# Generally noise-resilient quantum gates for trapped-ions

Modesto Orozco-Ruiz  Blackett Laboratory, Imperial College London, SW7 2AZ,
United Kingdom  Wasim Rehman  Department of Mathematics, Imperial College
London, London, SW7 2AZ, United Kingdom  Florian Mintert  Blackett Laboratory,
Imperial College London, SW7 2AZ, United Kingdom  Helmholtz-Zentrum Dresden-
Rossendorf, Bautzner Landstraße 400, 01328 Dresden, Germany

###### Abstract

We present an entangling gate scheme for trapped-ion chains that achieves
high-fidelity operations with excited motional states despite multiple error
sources. Our approach incorporates all relevant motional modes and exhibits
enhanced robustness against both motional heating effects and detuning errors,
critical features for building robust and scalable trapped-ion quantum
computers.

##  I Introduction

Trapped-ion quantum computing holds promise for building scalable and fault-
tolerant quantum computers due to the exceptional control that can be exerted
over individual qubits and their collective interactions [1]. This exquisite
control enables the implementation of the two-qubit entangling gate – a
fundamental building block for quantum algorithms. Relying on the Coulomb
interaction between ions, this gate utilizes the shared motion of the ions to
generate entanglement in their electronic states [2].

The Mølmer-Sørensen gate [3] has emerged as one of the most promising
implementations in trapped-ion quantum computing, with fidelities exceeding
$99\%$ in both optical and hyperfine qubits [4, 5, 6, 7]. Exploiting precisely
tuned bichromatic laser fields that couple internal and motional states, the
Mølmer-Sørensen gate proves to be insensitive to the initial motional state,
offering substantial advantages for scalability and error correction [8, 9,
10, 11, 12].

However, despite robustness to initial motional state, current entangling
gates remain vulnerable to various experimental imperfections, due to their
inherent coupling with the environment [8, 9, 3], as well as to systematic
errors resulting from imperfect calibration of system parameters. The complex
interplay of error mechanisms in entangling gates presents a significant
obstacle to the development of a robust scheme against multiple errors. While
some approaches have demonstrably enhanced resilience against specific noise
sources, optimizing resilience against one mechanism typically compromises
resilience against another. For example, increasing laser power for faster
gates can alleviate detuning errors but exacerbate off-resonant coupling to
undesired carrier transitions [13]. Similarly, suppressing motional mode
cross-talk in many-ion systems by slowing down the gate operation can lead to
reduced coherence time and imperfect coupling [14, 15]. These conflicting
effects highlight the need for control protocols capable of simultaneously
addressing diverse error sources.

Motional heating is arguably a primary source of errors in current
implementations, with electric and magnetic field noise leading to unwanted
phonon exchange that disrupts the motional states of the trapped ions [13].
Similarly, frequency fluctuations of the vibrational modes induce dephasing,
ultimately compromising the gate fidelity. To address these challenges,
several schemes have been proposed to enhance the motional robustness of
entangling gates by engineering optimal ion trajectories in phase space. One
approach involves modulating sideband field parameters (phase, frequency or
amplitude [16, 17]) to create phase-space trajectories with minimal
susceptibility to heating. Alternatively, a multi-tone driving strategy
employs a combination of detuned fields with varying amplitudes for each ion
sideband, achieving comparable robustness gains [18]. However, despite their
demonstrated success [11], these techniques are inherently limited by the
constraints of the Lamb-Dicke regime within which entangling gates operate
[18]. This regime, characterized by weak laser-ion interactions, requires low
laser power for high-fidelity operations, leading to slower gate times.
Additionally, shared motional states must be cooled close to their ground
state, posing a scalability bottleneck for proposals involving active ion
shuttling [19, 20], as such movements inherently excite these states.
Deviations from these conditions, such as using stronger laser fields or
excited motional states, result in compromised fidelities.

Logic gates beyond the Lamb-Dicke regime offer significant improvements,
including high fidelities with high motional states and significantly faster
quantum operations. One approach employs a series of complex pulses inducing
state-dependent kicks exceeding the motional frequencies [21]. While this
method demonstrably enhances speed, current fidelities remain moderate due to
experimental complexity and unwanted coupling to additional motional modes.

An alternative strategy employs simpler driving fields and higher-order
sideband transitions that are designed to cancel undesired contributions from
the non-linear spin-motion coupling beyond the Lamb-Dicke regime [18]. This
approach holds promise for achieving high fidelities in the strong-coupling
regime, but relying on driving more sideband transtions is more vulnerable to
motional heating.

Another significant challenge faced by current entangling gates involves
scaling to larger ion chains with a complex multi-mode structure [22, 23, 24].
This not only amplifies noise sensitivity due to increased crosstalk between
ions [15] but also significantly complicates control protocols, specially far
from the Lamb-Dicke regime. This occurs because the coupling strength
corresponding to each motional mode becomes comparable in magnitude and it
varies across different ions in the chain. Selectively driving a single mode
in such systems inevitably triggers higher-order phonon exchange processes,
unintentionally coupling non-targeted spectator modes to the qubit states.
This presents a significant challenge in achieving high-fidelity gates,
requiring control strategies distinct from those employed in paradigms with
fewer modes, where precise control over individual modes is sufficient.

This manuscript introduces a gate scheme that integrates multiple strategies
to combat diverse error sources in trapped-ion quantum computing. Our control
framework achieves high-fidelity entanglement in hot trapped-ions within a
multi-mode system, demonstrating inherent resilience against motional heating
and increased robustness against fluctuations in both qubit frequency and
normal-mode frequencies.

##  II Hamiltonian model

The dynamics of a system comprised of the internal degrees of freedom of $N$
ions and $M$ collective motional degrees of freedom is induced by the
Hamiltonian

$\displaystyle
H_{0}=\sum_{j=1}^{N}\frac{\omega_{j}}{2}\sigma_{z}^{(j)}+\sum_{l=1}^{M}\nu_{l}a_{l}^{\dagger}a_{l}\
,$ (1)

where $\sigma_{z}^{(j)}$ is the Pauli-Z operator for ion $j$ with resonance
frequency $\omega_{j}$, and $a_{l}^{\dagger}(a_{l})$ are the creation
(annihilation) operators for the collective motional degree of freedom $l$
with resonant frequency $\nu_{l}=\kappa_{l}\nu$ in terms of the trapping
frequency $\nu$.

The interaction with an external light field, such as a laser field, can
excite both internal and motional degrees of freedom of the ions. The
respective interaction Hamiltonian reads

$\displaystyle H(t)$
$\displaystyle=\sum_{j=1}^{N}f_{j}(t)\sigma_{+}^{(j)}\prod_{l=1}^{M}e^{i\eta_{jl}\left(a_{l}+a_{l}^{\dagger}\right)}+h.c.\
,$ (2)

in terms of the raising operator $\sigma_{+}^{(j)}$ of the internal degree of
freedom of ion $j$, and the driving functions $f_{j}(t)$ that encapsulate the
generally time-dependent Rabi-frequency and a time-dependence due to carrier
frequencies of the light field. The Lamb-Dicke parameters
$\eta_{jl}=\chi_{jl}\Lambda$ characterize the coupling between the state
transition in ion $j$ and each motional state transition in mode $l$ due to
light field interaction. They depend on the expansion coefficients $\chi_{jl}$
and the coupling factor
$\Lambda=\frac{2\pi}{\lambda}\sqrt{\frac{\hbar}{2m\nu}}$ defined in terms of
the wavelength of the light field $\lambda$ and the mass $m$.

Since entangling gates are realized via the exchange of virtual phonons in the
collective motion of the ions, it is instructive to express the interaction
Hamiltonian as

$\displaystyle H(t)$
$\displaystyle=\sum_{j=1}^{N}\tilde{f}_{j}(t)\sigma_{+}^{(j)}\prod_{l=1}^{M}\sum_{k=-\infty}^{\infty}\mathcal{D}_{l,k}(\eta_{jl})+h.c.\
,$ (3)

in terms of the operators

$\displaystyle\mathcal{D}_{l,k}(\eta)=\sum_{n=0}^{\infty}\left(i\eta\right)^{2n+k}\frac{a_{l}^{\dagger
n+k}a_{l}^{n}}{(n+k)!n!}\ ,$ (4)

and

$\displaystyle\mathcal{D}_{l,-k}(\eta)=(-1)^{k}\mathcal{D}_{l,k}^{\dagger}(\eta)\
,$ (5)

for $k\geq 0$, that encapsulate all processes resulting in the creation (of
$k>0$) and annihilation ($k<0$) of $k$ phonons in a given mode, and a
renormalised driving function

$\displaystyle\tilde{f}_{j}(t)=f_{j}(t)\
e^{-\frac{1}{2}\sum_{l=1}^{M}\eta_{jl}^{2}}\ .$ (6)

The operators $\mathcal{D}_{l,k}$ capture the $k$-th order sideband
transitions of the motional mode $l$. A generic driving pattern
$\tilde{f}_{j}(t)$ can have spectral components that are close to resonance
with each of these (red and blue) sideband transitions. In standard rotating
wave approximation, each spectral component needs to be considered in the
Hamiltonian only in close-to-resonant terms, but it can be neglected in any
far-off-resonant term. A general parametrization of driving patterns is thus
given by

$\displaystyle\tilde{f}_{j}=-i\sum_{l}\frac{1}{\eta_{jl}}\sum_{k}F_{l,k}^{(j)}(t)\
e^{-i(k\nu_{l}+\omega_{j})t}\ ,$ (7)

where the factors $e^{-i(k\nu_{l}+\omega_{j})t}$ capture the time-dependence
necessary for resonance with a given sideband transition, as demonstrated by
expressing $H(t)$ (Eq. (3)) in the interaction picture with respect the free
Hamiltonian $H_{0}(t)$ (Eq. (1)). The factors $F_{l,k}^{(j)}(t)$ have a slow
time-dependence compared to $\nu_{l}$ and account for finite detuning and
temporal modulation that can be used to achieve the desired robustness.
Finally, the factor $\eta_{jl}^{-1}$ reflects the fact that weaker interaction
(smaller $\eta_{jl}$) requires proportionally stronger driving fields to
maintain constant interaction strength.

A coherent gate requires that there are no changes to the motional state that
depend on the internal state of the ions, and achieving this is facilitated in
terms of driving patterns in which the driving amplitude $F_{l,-k}^{(j)}$ for
any blue sideband transition ($k>0$) is given by the driving amplitude
$F_{l,k}^{(j)}$ for the corresponding red sideband transition following the
relation

$\displaystyle F_{l,-k}^{(j)}=(-1)^{k}\left(F_{l,k}^{(j)}\right)^{\ast}\ ,$
(8)

where the factor $(-1)^{k}$ has its origin in Eq. (5).

With any such driving pattern, the interaction Hamiltonian reduces to

$\displaystyle
H(t)=\sum_{j=1}^{N}\sigma_{y}^{(j)}\sum_{l,k>0}\frac{F_{l,k}^{(j)}(t)}{\eta_{jl}}\
\mathcal{D}_{l,k}\prod_{l^{\prime}\neq l}\mathcal{D}_{l^{\prime},0}+h.c.\ ,$
(9)

with $\sigma_{y}=i(\sigma_{-}-\sigma_{+})$, and with factors

$\displaystyle\mathcal{D}_{l,0}=\sum_{n=0}^{\infty}\frac{\left(i\eta\right)^{2n}}{n!^{2}}a_{l}^{\dagger
n}a_{l}^{n}$ (10)

that describe phonon-conserving, but phonon-number-dependent energy shifts.

##  III Driving patterns

A light field modulated as described in Eq. (7) allows the engineering of a
time-dependent Hamiltonian (Eq. (9)) with desired dynamics. By carefully
choosing the driving functions $F_{l,k}^{(j)}(t)$, the interaction with ions
can be tailored to induce specific processes necessary for entangling state
generation.

Crucially, not all motional modes of the ion chain are required for
entanglement generation. Only a subset of these modes, along with a finite
number of their sideband transitions, play a crucial role. The Mølmer-Sørensen
(MS) entangling gate [3, 25] exemplifies this principle, where entanglement is
achieved by driving solely the first sideband of a chosen set of motional
modes.

This specific solution to Eq. (9) requires the weak coupling regime or
motional states near their ground states as this allows approximating the
displacement operators for the driven modes $d$ in Eq. (9) to the lowest order
in $\eta$ as $\mathcal{D}_{d,1}\approx i\eta_{d}a_{d}^{\dagger}$, while those
for the spectator modes reduce to the identity
$\mathcal{D}_{l^{\prime},0}\approx 1\\!\\!1$. This first-order simplification
implies that spectator modes do not couple with the driven modes (see Eq. (9))
and can therefore be safely neglected, thus resulting in the simplified
Hamiltonian
$H_{MS}(t)=iS_{y}\sum_{d}\left(F_{d}(t)a_{d}^{\dagger}-F_{d}^{*}(t)a_{d}\right)$,
with the collective spin operator $S_{y}=\sum_{j}\sigma_{y}^{(j)}$. With this
simplified Hamiltonian, a carefully chosen set of driving functions $F_{d}(t)$
enables the implementation of the entangling gate $U=e^{i\phi_{T}\sum_{j\neq
k}\sigma_{y}^{(j)}\sigma_{y}^{(k)}}$ acting on all ion pairs.

###  III.1 Non-identical ion species and multi-mode systems

Although the Hamiltonian $H_{MS}$ offers a straightforward approach for
creating entangled states, it comes with certain limitations. While a global
driving field ($f_{j}(t)=F(t)$ for all ions $j$) works for identical ion
species, two key challenges arise when dealing with non-identical ions.
Firstly, their differing resonance frequencies $\omega_{j}$ lead to the
acquisition of distinct phase factors $e^{i\omega_{j}t}$ during their
dynamics. Secondly, non-identical ions couple to the mediating modes with
varying strengths $|\eta_{j,l}|$. This dependence on the specific ion $j$
results in different scaling factors $e^{\frac{1}{2}\eta_{j,l}^{2}}$ arising
in the expansion of the interaction Hamiltonian in Eq. (2).

These challenges related to non-identical ions can be addressed using a more
sophisticated individual addressing approach instead of a single, global
field. However, a more fundamental limitation lies with the Lamb-Dicke
approximation used to obtain the simplified first-order Hamiltonian $H_{MS}$.
This approximation restricts the applicability of $H_{MS}$ to systems
operating within the weak coupling regime or with motional states close to
their ground states, as dictated by the inequality
$\frac{\eta_{l}^{2}}{2}n_{l}\ll 1$ [25, 18]. Systems that do not satisfy this
constraint necessitate alternative approaches to obtain high-fidelity
operations.

###  III.2 Beyond the Lamb-Dicke approximation

Moving beyond the Lamb-Dicke approximation, either due to stronger ion-motion
interactions or higher motional states, necessitates considering higher-order
phonon-exchange processes within the displacement operators Eq. (4). This
leads to more complex spin-motion interactions for driven modes
($\mathcal{D}_{l,k}\approx\frac{(i\eta)^{k}}{k!}a_{l}^{\dagger
k}+\frac{(i\eta)^{k+2}}{(k+1)!}a_{l}^{\dagger(k+1)}a_{l}+\cdots$) and,
crucially, indirect coupling between driven and spectator modes via the non-
trivial products $\mathcal{D}_{l,k}\prod_{l^{\prime}\neq
l}\mathcal{(}1\\!\\!1-\eta_{l^{\prime}}^{2}a_{l^{\prime}}^{\dagger}a_{l^{\prime}}+\cdots)$
in Eq. (9).

Consequently, control schemes beyond the Lamb-Dicke regime must account not
only for higher-order phonon processes in the driven modes but also for the
phonon exchange processes in the remaining non-addressed collective
vibrational modes. To illustrate this phenomenon, consider a two-ion system
where driving the first sideband of the center-of-mass mode induces phonon
exchanges like $(\eta_{1}^{3}/2)a_{1}^{\dagger 2}a_{1}$ beyond the first-order
term $\eta_{1}a_{1}$. Similarly, even though the stretch mode is not directly
addressed, it couples through this driving and contributes to dynamics with
processes like $\eta_{2}^{2}a_{2}^{\dagger}a_{2}$.

Therefore, capturing these inter-mode dependencies and designing high-fidelity
entangling gates for many-mode ion chains beyond the Lamb-Dicke regime
necessitates a higher-order expansion of the interaction Hamiltonian $H(t)$.
This expansion reveals exchange processes neglected in the standard treatment,
requiring the development of a new control strategy that eliminates their
unwanted contributions at the gate time.

Following this crucial first step, designing a robust two-qubit entangling
gate involves selecting a suitable set of driven motional modes $L$ and the
specific sidebands for each of those modes $K(l)$. These choices determine
which processes are driven resonantly. The temporal modulation of the driving
functions, $F_{l,k}^{(j)}$ is then tailored to achieve entanglement at the
desired gate time $T$ while actively suppressing the unwanted processes and
maximizing the gate’s overall robustness.

###  III.3 Driven sidebands

The specific construction presented here enables the implementation of an
entangling gate in a multi-mode system of $N$ trapped ions with high motional
states and offers enhanced resilience against both motional heating and
combined errors in qubit and vibrational frequencies. While alternative
choices exist, the proposed configuration minimizes the number of driven
sidebands required for achieving the desired robustness, offering a practical
advantage for experimental implementation.

In a system with $M$ motional modes, the proposed gate implementation involves
simultaneous driving of all modes ($L=M$). The first-order sideband ($k=1$) of
a designated central mode, typically chosen for its stronger spin-motion
coupling, acts as the primary bus mode of the entangling process. Meanwhile,
other modes (together with the central one) contribute through second-order
sidebands ($k=2$) to actively suppress unwanted contributions arising from
higher-order terms in the Hamiltonian.

With such choice, the Hamiltonian Eq. (9) simplifies to

$\displaystyle H_{c}(t)$
$\displaystyle=\sum_{j=1}^{N}\sigma_{y}^{(j)}\left(\frac{F_{1,1}^{(j)}}{\eta_{j1}}\mathcal{D}_{1,1}D_{1}+\sum_{l=1}^{M}\frac{F_{l,2}^{(j)}}{\eta_{jl}}\mathcal{D}_{l,2}D_{l}\right)+h.c.,$
(11)

with $D_{k}=\prod_{l^{\prime}\neq k}\mathcal{D}_{l^{\prime},0}$ being the
product of phonon-conserving operators in the spectator modes.

Building upon Eq. (11), one can derive time-dependent functions
$F_{1,1}^{(j)}$ and $F_{l,2}^{(j)}$ specifically to achieve an exact solution
of the driven dynamics up to a particular order of $\eta$. Although higher-
order solutions are theoretically feasible, the present solution results from
an expansion to order $\order{\eta^{3}}$, significantly enhancing first-order
expansions while minimizing the number of sidebands required for achieving the
desired robustness. With their detailed derivation available in Sec. V, these
functions are given by

$\displaystyle F_{1,1}^{(1)}$ $\displaystyle=$
$\displaystyle\,F_{1,1}^{(2)}=\Omega\left(e^{2i\delta
t}-\frac{3}{2}e^{3i\delta t}\right)$ (12a) $\displaystyle F_{l,2}^{(1)}$
$\displaystyle=$
$\displaystyle\,s_{1l}\Omega\frac{\tilde{\eta}_{l}}{\eta_{1l}}e^{i\delta t}$
(12b) $\displaystyle F_{l,2}^{(2)}$ $\displaystyle=$
$\displaystyle\,s_{1l}\Omega\frac{\tilde{\eta}_{l}}{\eta_{2l}}e^{i\delta t}$
(12c)

where $l=1,\cdots,M$ refers to the motional mode,
$s_{jl}=\mbox{sign}(\eta_{jl})$ and
$\tilde{\eta}_{l}:=\frac{\sqrt{5}}{2}\sqrt{\eta_{1l}^{2}+\eta_{2l}^{2}}$.

Figure 1: Energy level diagram for two identical ions and two motional modes.
Note that only energy levels involved in the driving process are shown for
simplicity. Solid blue (red) sidebands represent transitions with a phonon
gain (loss) in motional mode 1. Dashed blue (red) sidebands represent similar
processes for motional mode 2.

Consider a specific pair of ions selected for entanglement generation within
the $N$-ion chain. According to Eq. (12a), both ions receive identical
bichromatic modulation with detunings $2\delta$ and $3\delta$ ions targeting
the first sideband of the primary motional mode (see Fig. 1 for the case of
two identical ions). With such a modulation, resilience against motional
heating is enhanced, while the specific detuning ratio optimizes gate speed.
Eqs. (12b) and (12c), in turn, specify different drivings for each ion in the
pair targeting second order sidebands of all vibrational modes (see Fig. 1).
These suppress unwanted processes that emerge as higher-order (in $\eta$)
contributions to the system’s dynamics and include mode-dependent amplitudes
to compensate for the different coupling strengths across the different modes.
Finally, the Rabi amplitude $\Omega$ is specified in terms of the detuning
$\delta$ as a solution to the equation

$\displaystyle-6\frac{\Omega^{4}}{\delta^{4}}\left(\sum_{j}\eta_{j1}^{2}\right)+\frac{\Omega^{2}}{\delta^{2}}\left(2+\sum_{l,j}\eta_{jl}^{2}\right)=\frac{2\phi_{T}}{5\pi}\
,$ (13)

defined in terms of the entangling phase $\phi_{T}$ in the unitary gate $U$.
In the weak-coupling regime, where $\eta_{jl}\approx 0$, the higher-order
terms in the equation become negligible and the entangling condition reduces
to the standard MS case with a bichromatic driving of the form Eq. (12b),
i.e., $\Omega^{2}/\delta^{2}=\phi_{T}/5\pi$.

(a)

(b)

Figure 2: Infidelity $1-F$ of the entangling gate as a function of the initial
Fock state occupation $n$ for a 4-qubit (left panel) and 2-qubit (right panel)
systems. In the left panel, each data point represents the average infidelity
across all qubit pairs, with the shaded area showing the range of values found
across different pairs. The infidelity corresponds to the standard MS gate
(green), the MS gate applied to all four modes (orange), and the present gate
(blue). No motional heating ($\gamma=0$) is assumed and different values of
the coupling strength $\Lambda$ are shown with different markers. In the right
panel, the impact of motional heating is evaluated for a fixed spin-motion
coupling $\Lambda=0.1$. Decay rates are represented by colors ranging from
light green (low) to purple (high) and marker styles differentiate the
standard MS gate (crosses) from the proposed robust scheme (circles). In all
simulations, the amplitude of the driving fields in the present scheme is
limited to match the amplitude of the MS gate, ensuring a fair comparison.

##  IV Results

The proposed control strategy enables the implementation of high-fidelity
entangling gates for trapped-ions in noisy environments, even when the ions
share multiple collective vibrational modes and are in high motional states.
Gate performance can be quantified using the gate fidelity, which is given by

$\displaystyle
F(\rho_{M})=\frac{1}{d^{2}}\sum_{\alpha,\beta=1}^{d}\matrixelement{\alpha}{U^{\dagger}\Tr_{M}\left(\mathcal{E}(\outerproduct{\alpha}{\beta}\otimes\rho_{M})\right)U}{\beta},$
(14)

where the states $\ket{\alpha},\ket{\beta}$ span the $d$-dimensional Hilbert
space of spin states (with $d=2^{N}$), $\rho_{M}$ is the initial density
matrix of motional states, $U$ represents the target unitary and
$\mathcal{E}(\cdot)$ the actual (potentially non-unitary) channel
corresponding to the implemented gate. Evaluating the fidelity $F(\rho_{th})$
for thermal states ($\rho_{th}=\sum_{{\bf n}}P({\bf n})\outerproduct{{\bf
n}}{{\bf n}}$, with $\ket{{\bf n}}=\ket{n_{1},n_{2},\cdots n_{M}}$) is
crucial, as these states are commonly encountered in real experiments. By
exploiting the linearity of the trace, one can avoid the construction of the
density matrix and instead compute the fidelity for a sufficiently large set
of Fock states. Averaging these individual fidelities yields the thermal state
fidelity $F(\rho_{th})=\sum_{{\bf n}}P({\bf n})F(\outerproduct{{\bf n}}{{\bf
n}})$.

The following discussion delves into the dependence of the fidelity $F$ on
several key parameters relevant to an experimental implementation, including
chain size $N$, initial motional occupations $\ket{{\bf n}}$, and the impact
of non-idealities like motional heating and frequency errors in the
vibrational modes $\epsilon_{\nu}$ and spin transition frequencies
$\epsilon_{\omega}$.

Figure 3: Combined impact of vibrational and spin frequency errors on the
performance of the present entangling scheme (Robust) compared to the standard
MS gate in a two-ion system. Each panel shows the ratio of MS gate infidelity
to SC gate infidelity $I_{MS}/I_{Robust}$ for a specific initial motional
state $\ket{n}$ and decay rate $\gamma$, with fixed coupling strength
$\Lambda=0.1$. The $x$-axis represents (in log scale) errors in vibrational
frequency $\epsilon_{\nu}$; while the $y$-axis depicts symmetric errors in
spin frequencies $\epsilon_{\omega}$. Contour lines reveal actual fidelities
achievable by each method. Solid lines enclose regions where infidelities
remain below $10^{-4}$, dashed lines correspond to $10^{-3}$ and dotted lines
to $10^{-2}$. Green lines represent the MS gate, while white ones represent
the robust scheme.

###  IV.1 Multi-mode systems with no dissipation or frequency errors

Unlike simpler two-ion systems, designing entangling gates for larger, multi-
mode, systems requires accounting for variations in coupling strengths
$\eta_{jl}$ across ion pairs. This key challenge is directly addressed by the
proposed control scheme. Fig. 2(a) exemplifies this point by depicting the
infidelity $1-F$ of the present gate in a 4-ion system under ideal conditions
(no dissipation or frequency errors). The x-axis represents the initial
motional Fock state $\ket{n}^{\otimes 4}$, and the shaded area shows the range
of infidelities across all ion pairs, with marker points indicating the
average value. Colors distinguish the present gate (in blue), the standard MS
gate applied to the center-off-mass mode (in green), and the MS gate applied
to all four motional modes [26] (in orange); while different marker styles
represent varying coupling strengths $\Lambda$.

Crucially, under ideal conditions, the present gate consistently outperforms
both MS variants across all initial states and coupling strengths. This
advantage is particularly evident in weak-coupling regimes ($\Lambda=0.05$,
circles). Even with high motional states ($n=10$), it achieves infidelities
below $10^{-4}$, a remarkable two orders of magnitude lower than any MS
variant. Notably, this performance is independent of the specific ion pair
within the four-ion system, demonstrating its robustness across various
configurations, a crucial feature for practical applications.

While the current analysis is limited to Fock states with $n=10$ for a four-
ion system, numerical evidence suggests this trend persists for significantly
higher levels of motional excitation, indicating the gate’s high performance
even under substantial thermal noise. Indeed, fidelities exceeding $99\%$ are
still achievable even at mean phonon occupations as high as $\bar{n}\sim 100$,
as shown in Supp. Mat. B for a two-ion system.

###  IV.2 Motional heating

The previous results suggest that the present entangling protocol allows high-
fidelity entanglement without near-ground-state motional cooling, particularly
advantageous for systems with weak spin-phonon coupling. However, a critical
consideration is whether this benefit comes at the cost of increased
susceptibility to motional decoherence due to the additional side-band
driving. Crucially though, the temporal modulation of the driving fields, as
defined in Eq. (12), strategically mitigates heating effects, thereby enabling
robust entanglement generation even in the presence of thermal noise. This
protection, as detailed in Sec. V, stems from the bichromatic driving in
$F_{1,1}^{(1)}$ and $F_{1,1}^{(2)}$, which minimizes the impact of incoherent
processes arising from motional heating to first order in $\eta$. Further
enhanced robustness can be achieved by implementing a polychromatic version
for all drivings in Eq. (12), although at the cost of a more intricate driving
scheme.

Fig. 2(b) illustrates the performance of the proposed gate in the presence of
motional heating for a two-ion system of identical species. In contrast to the
prior scenario, the coupling strength remains uniform across all motional
modes and ions (i.e. $|\eta_{jl}|$ is independent of $j$ and $l$) and,
therefore, local drivings can be relaxed to be global. The infidelity $1-F$ is
represented as a function of the motional occupation for a fixed coupling
strength $\Lambda=0.1$ and different heating decay rates $\gamma$ (see Eq.
(19)). The crosses correspond to the performance of the MS gate applied to the
center-off-mass mode, while circles represent solutions derived from the
parametrization in Eq. (12). The results demonstrate a significant performance
advantage of the present scheme over the MS approach across all decay rates,
particularly in low-dissipative environments. This enhanced robustness
achieved with bichromatic driving enables fidelities close to $10^{-3}$ for
motional occupations of up to $n=10$ in low and medium dissipative
environments ($\gamma\leq 0.01$), compared to an infidelity close to $10^{-1}$
for the MS gate.

###  IV.3 Detuning errors

Even minimal errors in vibrational and spin frequencies can significantly
affect the gate fidelity. These errors, typically small compared to the
resonant frequencies (i.e., $\epsilon\ll\omega$), can be modeled as shifts of
the form $\omega\rightarrow\omega+\epsilon$. Specifically, errors in motional
frequencies manifest as shifts for each motional mode $l$,
$\nu_{l}\rightarrow\nu_{l}+\epsilon_{\nu_{l}}$, whereas (symmetric) spin
frequency errors are modeled as
$\omega_{j}\rightarrow\omega_{j}+\epsilon_{\omega}$, affecting the resonant
frequencies of the spin transitions. These small frequency drifts in
vibrational and spin transitions lead to a crucial mismatch between the actual
transition frequencies and the driving frequencies defined in Eq. (7),
introducing an undesired detuning error that modifies the actual dynamics and
may lead to potential performance degradation.

Crucially, the properties that enhance resilience to motional heating (mainly
closed phase-space trajectories and vanishing time-integral, as detailed in
Supp. Mat. A) are also relevant for mitigating the impact of motional detuning
errors [8]. Therefore, the proposed gate exhibits robustness against these
errors by design. On the other hand, while not explicitly designed to address
spin frequency detuning errors, the proposed gate still exhibits a relevant
degree of robustness to them, as shown in Fig. 3.

Fig. 3 depicts the combined impact of detuning errors in vibrational and spin
frequencies for a two-ion system also experiencing motional heating. Each
panel represents the ratio in infidelities between the MS gate (applied to the
center-off-mass mode) and the present robust protocol, $I_{MS}/I_{Robust}$,
using a color code ranging from purple (similar performance) to yellow
(significant improvement). The panels vary across different initial motional
states $\ket{\mathbf{n}}$ and decay rates $\gamma$; and the $x$ and $y$ axes
correspond to errors in the vibrational $\epsilon_{\nu}$ (assumed to be the
same across all modes for simplicity) and spin frequencies
$\epsilon_{\omega}$, respectively.

The first row of panels shows the infidelity ratio for the initial motional
ground state ($n=0$) with increasing decay rates from left to right. Here, the
present scheme (robust) exhibits substantial improvement over the MS gate for
small and moderate errors in motional frequencies ($\epsilon_{\nu}<0.5\%$),
even with spin frequency shifts of up to $\epsilon_{\nu}<0.1\%$. Notably,
reductions in infidelity of up to two orders of magnitude are achieved in the
case with no motional heating and small frequency errors (yellow region in
leftmost panel). While this advantage diminishes with increasing motional
heating (moving right within the first row), the present scheme maintains
superior performance across all decay rates depicted, within the studied range
of frequency errors.

Overall, when restricting motional frequency errors to a maximum of $1.5\%$,
the proposed gate achieves fidelities exceeding $99.99\%$ (as delimited by the
green contour line), which represents an improvement of at least a factor of
five, and up to two orders of magnitude, over the standard MS approach.

Moving to higher motional states ($n=5$), the present scheme demonstrates a
more significant advantage. The region with very substantial improvement (in
red and yellow tones) expands to encompass larger errors across both axes.
This improvement also decays more gradually with higher decay rates compared
to the case with motional ground states (first row). Importantly, the MS gate
fails to achieve fidelities exceeding $99\%$ for any frequency error within
the studied range. In contrast, the robust scheme maintains high fidelities
exceeding $99\%$ within a broad range of errors (up to $\epsilon_{\nu}<3\%$
and $\epsilon_{\omega}<2\%$) and even achieves exceptional fidelities
exceeding $99.9\%$ with frequency errors of any type of up to $0.7\%$. This
advantage persists even for high motional occupations ($n\sim 100$), where the
robust scheme still maintains fidelities exceeding $99\%$ with moderate
frequency errors.

##  V Methods

This section details the derivation of the driving protocol in Eq. (12) that
implements a robust entangling gate operation. The approach involves first
designing the general driving pattern for the desired operation and then
refining it to enhance robustness against noise and imperfections.

###  V.1 Entangling gate

While Eq. (9) describes the ion-laser interaction Hamiltonian with a driving
pattern $\tilde{f}_{j}(t)$ near-resonant with all sideband transitions, only a
finite subset of those sidebands is needed for the implementation of the
entangling gate. This selection process involves specifying the time-dependent
functions $F_{l,k}^{(j)}(t)$ for specific side-bands ($k$) of specific
motional modes ($l$). Crucially, achieving a successful implementation relies
on both a well-designed set of functions $F_{l,k}^{(j)}(t)$ and an appropriate
choice of sidebands. If the chosen set of sidebands is inadequate, there may
not exist a modulation strategy (i.e., a choice of $F_{l,k}^{(j)}(t)$) that
fulfills both requirements at the gate time: (i) eliminating undesired spin-
motion coupling terms and (ii) reaching the desired Rabi angle $\phi_{T}$ for
the entangling term.

Once a sideband set and modulation strategy are chosen, one can analyze the
system’s evolution. An initial spin-motion state
$\rho=\rho_{S}\otimes\rho_{M}$ evolves under the influence of the time-
dependent Hamiltonian $H(t)$ in Eq. (9) to a final state $\rho(T)=U(T)\rho
U^{\dagger}(T)$, with the unitary evolution operator
$U(T)=\mathcal{T}\exp(-i\int_{0}^{T}dtH(t))$. Constructing $U(T)$ for a chosen
modulation strategy thus reveals the effective operators present at the gate
time and therefore the final state $\rho(T)$. The Magnus expansion [27]
provides an approximation of $U(T)$ using a finite number of terms,
effectively acting as an exact solution up to a specific order in the Lamb-
Dicke parameter $\eta$. This work employs a third-order solution in $\eta$,
surpassing the standard first-order approach, which leads to a more accurate
model of the system’s dynamics.

Specifically, the interaction Hamiltonian to order $\eta^{3}$ is given by

$\displaystyle H(t)$
$\displaystyle=\sum_{j}\left[iF_{1,1}^{(j)}\sigma_{y}^{(j)}a_{1}^{\dagger}\left(1\\!\\!1-\frac{1}{2}\eta_{j1}^{2}a_{1}^{\dagger}a_{1}-\sum_{l=2}^{M}\eta_{jl}^{2}a_{l}^{\dagger}a_{l}\right)\right.$
$\displaystyle\left.\qquad-\frac{1}{2}\sum_{l=1}^{M}\eta_{jl}F_{l,2}^{(j)}\sigma_{y}^{(j)}a_{l}^{\dagger
2}\right]+h.c.\ ,$ (15)

with the creation and annihilation operators $a_{1}^{\dagger}$ and $a_{1}$
referring to the designated central motional mode. Accordingly, the unitary
evolution operator induced by this Hamiltonian can be found analytically with
the Magnus expansion truncated to fourth order, i.e.,
${U(t)=\exp(\sum_{i=1}^{4}\mathfrak{H}_{i}(t))}$, where

$\displaystyle\mathfrak{H}_{1}$ $\displaystyle=-i\int\mathbf{dt}_{1}\,H_{1}$
(16a) $\displaystyle\mathfrak{H}_{2}$
$\displaystyle=\frac{-1}{2}\iint\mathbf{dt}_{2}\,\commutator{H_{1}}{H_{2}}$
(16b) $\displaystyle\mathfrak{H}_{3}$
$\displaystyle=\frac{i}{6}\iiint\mathbf{dt}_{3}\,\Bigl{(}\commutator{H_{1}}{\commutator{H_{2}}{H_{3}}}+\commutator{H_{3}}{\commutator{H_{2}}{H_{1}}}\Bigr{)}$
(16c) $\displaystyle\mathfrak{H}_{4}$
$\displaystyle=\frac{1}{12}\iiiint\mathbf{dt}_{4}\,\Bigl{(}\commutator{\commutator{\commutator{H_{1}}{H_{2}}}{H_{3}}}{H_{4}}$
$\displaystyle\quad\qquad\qquad\qquad+\commutator{H_{1}}{\commutator{\commutator{H_{2}}{H_{3}}}{H_{4}}}$
$\displaystyle\quad\qquad\qquad\qquad+\commutator{H_{1}}{\commutator{H_{2}}{\commutator{H_{3}}{H_{4}}}}$
$\displaystyle\quad\qquad\qquad\qquad+\commutator{H_{2}}{\commutator{H_{3}}{\commutator{H_{4}}{H_{1}}}}\Bigr{)}\
,$ (16d)

with the short-hand notations $H_{i}\equiv H(t_{i})$ and
$\mathbf{dt}_{j}\equiv dt_{1}\cdots dt_{j}$. This expansion reveals that
$U(T)$ becomes an exponential of a sum of operators of the form
$g(T)\left(\prod_{l}a_{l}^{\dagger
p_{l}}a_{l}^{q_{l}}\right)\left(\sigma_{y}^{(1)}\right)^{r}\left(\sigma_{y}^{(2)}\right)^{s}$
for some integers $p_{l},q_{l},r,s$ and function $g(T)$ dependent on the
integrals over the driving functions $F_{l,k}^{(j)}(t)$.

Crucially, not all these terms contribute to the desired entangling operation.
To achieve pure entanglement at the gate time $T$, the driving functions
$F_{l,k}^{(j)}$ need to be carefully designed to eliminate all terms except
for the one responsible for the entangling operation
$i\phi_{T}\sigma_{y}^{(1)}\sigma_{y}^{(2)}$. This selective elimination
translates to a set of constraints on the driving functions, that have been
provided in electronic format for convenience [28].

One can then verify that the monochromatic driving functions

$\displaystyle F_{1,1}^{(1)}$ $\displaystyle=F_{1,1}^{(2)}=\Omega e^{2i\delta
t}$ $\displaystyle F_{l,2}^{(1)}$
$\displaystyle=s_{1l}\Omega\frac{\tilde{\eta}_{l}}{\eta_{1l}}e^{i\delta t}$
$\displaystyle F_{l,2}^{(2)}$
$\displaystyle=s_{1l}\Omega\frac{\tilde{\eta}_{l}}{\eta_{2l}}e^{i\delta t},$
(17)

with $\tilde{\eta}_{l}:=\frac{\sqrt{2}}{2}\sqrt{\eta_{1l}^{2}+\eta_{2l}^{2}}$
and the entangling condition
$-3\frac{\Omega^{4}}{\delta^{4}}\left(\sum_{j}\eta_{j1}^{2}\right)+\frac{\Omega^{2}}{\delta^{2}}\left(2+\sum_{l,j}\eta_{jl}^{2}\right)=\frac{\phi_{T}}{\pi}$
are sufficient to satisfy all those constraints and, therefore, to eliminate
undesired contributions, as well as to achieve entanglement at the gate time
$T$. Notably, while the driving scheme based on Eq. (V.1) achieves
entanglement, it offers limited robustness compared to the solution presented
in Eq. (12).

###  V.2 Robustness to motional heating

Motional heating, caused by interaction with the environment, significantly
degrades gate fidelity by introducing correlations between qubits and motional
degrees of freedom. Thermalisation and dephasing, the leading sources of
error, can be modeled by the master equation [29]

$\displaystyle\frac{d}{dt}\rho=-i\commutator{H(t)}{\rho}+\sum_{l=1}^{M}\mathcal{L}^{(l)}(\rho)\
,$ (18)

with the density matrix $\rho=\rho_{S}\otimes\rho_{M}$, the Hamiltonian $H(t)$
in Eq. (9) and the Lindbladian $\mathcal{L}^{(l)}(\rho)$ that includes the
jump operators $C_{-}^{(l)}=a_{l}$ and $C_{+}^{(l)}=a_{l}^{\dagger}$ for
thermalization in the $l$-th mode and $C_{n}^{(l)}=a_{l}^{\dagger}a_{l}$ for
dephasing:

$\displaystyle\mathcal{L}^{(l)}(\rho)=\sum_{j=-,+,n}\gamma_{j}^{(l)}\left(C_{j}^{(l)}\rho
C_{j}^{(l)\dagger}-\frac{1}{2}\\{C_{j}^{(l)\dagger}C_{j}^{(l)},\rho\\}\right).$
(19)

Here, $\gamma_{j}^{(l)}$ are the decay rates associated to the $j$-th operator
of the $l$-th motional mode and $\\{\cdot,\cdot\\}$ the anticommutator.

In the absence of dissipation, the target gate $U$ is perfectly implemented,
i.e. $\rho(T)=U\rho(0)U^{\dagger}$, but the presence of dissipating processes
$\mathcal{L}^{(l)}(\rho)$ introduces noise, resulting in an imperfect
(potentially non-unitary) gate
$\tilde{\mathcal{E}}:\rho\rightarrow\tilde{\mathcal{E}}(\rho)$. Since this
distortion depends on the evolution of the motional state, the choice of the
driving functions $f(t)$ in $H(t)$ significantly impacts the susceptibility of
$U$ to these dissipative effects. The Mølmer-Sørensen gate is particularly
vulnerable to motional heating due to its non-centered phase-space
trajectories, but alternative choices of the driving functions $F_{l,k}(t)$
can significantly enhance gate’s resilience to heating [9].

In the time-dependent frame defined by the non-dissipative unitary $U(t)$
(defined before Eq. (V.1)), as long as the entangling conditions on the
functions $F_{l,k}^{(j)}(t)$ are met, the coherent evolution reduces to the
identity $1\\!\\!1$, and the the master equation in Eq. (18) becomes purely
dissipative $\frac{d}{dt}\rho=\sum_{l}\tilde{\mathcal{L}}^{(l)}(\rho)$ with

$\displaystyle\tilde{\mathcal{L}}^{(l)}(\rho)=\sum_{j=-,+,n}\gamma_{j}^{(l)}\left(\tilde{C}_{j}^{(l)}\rho\tilde{C}_{j}^{(l)\dagger}-\frac{1}{2}\\{\tilde{C}_{j}^{(l)\dagger}\tilde{C}_{j}^{(l)},\rho\\}\right),$
(20)

where the time-dependent jump operators are now expressed in the rotated frame
${\tilde{C}_{j}(t)=U^{\dagger}(t)C_{j}U(t)}$, and their explicit expressions
up to order $\eta^{3}$ can be found in electronic format in [28]. Given that
the decay rates satisfy $\gamma_{j}T\ll 1$, the dynamics of $\rho(t)$ can be
described at first order by the propagator $e^{\tilde{\mathcal{L}}t}\approx
1\\!\\!1+\sum_{l}\int_{0}^{t}dt^{\prime}\tilde{\mathcal{L}}(t^{\prime})$.
Specifically, the spin state evolves to a final state at the gate time
$\rho_{S}(T)=\rho_{S}(0)+\Delta\rho_{S}$, with

$\displaystyle\Delta\rho_{S}=\tr_{M}\int_{0}^{T}dt^{\prime}\tilde{\mathcal{L}}(t^{\prime})[\rho_{S}\otimes\rho_{M}]\
,$ (21)

and $\tr_{M}$ denotes trace over the motion. While in the absence of
dissipation Eq. (21) vanishes, thermalization and dissipation result in a non-
zero $\Delta\rho_{S}$, which can be generally expressed as

$\displaystyle\Delta\rho_{S}=\sum_{m,n}\chi_{mn}A_{m}\rho_{S}(0)A_{n}^{\dagger}\
,$ (22)

with the operators $A_{m}\in\\{\
1\\!\\!1,\sigma_{y}^{(1)},\sigma_{y}^{(2)},\sigma_{y}^{(1)}\sigma_{y}^{(2)}\\}$.
Here, $\chi_{mn}$ are the process matrix elements whose norm characterizes the
infidelity of the gate and depend on the motional properties of the state
through elements of the form $\tr_{M}\left(\rho_{M}\prod_{l}a_{l}^{\dagger
p_{l}}a_{l}^{q_{l}}\right)$ (for some integers $p_{l},q_{l}$) as well as on
the temporal shaping of the driving functions $F_{l,k}^{(j)}(t)$. Therefore,
since each element $\chi_{mn}$ is weighted with a factor of $\eta$, it is
possible to devise schemes that are robust to a particular order of $\eta$.
Formulated as an optimisation problem, this entails determining the form of
the drivings $F_{l,k}^{(j)}(t)$ that minimize a set of contraints, as shown in
Supp. Mat. A.

While the parametrization in Eq. (V.1) offers limited opportunities to improve
its robustness, a bichromatic drive with minimal modifications to the protocol
of Eq. (V.1), which results in Eq. (12), can significantly improve robustness
to the leading order $\order{\eta^{0}}$ by minimizing the functionals in Eq.
(23) of the Supp. Mat. A.

##  VI Conclusions

Achieving high-fidelity entangling gates is a critical bottleneck for trapped-
ion quantum computation. Current gate implementations, while often tailored to
address specific error sources, struggle when confronted with the combined
effects of thermal excitations, control field imperfections, and environmental
coupling. These factors lead to unwanted spin-motion couplings, detuning
errors, and motional heating, ultimately degrading gate performance.
Consequently, high-fidelity operations often require highly controlled
environments and sophisticated control techniques, even to achieve moderate
entanglement fidelities with small ion chains.

This work presents an entangling gate scheme that tackles the fundamental
challenge of mitigating diverse error sources simultaneously. Our proposed
gate utilizes a specifically designed driving protocol that addresses multiple
sidebands of each motional mode. This approach effectively suppresses
inhomogeneous spin-motion couplings, leading to significant improvements in
gate fidelity, even for large ion chains with numerous motional modes.
Furthermore, the driving patterns enable high-fidelity operations even under
substantial thermal excitations, a crucial factor for scalability.

The proposed scheme demonstrates remarkable robustness against motional
heating, vibrational frequency fluctuations, and detuning errors. This
combined robustness and scalability pave the way for significant advancements
in trapped-ion quantum computation. By enabling high-fidelity entanglement
under noisy conditions, this work brings us closer to the realization of
practical quantum computers.

##  VII Acknowledgements

We gratefully acknowledge stimulating discussions with Jake Lishman, Nguyen
Le, Winfried K. Hensinger and Samuel J. Hile. This work was supported by the
U.K. Engineering and Physical Sciences Research Council via the EPSRC Hub in
Quantum Computing and Simulation (EP/T001062/1).

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SUPPLEMENTARY MATERIAL

##  Appendix A Functionals for resilience against motional heating

The gate fidelity $F$ (Eq. (14)) in the presence of motional heating depends
on the temporal modulation of the driving functions $F_{l,k}^{(j)}$.
Specifically, the fidelity $F$ can be maximized by selecting these functions
$F_{l,k}^{(j)}$ to minimize the matrix elements $\chi_{m,n}$ in Eq. (22), or
equivalently, a set of cost functions $\\{c_{i}\\}$ evaluated at the gate time
$t=T$, which naturally arise at various orders in the Lamb-Dicke parameter
$\eta$.

#### Order $\eta^{1}$

The first-order-in-$\eta$ contributions to the motional heating are
functionals of the driving functions corresponding to the central motional
mode, $F_{1,1}^{(j)}$, with $j=\\{1,2\\}$. Specifically, the cost functions
read

$\displaystyle c_{1}=\\{\alpha_{i}\\}\ ,$ $\displaystyle
c_{2}=\\{\alpha_{i}^{2}\\}\ ,$ $\displaystyle
c_{3}=\\{\alpha_{i}\alpha_{j}\\}\ ,$ $\displaystyle
c_{4}=\\{|\alpha_{i}|^{2}\alpha_{j}\\}\ ,$ $\displaystyle
c_{5}=\\{\alpha_{i}^{2}\alpha_{j}^{*2}\\}\ ,$ $\displaystyle
c_{6}=\\{|\alpha_{i}|^{2}\\}\ ,$ $\displaystyle
c_{7}=\\{\alpha_{i}\alpha_{j}^{*}\\}\ ,$ $\displaystyle
c_{8}=\\{|\alpha_{i}|^{2}|\alpha_{j}|^{2}\\}\ ,$ $\displaystyle
c_{9}=\\{\alpha_{i}^{2}\alpha_{j}^{*}\\}\ ,$ (23)

with $i\neq j$, $\alpha_{j}:=\\{F_{1,1}^{(j)}\\}$ and with the short-hand
notation $\\{f\\}=\int_{0}^{t}dt^{\prime}f(t^{\prime})$.

Crucially, a Fourier series parametrization of the form
$f=\sum_{m}A_{m}e^{ia_{m}t}$ with the constraint
$\sum_{m}\left(A_{m}\prod_{k\neq m}a_{k}\right)=0$ for the functions
$F_{1,1}^{(j)}$ ensures that all conditions vanish $c_{i}=0$, except for
$i=\\{6,8\\}$. Consequently, optimal motional heating resilience at first
order in $\eta$ can be achieved through the minimization of the functional
$C=\max(c_{6},c_{8})$.

#### Orders $\eta^{2}$ and $\eta^{3}$

Second and third-order contributions to motional resilience involve numerous
functionals, becoming significant only in the regime of large Lamb-Dicke
parameters. For $2$ ions, there are $186$ and $2543$ independent elements for
the respective orders, available conveniently in electronic format in [28].

##  Appendix B Highly excited thermal states

While Figure 2 in the Main text showcases the performance of the proposed gate
for various ion chain sizes and scenarios including ideal conditions or
motional heating, the results are limited to motional Fock states
$\ket{\mathbf{n}}$ with occupation numbers of up to $n=10$.

Fig. 4 depicts the infidelity of a two-ion system for thermal states with
average phonon occupations (thermal excitation) of up to $\bar{n}=100$ and
different spin-motion coupling strengths $\Lambda$. The figure demonstrates
that the proposed scheme outperforms the standard MS approach for any value of
$\bar{n}$ and $\Lambda$. Remarkably, fidelities exceeding $99\%$ are achieved
with spin-motion couplings $\Lambda$ of $0.05$, a value commonly encountered
in experiments, which represents more than an order of magnitude improvement
compared to the standard MS approach.

Figure 4: Infidelity $1-F$ for $N=2$ qubits as a function of the mean
occupation $\bar{n}$ of a thermal state (on log scale). Increasing line
thickness represents increasing values of the coupling parameter $\Lambda$
(defined in the main text), while the color line distinguishes between the
conventional MS approach (orange) and the present robust gate scheme (blue).
No motional heating or detuning errors are considered.