {{\cal
J}}=0\ .$ (29)

Vanishing derivatives of $R$ thus result in the boundary condition

$R(t)=\left(\mathcal{M}(t)+\frac{D}{\sqrt{m_{1}m_{2}}}\right)^{-\frac{1}{4}}$
(30)

to be satisfied for $t=0$ and $t=T$.

##  V Ion separation

The capability of the present invariant for the construction of shuttling
protocols can be demonstrated with the example of separating two ions in one
spatial direction while they are moving along a second spatial direction. This
example will also demonstrate that despite the degenerate spectrum of the
invariant, the accuracy of state transfer is substantially higher than that
required to improve shuttling experimentally.

Initially, the ions are trapped in the same potential well centered at $[0,0]$
and separated only by their Coulomb repulsion. At the final time $T$, the ions
are trapped in two distinct potential wells centered at $[x,y]$ and $[-x,y]$
with a mutual separation $d=2x$.

The confinement

$M(0)=\begin{pmatrix}\omega_{t}^{2}&0\\\ 0&\omega_{r}^{2}\end{pmatrix}\ ,\ \
M(T)=\begin{pmatrix}\omega_{r}^{2}&0\\\ 0&\omega_{t}^{2}\end{pmatrix}\ ,$ (31)

with $\omega_{r}\gg\omega_{t}$ of the initial and final trapping potential is
chosen such that the direction of strong confinement rotates during the
separation process, as would be the case in a T-junction [29].

The trapping potential together with the Coulomb interaction determines the
initial and final positions $\vec{x}_{1}$ and $\vec{x}_{2}$ of the two ions.
The actual trajectories can be chosen freely as long the the required boundary
conditions are satisfied. One may thus take the trajectories

$\vec{x}_{i}(\tau)=(1-p(\tau))\vec{x}_{i}(0)+p(\tau)\vec{x}(T)\ ,$ (32)

with

$p(\tau)=10\tau^{3}-15\tau^{4}+6\tau^{5}\ ,\mbox{ and }\ \tau=t/T\ .$ (33)

Figure 2:  Two ions, initially trapped together in a T-junction, are separated
into two distinct trapping locations at an angle of 90 degrees, over a
duration $T=3\omega_{t}^{-1}$. The trap centre trajectories, plotted here in
purple and orange, have superimposed on them level sets of the trapping
frequencies in light blue and dark blue. The ions themselves travel on
straight line trajectories shown in green and purple.

The time-evolving confinement $M(t)$ is determined in terms of the positive
semi-definite matrix $R(t)$ via Eq. (26). The boundary conditions

$R(0)=\left(\mathcal{M}(0)+\frac{D}{m}\right)^{-\frac{1}{4}}$ (34)

and

$\displaystyle R(T)$ $\displaystyle=$
$\displaystyle\left(\mathcal{M}(T)+\frac{D}{m}\right)^{-\frac{1}{4}}$ (35)
$\displaystyle\simeq$
$\displaystyle\begin{pmatrix}\omega_{r}^{-\frac{1}{2}}&0\\\
0&\omega_{t}^{-\frac{1}{2}}\end{pmatrix}$ (36)

together with the choice $R(\tau)=(1-p(\tau))R(0)+p(\tau)R(T)$ are consistent
with Eq. (31).

The following numerical examples are based on the explicit choices
$\omega_{r}=10\omega_{t}$, $\omega_{t}=2\pi\times 1$\text{\,}\mathrm{MHz}$$,
and where $m$ is chosen to be the mass of a ytterbium-171 ion. The ions are
separated to a distance $d=200$\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, at
which point the effect of the Coulomb interaction becomes negligible and the
ions are effectively non-interacting.

Fig. 1 depicts three examples of shuttling protocols of different duration,
with a fast ($T=3\omega_{t}^{-1}$) protocol in inset $a)$, a protocol of
intermediate duration ($T=5\omega_{t}^{-1}$) in inset $b)$ and a close-to-
adiabatic protocol ($T=10\omega_{t}^{-1}$) in inset $c)$.

The trajectories of the centers of the trapping potentials are depicted by
blue lines, and the trajectories of the ions are depicted by purple lines.
Each inset depicts a zoom into the initial part of the trajectories. In this
part, the positions of the ions do not coincide with the centers of the
trapping potential because of their mutual Coulomb interaction. At the final
positions, however, the Coulomb interaction is negligibly small, and there is
no discernible difference between center of trapping potential and ion
position. With increasing adiabaticity, the trajectories of the ions get
closer to the trajectories of the trapping potential, and, in particular, in
inset $a)$ the ions get substantially displaced from their equilibrium
positions.

Fig. 2 depicts the dynamics of the confinement of the trapping potential in
terms of equipotential lines at different positions in the plane. The initial
and final potentials are rotated with respect to each other by $90\degree$ as
specified by the boundary condtions in Eq. (31). In addition to the rotation
of the trapping potential, the potential also becomes isotropic during the
dynamics before reaching the final an-isotropic potential specified by the
boundary conditions. This deformation of the confining potential results in
the desired dynamics of the covariances of the ions shown in Fig. 4 that
ensures that the ions end up close to their motional ground state.

Figure 3: Fidelity of the ion separation protocol as a function of the
protocol duration $T$. For adiabatic dynamics ($T\to\infty$), the adiabatic
theorem guarantees a perfect fidelity, but for fast protocols there is no such
guarantee. The present protocol realise high fidelities even for fast
protocols, far outside the range of adiabatic dynamics.

Since the matrix $G$ defined in Eq. (15) is rectangular, the matrix $\Gamma$
in Eq. (13) has a non-vanishing null-space, resulting in a degenerate spectrum
of the invariant ${\cal I}$ (Eq. (8)). As mentioned above, the rigorous
footing for the working of the ground-state-to-ground-state is thus not given,
but as shown in Fig. 3 the trapping potentials derived from this invariant
result in transfers with very high fidelities. Fig. 3 depicts the state-
fidelity of the above protocol as function of the duration $T$. In the limit
of slow protocols ($T\to\infty$) the success of the shuttling protocol is
guaranteed due to adiabatic dynamics, and indeed the fidelities approach the
ideal value of unit fidelity with increasing duration $T$. Even for fast
protocols, however, the fidelities are very high and clearly exceed a value of
$96\%$. Since the degeneracy is attributed to the quadratic part of the
invariant, it is guaranteed that the ions trajectories satisfy the desired
properties; the expected positions and momenta of the ion do thus match the
desired values at the end of the trapping potential, but the covariances of
the ions do not necessarily end up in exactly their desired values.

(a) $t=0$

(b) $t=\frac{1}{2}T$

(c) $t=T$

(d) Ground state at $t=T$

Figure 4:  Covariance matrices at different instances of the fast shuttling
protocol of duration $T=3\omega_{t}^{-1}$. The individual matrix elements are
displayed in color-scale in harmonic oscillator units: position-position
correlations have the unit $m\hbar\omega_{t}$, position-position correlations
have the unit $\hbar/(m\omega_{t})$, and position-momentum correlations have
the unit $\hbar$. The starting state of the system depicted in (a) contains
some position-position and momentum-momentum correlations between the ions
resulting from the sizeable Coulomb interaction. In particular the position-
momentum correlations undergo some transient dynamics (inset b) which are
typical of states far from the ground state of a harmonic potential. Most of
these correlations are negligibly small at the end of the protocol (inset c),
but the correlation of the $p_{x}$-momenta between the two ions remains
finite, whereas it vanishes in the correlation matrix of the ground state of
the final trapping potential. The complete time-dependence of the
$p_{x_{1}}$-$p_{x_{1}}$ correlation and the
$p_{x_{1}}$-$p_{x_{2}}$-correlation is depicted in Fig. 5.

Fig. 4 depicts instances of the covariance matrix for the fast shuttling
protocol of duration $T=3\omega_{t}^{-1}$ depicted in inset $a)$ of Fig. 1.
The initial covariance ($t=0$) is shown in inset $a)$; the covariance matrix
in the middle of the shuttling protocol ($t=\frac{T}{2}$) is shown in inset
$b)$; the final ($t=T$) covariance matrix and the desired, final covariance
matrix are shown in insets $c)$ and $d)$.

The uncertainties in position and momentum reflect the properties of initial
and final trapping potential: at $t=T$ the spatial uncertainty in the $x$
direction is larger than in the $y$ direction, whereas the opposite is the
case at $t=T$, and a similar pattern exists for the uncertainty in the
momenta. The initial covariance matrix (inset $a$) has a sizeable
$x_{1}$-$x_{2}$-covariance and $p_{x_{1}}$-$p_{x_{2}}$-covariance as a result
of the strong Coulomb interaction between the ions. In the ideal final state
(inset $d$) all inter-ion covariances are negligibly small. In the actual
final covariance matrix (inset $c)$), also the $x_{1}$-$x_{2}$ correlation has
become negligible, but the magnitude of the $p_{x_{1}}$-$p_{x_{2}}$ covariance
has increased as compared to its initial value. The covariance matrix at
$t=\frac{T}{2}$ (inset $b$) has several finite elements that are vanishing or
negligibly small at both $t=0$ and $t=T$, which highlights the non-monotonic
dynamics of the covariances.

(a) $\Sigma_{p_{x_{1}}p_{x_{1}}}$

(b) $\Sigma_{p_{x_{1}}p_{x_{2}}}$

Figure 5:  Time-dependence of the $p_{x_{1}}$-$p_{x_{1}}$ correlation (inset
a) and the $p_{x_{1}}$-$p_{x_{2}}$ correlation (inset b) for the fast
shuttling protocol of duration $T=3\omega_{t}^{-1}$. In particular due to the
rotation between initial and final trapping potential the initial and final
value of the $p_{x_{1}}$-$p_{x_{1}}$ correlation are different. The shuttling
protocol realises a non-monotonic dynamics that ensures that the final value
of this correlation matches the desired value. The desired value of the
$p_{x_{1}}$-$p_{x_{2}}$ correlation is negligibly small because of the weak
interaction between the separated ions. The shuttling protocol ensures that
the actual final value of the $p_{x_{1}}$-$p_{x_{2}}$ correlation is smaller
than the values adopted during the dynamics, but due to the invariant’s
degeneracy there is a small mismatch between the actual and the desired value.

Fig. 5 depicts the actual time-dependence of two elements of the covariance
matrix. Inset $a)$ depicts the time-dependence of the $p_{x_{1}}$-$p_{x_{1}}$
correlation. This correlation is non-vanishing at both the start and the end
of the protocol. Over time, it develops some oscillations, and eventually
settles on a much higher value than it had at the start of the protocol.
Nevertheless, some deviation with the expected value is observed at the end of
the protocol. Inset $b)$ depicts the time-dependence of the
$p_{x_{1}}$-$p_{x_{2}}$-covariance. This value is expected to become
negligible at the end of the separation procedure, as may be seen by
inspecting Fig. 4. Physically, this is due to the fact at large ion
separations the Coulomb potential is negligible and one may expect there to be
no correlations between the two ions in their ground state. Nevertheless, as
one can see from inset $b)$, some oscillations build up over time and at the
end of the protocol, the covariance does not attain a value close to 0 as it
should.

The deviations of the covariances from their desired values implies that the
ions do not end up exactly in their motional ground state, but that there are
some motional excitations. The dominant contributions are occupations of the
states $\ket{{N}_{x_{1}}{N}_{y_{1}}{N}_{x_{2}}{N}_{y_{1}}}=\ket{1010}$
($1.57\%$), $\ket{2000}$ ($0.79\%$) and $\ket{0020}$ ($0.79\%$), with all
occupation numbers of higher states becoming negligible. These excitations are
substantially below what is required for the realization of quantum gates [30,
31].

##  VI Conclusion

With the rapid advances in controlling trapped ions and current activities
towards scaling up hardware for quantum information processing, shuttling of
trapped ions requires practical control techniques. The invariant developed
here offers an efficient approach for the construction of ground-state-to-
ground-state shuttling protocols that work far outside the validity of the
adiabatic approximation, and the ability to go beyond the one-dimensional
approximation enables control also in challenging trap geometries as
encountered for example around junctions [32, 33].

Similar to experiments on control of motional quantum states, extensions to
the theoretical framework of suitable invariants are also a great challenge.
Dynamical mode expansions [34, 16, 18] offer an alternative to the wave-packet
dynamics used in this paper; they offer the prospect to control mixed ion
species [34] and have proven helpful for separation of trapped ions [16]. Even
though symmetries – as assumed here for the trap geometry – can be very
helpful for explicit solutions to control problems, a framework to support
laboratory experiments also needs the capability to address problems without
simplifying symmetries. Finding invariants that satisfy all these wishes has
proven very difficult, but the rapid advances in artificial intelligence over
the last years gives hope for accelerated progress in the development of
practically useful quantum invariants.

##  VII Acknowledgements

We are indebted to stimulating discussions with Adam Callison, Modesto Orozco
Ruiz, Sam Hile, David Bretaud, Alex Owens, Pedro Taylor-Burdett, Sebastian
Weidt and Winnie Hensinger. This work was supported through a studentship in
the Centre for Doctoral Training in Controlled Quantum Dynamics at Imperial
College London funded by EPSRC(EP/L016524/1).

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