# Scalable architecture for trapped-ion quantum computing using RF traps and
dynamic optical potentials

David Schwerdt1,∗,† Lee Peleg1,∗ Yotam Shapira1,2,∗ Nadav Priel2,∗ Yanay
Florshaim2 Avram Gross2 Ayelet Zalic2 Gadi Afek2 Nitzan Akerman1 Ady Stern3
Amit Ben Kish2 Roee Ozeri1,2 1Department of Physics of Complex Systems,
Weizmann Institute of Science, Rehovot 7610001, Israel
2 Quantum Art, Ness Ziona 7403682, Israel
3 Department of Condensed Matter Physics, Weizmann Institute of Science,
Rehovot 7610001, Israel
∗ These authors contributed equally to this work
† Corresponding author

###### Abstract

Qubits based on ions trapped in linear radio-frequency traps form a successful
platform for quantum computing, due to their high-fidelity of operations, all-
to-all connectivity and degree of local control. In principle there is no
fundamental limit to the number of ion-based qubits that can be confined in a
single 1d register. However, in practice there are two main issues associated
with long trapped ion-crystals, that stem from the ‘softening’ of their modes
of motion, upon scaling up: high heating rates of the ions’ motion, and a
dense motional spectrum; both impede the performance of high-fidelity qubit
operations. Here we propose a holistic, scalable architecture for quantum
computing with large ion-crystals that overcomes these issues. Our method
relies on dynamically-operated optical potentials, that instantaneously
segment the ion-crystal into cells of a manageable size. We show that these
cells behave as nearly independent quantum registers, allowing for parallel
entangling gates on all cells. The ability to reconfigure the optical
potentials guarantees connectivity across the full ion-crystal, and also
enables efficient mid-circuit measurements. We study the implementation of
large-scale parallel multi-qubit entangling gates that operate simultaneously
on all cells, and present a protocol to compensate for crosstalk errors,
enabling full-scale usage of an extensively large register. We illustrate that
this architecture is advantageous both for fault-tolerant digital quantum
computation and for analog quantum simulations.

##  I Introduction

Trapped ions have ideal properties to be used as qubits for quantum computing
(QC); they feature long coherence times, efficient state preparation and
detection techniques, and a high degree of connectivity [1, 2, 3, 4, 5]. A
quantum register of 1000s of qubits, or more, can be formed, for example, by
utilizing an equally spaced crystal of ions in a linear RF Paul trap. Indeed
recent years have seen many experimental attempts to work with increasingly
larger trapped-ion registers [6, 7, 8, 9].

However, there are two practical issues associated with large ion-crystals
that impede progress in this direction. The first is heating rates; as the
number of ions in the crystal, $N$, increases, heating of the ions’ motional
modes due to electric field noise drastically increases. Of particular concern
is the axial center-of-mass (COM) mode, whose frequency typically decreases as
$1/N$. This mode is especially vulnerable, as electric field noise tends to be
spatially uniform and to target low-frequency modes [10, 11]. The resulting
significant heating rates prohibit the implementation of high fidelity qubit
operations, and might destabilize the ion-crystal.

The second issue with large ion-crystals is spectral crowding. As the size of
the crystal increases the frequency of adjacent motional modes becomes tightly
spaced. For large ion-crystals with a dense mode spectrum, resolving
individual modes becomes challenging; this complicates the implementation of
entangling gates between two or more ions, which involves exciting their
common motion. Granted, spectral control methods [12, 13] allow for
simultaneously targeting multiple modes to achieve desired qubit couplings
with high fidelity. While such methods are promising for a moderate, up to
100’s, number of ions, they do not generally provide a scalable solution for
arbitrarily large ion-crystals. First, the optimal control problems that must
be solved to implement these methods becomes intractable for large $N$.
Moreover, there is strong evidence that the minimum achievable gate time is
set by the smallest frequency spacing among the motional modes [14]. This
implies that gate times scale at least as $N^{2}$, making large ion-crystals
prohibitively slow for quantum computations.

One active direction for scaling ion traps is the quantum charge-coupled
device (QCCD) architecture [15, 16]. This setup involves many spatially
separated trapping sites, each containing a small number of ions, where
communication between sites is done by shuttling individual ions. Inevitably,
this comes at the expense of high overhead in hardware and long circuit
duration dominated by ion-shuttling and ion-cooling times. An alternative
scale-up approach is using photonic-interconnects in order to link small-scale
ion-crystals [17, 18, 19]. This method as well involves a high overhead, due
to the currently low entanglement rate via the interconnect, leading to slow
operations.

Here we propose a scalable architecture for QC based on trapped ions qubits,
that maintains the advantages of a long ion-crystal, while circumventing its
challenges. In our proposal, an arbitrarily long ion-crystal is segmented into
cells by means of dynamically operated optical potentials, e.g. optical
tweezers. In this way, the crystal’s motional mode structure is modified such
that heating rates reflect only the cell size, and not $N$. Furthermore,
programmable high-fidelity multi-qubit entangling gates can be implemented
independently within each cell simultaneously. In addition, we show that mid-
circuit measurements, a cornerstone of quantum error correction (QEC)
techniques, as well as other central quantum computational tools, are
straightforward to implement in an optically segmented ion-crystal.

In this work we generalize the method presented in Ref. [14], and employ
qubit-local driving fields that implement programmable multi-qubit entangling
gates. The method in Ref. [14] provides a clear physical intuition, an
efficient procedure for designing gate drive parameters and a thorough scaling
analysis, all of which are relevant to this proposal.

The integration of optical tweezers into ion traps is an active area of
research [20, 21, 22, 23, 24, 25]. Specifically Refs. [22, 23, 24, 25] make
use of optical tweezers in order to generate target entanglement operations.
In Ref. [20] the authors use tweezers for parallel two-qubit entangling gates,
the system’s scalability is discussed however there is no treatment of large
ion-segments or ion-crystal heating rates. In Ref. [21] the authors consider
segmenting an ion-crystal using tweezers, however do not analyze generating
large-scale entanglement or connectivity between the segments. Mid-circuit
measurements are not discussed in any of the above works. Unlike previous
studies, here we detail how to achieve a holistic architecture, incorporating
parallel and large-scale entangling operations and mid-circuit measurements,
that are needed for large scale QC.

The remainder of this paper is organized as follows. In section II we present
an overview of the proposed architecture. In section III we sketch two
examples of applications of quantum information processing tasks that are
amenable to our architecture: quantum simulation and quantum error correction.
An in-depth analysis of our proposed architecture follows. Namely, in section
IV we derive the spectral properties of optically segmented traps. In section
V we analyze the implication of optical segmentation on the ion-crystal’s
heating rate. In section VI we present our method for designing high-fidelity
multi-qubit logical operations that are not hindered by unwanted crosstalk.
Lastly, in section VII we present a protocol for performing mid-circuit
measurements.

##  II Proposed architecture

There are well-established techniques for preparation, control and measurement
of trapped-ions based qubits for small-scale registers [26, 27]. We therefore
focus mostly on unique aspects of our architecture. We propose a QC
architecture in which a long trapped ion-crystal, made of $N$ ions, is
segmented into $S$ cells, with each cell containing $C$ computational qubits
available for quantum computation and simulation, generating a $C\cdot S$
qubit register. The segments are formed by placing $B_{A}$ barrier ions
between adjacent cells, that are illuminated by an optical trapping potential,
i.e. optical tweezers, made of tightly focused laser beams [28, 29], that
provide an additional local confinement for the barrier ions.

The key point enabling scalability is that due to the optical segmentation the
cells behave as nearly independent quantum registers. Indeed, we show below
that both challenges of scaling-up trapped ions based QCs, i.e., ion-crystal
stability and the performance of logical operations, both scale with the cell
size, $C$, and not the total number cells, $S$. Thus the dynamically
reconfigurable segmentation removes the fundamental limits on the number of
cells and on the number of qubits in the ion-crystal, and accordingly on
quantum circuit and simulation size.

Figure 1: Architecture for scalable trapped-ion quantum computing using RF
traps and dynamic optical potentials. The ion-crystal (vertical direction) is
segmented to cells by dynamically applying optical trapping potentials. This
generates $S$ segments, each containing $C$ computational qubits (green) that
are separated by $B_{A}$ consecutive optically confined ‘barrier ions (dark
purple). Additional $B_{B}$ ions in each cell (light-green) are used to
support additional segmentation configurations of the ion-crystal, and mid-
circuit measurements (‘Meas.’) that enable classical feedback (orange arrows)
for the implementation of e.g. quantum error correction. For simplicity the
schematic shows only 4 segments with 8 computational qubits per cell, however
in practice our proposal supports 10s of qubits per cell and an arbitrary
number of segments (vertical dots). The horizontal direction shows a typical
mode of operation of the QC, namely logical operations, involving multi-qubit
logical gates, $U_{s,n}^{\left(A\right)}$, are performed in each cell, $s$, at
the $n$th computational step. The optical potentials are then switched to a
different segmentation configuration, allowing for large-scale connectivity
between cells of the previous configuration, $U_{s,n}^{\left(B\right)}$. Thus
ultimately guarantying connectivity between all computational qubits. An
intermediate segmentation configuration, in which all $B_{A}$ and $B_{B}$ ions
are optically confined, is used to accommodate for mid-circuit measurement,
cooling of the ion-crystal and state re-initialization.

A general overview of our scalable architecture, is presented schematically in
Fig. 1. In favor of a simple presentation we show segmentation of only $S=4$
cells, and explicitly highlight $C=8$ computational qubits (green) per cell.
In practice 10s of qubits may be placed in each cell, and the number of cells
is theoretically unlimited (vertical dots), forming a large ion-crystal. The
cells are separated by $B_{A}$ barrier ions (dark purple), with the schematic
showing $B_{A}=2$. $B_{B}$ additional ions (light green) are placed in the
center of each cell and are used for segment re-configuration and mid-circuit
measurements, with the schematic showing $B_{B}=2$. Thus, the ion-crystal
contains $N=\left(C+B_{A}+B_{B}\right)S+B_{A}$ trapped ions.

The figure shows a general mode of operation of the QC, with parallel
operations along the vertical axis and sequential operations, ordered from
left to right, that form computational steps. Specifically, barrier ions are
strongly confined with optical trapping potentials (dark purple), generating
the $S=4$ segments in this example (configuration ‘A’). At the $n$th
computational step, local and global driving fields simultaneously implement
unitary operations, $U_{s,n}^{(A)}$ (blue), with $s=1,2,...,S$. These unitary
operators implement components of the overall quantum algorithm, and may
involve all $C$ (green) computational qubits and $B_{B}$ (light-green)
auxiliary ions which are not optically confined in this configuration.

Next all $B_{A}$ and $B_{B}$ barrier ions are optically confined, enabling
mid-circuit measurement of the auxiliary ions as well as recooling of the ion-
crystal and preparation of ions to be used as auxiliary ions (detailed below).
Classical feedback (orange arrows), based on the mid-circuit measurement
results, may be implemented at this step in order to influence the next layer
of unitary evolution.

Next, the segmentation is reconfigured by strongly confining previously
unconfined $B_{B}$ barrier ions and removing the confinement of the $B_{A}$
barrier ions, thus interchanging their roles. This dynamically and quickly
generates a different configuration of the segments (configuration ‘B’).
Unitary operators, $U_{s,n}^{(B)}$, can now be implemented and connect
previously uncoupled qubits. Additional measurement and re-configuration steps
can be applied as required. We remark that here we have shown only two basic
ion-crystal configurations (A and B), and have further considered here and
below $B_{A}=B_{B}$, however various additional segmentation configurations,
with different parameters, can be flexibly generated provided relevant ions
are allocated as barrier ions.

In addition to the configurable optical trapping potentials, we make use of
local independently and simultaneously applied coherent driving fields. As we
show below, using these local fields for controlling just one of the atomic
transitions of the trapped ions is enough in order to accommodate for
programmable multi-qubit logical gates [14], which occur simultaneously in all
cells ($U^{(A)}$s and $U^{(B)}$s in Fig. 1), as well as for the mid-circuit
measurements and the state preparation that they entail.

##  III Application demonstrations

We sketch examples of applications that showcase the utility of our
architecture. These examples rely on various features of our proposal, e.g.
implementing large-scale multi-qubit entangling operations and performing mid-
circuit measurements. The methods underlying these features are thoroughly
discussed and analyzed in the sections below.

We start by considering analog quantum simulations on our system, specifically
a three dimensional (3d) quantum-spin model. In general, quantum simulations
are considered a well-suited task for noisy intermediate-scale quantum (NISQ)
era QCs [30], as simulation of quantum systems is a notoriously challenging
task for classical computers, while quantum computers are considered naturally
suited for it [31]. Indeed, numerous quantum systems have recently implemented
impressive demonstrations of quantum simulations [32, 33, 34].

Quantum simulations using trapped-ions based qubits, have also been recently
demonstrated [35, 36, 37, 38, 39, 40, 41, 42], many of which take advantage of
the unique all-to-all coupling present in these systems. In these works Ising-
type interactions, that are inherent to trapped ions systems, can be
straightforwardly iterated and combined with single-qubit rotations in order
to generate arbitrary XYZ-type spin-models [43, 44].

Our architecture is well-suited for quantum simulations, as our ability to
design programmable multi-qubit entanglement manifests as the dimensionality
and geometry of the simulated model [45], and the optical segmentation and
reconfiguration provide a straightforward approach to simulating 3d bulk
systems. Such 3d systems are challenging to simulate on linear or planar
quantum processors with short-range interactions, due to a large gate overhead
required for embedding the 3d geometry.

Figure 2: Quantum simulation of a 3d rectangular Ising spin-model on an
optically segmented ion-crystal. Computational qubits (green) are coupled in
cells (purple brackets) such that each cell holds two planes of the
rectangular spin-model, and implements inter-plane (dark-blue and bright-blue
links) and intra-plane (orange links) couplings (a) Couplings of a single
cell, holding two planes of the model, with $d=4$, used for both segmentation
configurations. (b) Multiple cells form the 3d Ising model. The two optical
configurations (‘A’, ‘B’ and purple brackets) enable interlaced coupling of
the planes (solid orange and dashed yellow, respectively), such that two
entanglement operations implement the model’s Hamiltonian. (c) We design the
control pulses that implement the nearest-neighbor couplings required for the
Ising model, here for $d=4$ and $S=5$. The resulting coupling between all
qubit pairs (horizontal and vertical axes) is shown (color), exhibiting a
block-diagonal structure that implements the links shown in (a), for each
cell. (d) Deviation of the qubit-qubit coupling from the ideal structure (log
scale), showing a low error, that is not limited by crosstalk between adjacent
cells, and enables scaling up. The implementation’s infidelity scales as the
deviations squared, and is here evaluated to $10^{-4}$ per cell.

For simplicity, we focus here on a 3d rectangular lattice of spins, though
many other geometries can be realized, e.g. hexagonal. Figure 2 shows an
implementation of such a $d\times d\times 2S$ rectangular Ising model.
Specifically, Fig. 2(a) shows a single cell of the ion-crystal, that contains
$2d^{2}$ computational qubits (green), and encodes a pair of two-dimensional
planes of the simulated system. Programmable entanglement gates implement the
required couplings that generate the model’s geometry, i.e. inter-plane
couplings (dark-blue and bright-blue) and intra-plane couplings (orange). The
model is made three dimensional, as shown in Fig. 2(b), by interlacing the
vertical couplings between adjacent layers (solid orange and dashed yellow),
using the two segmentation configurations (purple brackets).

As an example, we consider a specific realization of our architecture, in
trapped ${}^{40}\text{Ca}^{+}$ ions, and utilize it for a $d=4$ and $S=5$
realization of a nearest-neighbor Ising model (see further system details in
the SM [46]). We design the control pulses that drive the computational
qubits, and generate the required entanglement operations for simultaneous
$XX$-coupling of the qubits, in the entire ion-crystal, similarly to the
couplings shown in Fig. 2(a). For simplicity, we use a uniform, unity coupling
between all nearest-neighbor qubits. Our design protocol is presented and
discussed below.

We calculate the coupling terms that result from the control pulses that we
designed, and plot them in Fig. 2(c), with each point on the plot representing
a qubit-qubit coupling between qubits of the corresponding indices (horizontal
and vertical axes). The block-diagonal structure that is seen reflects the
underlying segmented structure of the ion-crystal. The relative difference
between the ideal and the resulting qubit-qubit couplings is shown in Fig.
2(d) in log-scale exhibiting a low coupling error that indicates a high-
fidelity and accurate simulation. Indeed, the overall performance of our
implementation, simultaneously coupling $160$ computational qubits, is
evaluated with an infidelity that is better (lower) than $10^{-4}$ per cell.

We note that with this encoding, a time-step of an $XX$ nearest-neighbor 3d
rectangular Ising model can be implemented with only two sequential multi-
qubit entanglement gates, such as that shown in Fig. 2(c), which, for $d=4$
and $S=5$, would otherwise require the application of 384 sequential two-qubit
gates between arbitrary pairs in the quantum register (in general
$\mathcal{O}\left(d^{2}S\right)$ two-qubit gates). By considering more
elaborate models, e.g. next-nearest neighbor interactions within the two-
dimensional planes, non-uniform couplings between the planes, or larger
systems, the two-qubit gate count will increase, while with our method two
multi-qubit gates still suffice.

The second example showcases a path towards fault-tolerant quantum computation
[47, 48, 49]. As shown in 3(a), each segment (purple brackets, ‘A’) is
utilized to encode a single logical error-corrected qubit using a distance-
five surface code. The code uses 25 computational qubits to store the logical
state, and 12 auxiliary (ancilla) qubits to readout the values of both $X$ and
$Z$ stabilizer measurements. The entire set of $X$ stabilizer measurements can
be implemented in one step, using a single multi-qubit entanglement gate and
mid-circuit measurements. Then, the auxiliary qubits are reinitialized, after
which the entire set of $Z$ stabilizers is similarly performed. Clearly cell
segmentation does not require 12 barrier ions, thus here most of the 12
auxiliary qubits are implemented by computational qubits in the cell.

Figure 3: Quantum error correction code on an optically segmented ion-crystal.
(a) We implement several distance-five logical qubits, shown pictorially as
grids of $5\times 5$ computational qubits (green). Each cell in configuration
‘A’ (purple brackets) houses a single logical qubit (labeled 0 - 12 and 25 -
36) and twelve auxiliary qubits (labeled 13 - 24). Programmable multi-qubit
gates generate the required entanglement structure for stabilizer measurements
on each logical qubit in parallel. Here we show an implementation that
generates $X$-stabilizers (light-blue plaquettes and blue lines).
$Z$-stabilizers may be implemented with an additional gate (light-red
plaquettes). Connectivity across the entire register is enabled by
configuration ‘B’ (light-purple brackets) such that each cell in this
configuration holds data qubits of two adjacent logical qubits. This enables
logical entangling operations between any neighboring encoded qubits via
stabilizer measurements formed on the border between the two surface codes
(yellow plaqeuttes and orange lines), using the auxiliary ions in this
configuration (labeled 17’-19’). (b) Example of programmable multi-qubit gate
generating $X$ and $Z$ stabilizers on five logical qubits, implemented on a
segmented ion-crystal with $S=5$, $C=34$ and $B=3$. We design the control
pulses that implement the entanglement gates required for the various
stabilizers. The resulting qubit-qubit couplings between corresponding qubit
pairs (horizontal and vertical axes) is shown (color), exhibiting a block-
diagonal structure that reflects the formation of five logical qubits. (c)
Deviation of the qubit-qubit coupling from the ideal structure (log scale),
showing a low error, that is not limited by crosstalk between adjacent cells,
and enables scaling up. The gate’s infidelity scales as the deviations
squared, and is here evaluated to $10^{-4}$ per cell.

Naturally, these stabilizer measurements can be performed across all logical
qubits in parallel. Note that not all segments must implement the same
operation at any given time; some may, for example, implement $X$ stabilizer
measurements while others implement $Z$ stabilizer measurements. Figure 3(b)
shows a specific example involving five logical qubits, and a multi-qubit
entanglement gate, designed to perform simultaneous $X$ and $Z$ stabilizer
couplings, in different cells, throughout the entire ion-crystal. The relative
difference between the ideal and the resulting qubit-qubit couplings is shown
in Fig. 3(c) in log-scale, exhibiting a low error that indicates a high-
fidelity and accurate implementation.

Entanglement between two logical qubits, implemented on independent surface
codes, can be achieved, for example, using ‘lattice surgery’ [50, 51]. In this
method, stabilizers formed along the border between two surface codes are
measured, projecting the joint state of the logical qubits to an entangled
basis. In our case, this is naturally enabled by re-configuring the optical
segmentation (light-purple brackets, ‘B’) such that ions from neighboring
cells can interact, as shown schematically in Fig. 3 (orange lines and yellow
plaquettes). Following this operation, the segmentation reverts to its
original configuration, and additional stabilizers are measured. The overall
result of this procedure is an entangling logical $XX$ operation between the
adjacent surface codes.

We emphasize that even though $S=5$ is used in this example, our
architecture’s scalability enables using considerably more cells in order to
implement many logical qubits. This example shows one potential route to fault
tolerance using our architecture; however many different variants could be
considered. First the distance of the code, and thus the number of physical
qubits required, should be determined by the demands on the overall logical
error rate. Moreover, the choice of the surface code itself, while motivated
by its high fault tolerance threshold, is not the only possible choice. In
fact, there are other high-threshold codes, with potentially better encoding
rates, that may be suitable as well [52]. Constructing an optimal protocol for
fault tolerant QC within our architecture, including a prescription for
performing logical non-Clifford operations, is a subject for further study.

##  IV Spectral properties of optically-segmented ion-crystals

In general, similar to alternative scale-up methods, segmentation decouples
cells from each other, reshaping the well-known all-to-all coupling of trapped
ion-crystals to local all-to-all couplings within cells of manageable sizes of
tens of ions. As we show below, here this segmentation allows to maintain the
ion-crystal’s stability as well as to shape programmable multi-qubit quantum
gates that act simultaneously and independently within the different cells.
Another unique feature of our scale-up strategy, compared to other
segmentation approaches, stems from the ability to dynamically and quickly
reconfigure the applied segmentation, generating new cells that combine and
couple previously decoupled qubits, thus enabling fast and large-scale
connectivity within the whole ion-crystal.

We analyze the effect of optical segmentation on the ion-crystal, specifically
on collective modes of motion of the ion-crystal in the axial and transverse
directions. To this end we assume that the barrier ions are illuminated by
beams that generate an optical harmonic trapping frequency of the form [53],

$\omega_{\text{o.t.p}}=\sqrt{\frac{2\text{Re}[\alpha\left(\lambda\right)]}{m}}|E|,$
(1)

where $\alpha\left(\lambda\right)$ is the wavelength-dependent polarizibility
of an illuminated ion, $m$ is its mass, and $E$ is the field strength. The
impact of this potential is best quantified by comparing it to another
important parameter in the system, namely the characteristic frequency-scale
associated with Coulomb interaction between adjacent ions [20],

$\nu=\sqrt{\frac{e^{2}}{4\pi\epsilon_{0}md^{3}}},$ (2)

with $e$ the elementary electron charge, $\epsilon_{0}$ the vacuum
permittivity and $d$ is the inter-ion distance of an equidistant ion-crystal.
As we show below, a strong optical potential, $\omega_{\text{o.t.p}}>\nu$,
drastically changes the motional mode structure of the chain. By incorporating
the optical trapping potential with common methods to analyze ion-crystal
motion [54], we obtain the normal-mode frequencies and structure in the axial
and transverse directions (for further details see the SM [46]).

We first consider the effect of the optical trapping on the axial modes of
motion of the ion-crystal, that lie at the lower parts of the motional
spectrum and are typically more prone to heating. When considering the joint
motional mode structure of two identical ion-crystals, in two completely
separate traps, we observe a degenerate mode structure, with each motional
frequency appearing twice, and each motional mode being localized at one of
the ion-crystals. Thus we intuitively expect that the motional spectrum of a
well-decoupled optically segmented ion-crystal will form approximately
degenerate bands.

Figure 4 demonstrates this by showing the axial motional spectrum of a $N=231$
long ion-crystal for a varying optical trapping potential that segments the
ion-crystal into $S=6$ cells, separated by $B_{A}=3$ barrier ions, such that
each cell contains 35 ions (out of which $C=32$ are computational qubits and
$B_{B}=3$ acts as barrier ions for segmentation configuration ‘B’). As the
optical trapping potential increases the axial spectrum (blue) forms 35 bands
of $S$ modes each.

Figure 4: Axial spectrum of an ion-crystal with N=231 ions, segmented into
$S=6$ cells. Cells are separated by $B_{A}=3$ barrier ions and include
$C+B_{B}=35$ ions each. The axial frequencies (blue) are shown for various
trapping optical potentials, $\omega_{\text{o.t.p}}$ and are normalized by the
characteristic Coulomb scale, $\nu$. For comparison, an additional axial
spectrum of an independent ion-crystal with 35 ions is shown (dashed orange).
For $\omega_{\text{o.t.p}}>\nu$ the axial spectrum of the optically segmented
trap forms bands which are located at the frequencies of the independent cell.
As $\omega_{\text{o.t.p}}$ increases high-frequency bands form, which are due
to the $B_{A}$ barrier ions. The radial trapping frequency,
$\omega_{\text{rad}}/\nu\approx 4.2$ (vertical line), is relevant for
generating multi-qubit operations. The inset is a zoom-in on the center-of-
mass band (vertical axis in log-scale), showing that at
$\omega_{\text{o.t.p}}=2\nu$ the segmented crystal’s modes resemble the
independent cell, implying that heating is dictated by the properties of a
single cell rather than the whole, multi-cell, crystal.

Indeed the formation of bands heralds the decoupling of the motion of cells
from each other, such that band frequencies resemble those of an independent,
un-segmented, ion-crystal containing $C+B_{B}$ ions (orange), with each mode
in the band being a superposition of local excitations of the corresponding
mode of the independent cell. Thus, the width of each band marks the relevant
rate in which a motional excitation traverses from one cell to another. That
is, an excitation of a local mode in a single cell is composed of a
superposition of all modes of the band, which then disperse at a time scale
that is inversely proportional to the band width.

Figure 4 further shows $B_{A}$ high-frequency bands, containing $S+1$ modes,
associated with the motion of $\left(S+1\right)B_{A}$ barrier ions. Crucially
these bands are spectrally separated from the ‘bulk’ ions such that their
motion is essentially decoupled from the bulk. We exploit this fact below in
order to implement mid-circuit measurements and cooling of the barrier ions.

Figure 5 shows the transverse (radial) band structure of the same $N=231$ ions
systems with $\omega_{\text{o.t.p}}$ set to match the scale of the RF trapping
in the transverse direction, $\omega_{\text{rad}}$, yielding
$\omega_{\text{o.t.p}}\approx 4.2\nu$ (dashed vertical line in Fig. 4).
Similarly to the axial direction, optical segmentation results in the
formation of $C+B_{B}$ bulk-bands as well as additional $B_{A}$ bands
associated with the motion of barrier ions (blue). The average frequencies of
the bands resemble those of an independent cell (orange), also shown in the
inset (bottom right) that presents a zoom-in on the last four computational
qubit bands, showing narrow and well-separated bands. For transverse modes of
motion the band index, $b$ (horizontal axis), ordered according to the band’s
frequency, is opposite to the mode’s wave vector, i.e. large $b$’s represent
long-wavelength modes of motion.

Transverse modes of motion are used in order to generate entanglement between
computational qubits, by mediating qubit-qubit interaction via spin-dependent
motion. Typically the entanglement operation duration is inversely related to
the frequency difference between adjacent modes of motion [14].

Here, the frequency gap between bands,
$\Delta\omega_{b}=\omega_{b}-\omega_{b-1}$, with $\omega_{b}$ the mean
frequency of the modes of bulk bands, $b=1,...,C+B_{B}$, marks the typical
interaction rate between ions within the same cell. Thus a simple estimation
of unwanted crosstalk in the entanglement operations between cells is given by
$\varepsilon_{\text{BW},b}=\frac{\text{BW}_{b}}{2\Delta\omega_{b}}$, with
$\text{BW}_{b}$ the bandwidth of band $b$, that sets the coupling rate between
cells. The resulting estimate is presented in the inset (top left) of Fig. 5
(orange), showing that high $b$ bands, associated with long-wavelength motion
across cells, results in a higher crosstalk, at the few percent level. A more
precise estimate of the crosstalk is also shown (blue) and discussed below.

Figure 5: Radial modes spectrum of an optically-segmented ion-crystal
containing $N=231$ ions, and the resulting cell crosstalk. Due to the
segmentation the mode frequencies (blue) form $C+B_{B}$ bands, which imitate
the structure of an unsegmented independent cell (orange). Additional $B_{A}$
high-frequency bands associated with the barrier ions are formed as well. The
bottom-right inset shows a zoom-in on the last four bulk bands. The bands are
well separated and narrow in frequency. The top-left inset presents
estimations of unwanted crosstalk between cells during simultaneous multi-
qubit operations, mediated by the various bulk bands. The estimations are
formed by considering only spectral consideration, $\varepsilon_{\text{BW},b}$
(orange) or mode-structure considerations, $\varepsilon_{J,b}$ (blue). Both
predict a $\sim 2.5\%$ crosstalk due to mainly long-wavelength motion of the
cells, associated with high bands. Mitigation of the crosstalk is presented in
the main text.

Optically confined ions are not used as computational qubits since optical
confinement significantly degrades their coherence due to spontaneous photon
scattering and uncontrolled light-shifts. For example, we consider
${}^{40}\text{Ca}^{+}$ ions that are illuminated by a confining optical field
at $400$ nm with a beam that has a diameter of $1\text{ }\mu\text{m}$, driving
the $S\leftrightarrow P$ transition. In this setup, a harmonic confinement
frequency of $\omega_{\text{o.t.p}}=\omega_{\text{rad}}=3$ MHz, used
throughout our analysis, requires an optical power of $50$ mw per barrier ion.
The resulting photon scattering rate from the $P$ manifold is approximately
$10$ kHz and the frequency shift of the $S\leftrightarrow P$ transition is
approximately $2$ GHz. Photons scattered from barrier ions are significantly
detuned, by few THz, from this transition in the computational qubits,
resulting in a low, $<1$ Hz, decoherence rate.

##  V Heating rates of optically-segmented ion-crystals

A crucial criteria for successfully scaling-up ion-crystals is maintaining
crystal stability. In the standard approach to trapped ion-crystals this
challenge comes about as a heating rate of the crystal that scales unfavorably
with $N$. Specifically, in un-segmented ion-crystals as $N$ increases, the
frequencies of low-lying long-wavelength modes of motion decrease, and become
more susceptible to electrical noise, leading to degradation of coherent
operations, destabilization of the ion-crystal and its eventual melting.
Utilizing optical trapping potentials generates cell-local heating rates,
which only depend on the cell size, $C$, and not on the total number of ions
$N$ (or $S$), potentially allowing for arbitrarily long ion-crystals.

This is shown graphically in Fig. 4, since for $\omega_{\text{o.t.p}}\geq
2\nu$ the low-lying COM band is lifted and its frequency converges to the COM
mode of an independent cell (inset), implying cell-dependent (and not
$N$-dependent) heating rates. The COM modes remain gapped regardless of $N$.

We directly quantify the phonon excitation rate of the optically-segmented
architecture. For a given motional mode, the phonon excitation Fermi’s golden
rule rate due to the presence of some electric field noise, $\delta E(t,r)$,
is given by [11],

$\Gamma^{(k)}=\frac{e^{2}}{4m\hbar\omega_{k}}S_{E}^{(k)}(\omega_{k}),$ (3)

with $\omega_{k}$ the frequency of mode $k$, and $S_{E}^{(k)}(\omega_{k})$ is
the spectral density of the electric field noise. The mode’s heating rate is
$\hbar\omega_{k}\Gamma^{\left(k\right)}$.

The precise form of the spectral density function is a subject of much
theoretical and experimental study [55, 56, 57, 58, 10, 11, 59], is system-
dependent and its theoretical limits remains somewhat inconclusive. Yet it is
generally agreed that it scales as,
$S_{E}(\omega)\propto\omega^{-\alpha}D^{-\beta}T^{\gamma}$, with mode
frequency $\omega$, ion-electrode distance $D$, and temperature, $T$, and
$\alpha$, $\beta$ and $\gamma$ are scaling exponents. Motivated by many
experimental results, here we assume that the noisy electric field is
spatially uniform along the ion-crystal and that $\alpha=1$.

Using these consideration we obtain the system’s excitation rate (see details
in the SM [46]),

$\Gamma=\sum_{k}\Gamma^{(k)}\propto\sum_{k}\frac{e^{2}}{4m\hbar\omega_{k}^{1+\alpha}}\sum_{i,j}A_{k}^{(i)}A_{k}^{(j)},$
(4)

with $\Gamma^{(k)}$ the excitation rate of the $k$th mode and $A$ an
orthogonal axial mode-matrix, such that $A_{k}^{(i)}$ is the participation of
the $i$th ion in the $k$th axial mode of motion, obtained by analyzing the
axial mode structure [54, 46].

Due to the summation over $k$, our formulation of $\Gamma$ in Eq. (4) is
extensive, e.g. for two identical and independent copies of an ion-crystal,
$\Gamma$ will be evaluated as twice the value of a single copy of the ion-
crystal. Thus ideal decoupling between cells in the optically segmented ion-
crystal is expected to show-up as proportionality of $\Gamma$ to $S$.

Figure 6 shows the predicted excitation rate of the optically segmented ion-
crystal, due to Eq. (4), for various values of the cell size, $C$ (horizontal
axis), and number of segments, $S$ (colors). The barrier size is set to
$B_{A}=3$ and each cell contains $C+B_{B}$ ions, with $B_{B}=3$. Excitation
rates are calculated for ion-crystals reaching $N>1300$ ions. The figure
shows, $\Gamma/\left(\Gamma_{\text{ind}}S\right)$ (vertical axis), such that
the excitation rates are normalized by the number of segments, $S$, and by the
excitation rate of an independent unsegmented ion-crystal containing $C+B_{B}$
ions, $\Gamma_{\text{ind}}$. We observe that the normalized excitation rates
are all of order unity and converge towards 1 as $C$ increases, with a
negligible dependence on the number of segments. Thus we conclude that the
cell’s stabilization is dominated by cell properties, and is specifically
independent from other cells in the ion-crystal.

The main contribution to $\Gamma$ is given by the lowest frequency band, which
is a COM motional band. This is intuitive as COM modes have the lowest
motional frequencies, as well as a high overlap with the uniform electric
field noise considered here, as compared to other modes which have negligible
contributions. This is seen in the dashed lines of Fig. 6 in which we
calculate $\Gamma$ using exclusively this single band (while
$\Gamma_{\text{ind}}$ is still calculated in full).

Figure 6: Axial excitation rate of optically segmented ion-crystals. The
excitation rates (vertical axis), $\Gamma$, are shown for various values of
cell sizes, $C$ (horizontal axis) and number of segments, $S$ (colors). We
normalize $\Gamma$ by the excitation rate of an independent single cell,
$\Gamma_{\text{ind}}$ and by the number of segments, $S$. All resulting
normalized rates are of order unity, showing that the stability of the
optically segmented ion-crystal is dictated by $C$, and is largely independent
of $S$. We repeat this process but with only considering the lowest band of
motional modes in the expression for $\Gamma$ (dashed), without varying
$\Gamma_{\text{ind}}$. The resemblance between this result and the full
expression of $\Gamma$ (solid) shows that the lowest-lying band is the
dominant contributor to the heating. This result is intuitive as this band is
a COM motional band, having a high-overlap with a uniform electric field.

##  VI Parallel multi-qubit logic and crosstalk mitigation

In trapped ions based quantum computers, logical gates between qubits, i.e.
entanglement, is typically facilitated by spin-dependent forces acting on
normal-modes of motion of the ion-crystal [60, 61]. Thus adequate control of
the normal-mode structure is crucial for scaling-up the trapped ion quantum
register size. Specifically, utilizing many modes of motion simultaneously
enables programmable long-range interactions [12, 13, 14], which are
significant for efficient implementation of various quantum computational
tasks [62, 63, 64, 65, 52]. However utilizing multiple modes of motion becomes
increasingly challenging in large ion-crystals due to the overwhelming
complexity of the required qubit drive, which comes about as a NP-hard
optimization problem [12], as well as an apparent slow-down of the feasible
gate duration [14] which, in absence of segmentation, scales as $N^{2}$. Thus
segmentation plays here a crucial role as well in generating independent
large, yet manageable, computational qubit cells.

The heuristic estimation for crosstalk error, provided by
$\varepsilon_{\text{BW,b}}$ above, can be improved by a detailed consideration
of the transverse mode-structure. Specifically, based on the quadratic-form of
qubit-qubit entanglement discussed thoroughly in [14], the coupling term
between ions $c$ ($c^{\prime}$) in cells $s$ ($s^{\prime}$), mediated by band
$b$, is estimated as

$J_{b}^{\left(c,s\right),(c^{\prime},s^{\prime})}=\sum_{m=1}^{S}R_{\left(b,m\right)}^{\left(c,s\right)}R_{\left(b,m\right)}^{(c^{\prime},s^{\prime})},$
(5)

with $R$ an orthogonal radial mode-matrix (similar to $A$ above) such that
$R_{\left(b,m\right)}^{\left(c,s\right)}$ is the participation of ion $c$ of
cell $s$ in mode $m$ of band $b$ (specific examples of
$J_{b}^{\left(c,s\right),(c^{\prime},s^{\prime})}$ are shown in the SM [46]).
Using $J_{b}$ above, we form a more detailed crosstalk estimation, namely,

$\varepsilon_{J,b}=\frac{\max\limits_{c\neq c^{\prime},s\neq
s^{\prime}}\left|J_{b}^{\left(c,s\right),(c^{\prime},s^{\prime})}\right|}{\max\limits_{c\neq
c^{\prime},s}\left|J_{b}^{\left(c,s\right),(c^{\prime},s)}\right|},$ (6)

with $s,s^{\prime}$ maximized over all $S$ cells in the segmented ion-crystal
and $c,c^{\prime}$ maximized over all $C$ computational qubits in each of the
cells. In essence the estimation in Eq. (6) compares the coupling between ions
in the same cell ($s=s^{\prime}$) to coupling between ions in difference cells
($s\neq s^{\prime}$). This estimation is also shown in the inset (top-left) of
Fig. 5 (blue), exhibiting a similar behavior to that of
$\varepsilon_{\text{BW},b}$, i.e. $\sim 2.5\%$ crosstalk mainly due to long-
wavelength motional excitations of the cells. We also evaluate separately the
contribution of nearest-neighbor cells, $\varepsilon_{J,b}^{\text{nn}}$, which
follows the same definition as $\varepsilon_{J,b}$ in Eq. (6) but with
$s^{\prime}=s\pm 1$ in the numerator. This estimation is shown as well in the
inset (top-left) of Fig. 5 (green stars), showing an almost exact agreement
with $\varepsilon_{J,b}$. Thus we conclude that the $\sim 2.5\%$ crosstalk is
mainly due to unwanted coupling between adjacent segmented cells. Further
analysis shows that in general 2-3 barrier ions, each confined with an optical
trapping frequency of $\omega_{\text{o.t.p}}\approx\omega_{\text{rad}}$ are
sufficient for maintaining low crosstalk levels (see the SM [46]).

A few percent crosstalk level, reflected by both $\varepsilon_{\text{BW},b}$
and $\varepsilon_{J,b}$, marks an almost complete decoupling between cells.
Nevertheless, it is too high for high-fidelity quantum operations, and
restricts performing simultaneous logical gate on distinct cells. However this
crosstalk level is low enough such that it can be mitigated perturbatively in
a scalable manner. Here we specifically rely on the large-scale fast (LSF)
method, presented in Ref. [14].

In essence, LSF makes use of a multi-tone drive in order to generate
programmable multi-qubit gates, which naively involves solving a quadratically
constrained optimization problem of dimension $N^{2}$, and generates the
unitary gate,
$\exp\left(\sum_{n,m=1}^{N}\varphi_{n,m}\sigma_{x}^{\left(n\right)}\sigma_{x}^{\left(m\right)}\right)$,
with $\sigma_{x}^{\left(n\right)}$ a Pauli-$x$ operator acting on the $n$th
qubit and $\varphi_{n,m}$ a target multi-qubit coupling matrix. The hardness
of the optimization problem is mitigated in LSF with two key realizations.
First, for each fixed system any non-vanishing drive solution to a special
‘zero-phase’ version, $\varphi^{\left(0\right)}_{n,m}=0$, of the optimization
problem can be efficiently converted to a high-fidelity solution of any
general problem. Second, the zero-phase solutions (ZPS) can be obtained by
considering homogeneous illumination of the ion-crystal, resulting in a
dimension $N$ quadratic problem.

The operational approach of LSF is naturally adopted to our architecture,
making the design of multi-qubit entanglement, simultaneously in all cells,
fast and scalable. Specifically, we use LSF in order to generate ZPSs for an
effective ‘typical’ cell of the bulk of the ion-crystal (see the SM [46]). Up
to small deviations that are corrected below, this ZPS is adequate to all of
the ion-crystal’s cells. Using LSF, different multi-qubit gates are designed
for each cell by converting the same ZPS. This classical computation can be
performed fast and in parallel. After performing these steps we obtain a
specific drive spectrum, for all computational and auxiliary qubits in the
ion-crystal, that generates the required entanglement per-cell at once,
$\exp\left(\sum_{s=1}^{S}\sum_{n,m=1}^{C}\varphi_{n,m}^{\left(s\right)}\sigma_{x}^{\left(n,s\right)}\sigma_{x}^{\left(m,s\right)}\right)$,
with $\sigma_{x}^{\left(n,s\right)}$ a Pauli-$x$ operator acting on the $n$th
qubit in cell $s$ and $\varphi_{n,m}^{\left(s\right)}$ the desired multi-qubit
target of cell $s$.

As discussed above, the unwanted coupling between cells incurs a $\sim 2.5\%$
crosstalk error, which is dominated by nearest-neighbor cells. Here this
brings about a typical coupling of $\sim 0.025\varphi_{n,m}^{\left(s\right)}$
between ions in distinct cells. Therefore we make use of an additional
mitigation layer on top of LSF. Crucially, the nearest-neighbor structure of
the crosstalk enables our mitigation technique to scale favorably, in terms of
classical computation resources, and does not impose constraints on the ion-
crystal size.

Specifically, we perform an iterative optimization by linearizing the
quadratic constraints around the current solution, with the initial solution
given by LSF as discussed above. The resulting linear equations provide
conditions that resolve crosstalk as well as inaccuracies of the LSF solution
that originate from the assumption of a typical cell. We do so locally, i.e.
the linear conditions are formed for only two adjacent cells at a time,
generating $C\left(C-1\right)$ linear constraints that improve the LSF results
and $3C^{2}$ linear constraints that correspond to crosstalk terms. The
iterative optimization is stopped when a target infidelity is reached.
Crucially since the linear equations are local, relating only four adjacent
cells at a time, they involve only $\mathcal{O}\left(C^{2}\right)$ linear
constraints, allowing parallel optimization of the next four adjacent cells
[46].

An additional condition for a high-fidelity process is decoupling of the
qubits from the modes of motion at the end of the entanglement operation. This
is typically satisfied by forming constraints for each of the $N$ modes of
motion independently. Here however such an approach will be a hurdle to
scalability since it will be infeasible to satisfy a large $N$ number of
constraints. Instead, we make use of only $C+B_{A,B}$ linear constraints of
the typical cell, and supplement these with additional robustness properties
that make the decoupling insensitive to the mode frequency inaccuracies,
described in Ref. [66], such that each constraint enforces the decoupling of
an entire band at once.

Figure 7: Performance of the crosstalk mitigation method as a function of
optical trapping potential $\omega_{\text{o.t.p}}$, evaluated on the
entanglement used to implement a simulation of a 3d rectangular Ising model.
Top: Error per cell, using Eq. (7), as a proxy for the entanglement infidelity
of the entire multi-qubit operation. For
$\omega_{\text{o.t.p}}\gtrsim\omega_{\text{rad}}$ the crosstalk compensated
solution (dark-blue) shows a low, $<10^{-4}$, infidelity. As expected, the
initial LSF solution, based on the effective typical cell (orange) shows a
$\sim 10^{-2}$ infidelity. The mitigation degrades in the weak optical
confinement regime, due to long-range coupling between cells that is not
considered in the crosstalk mitigation method. Indeed this degradation appears
when considering long-range cells (dashed purple), and not seen when only
considering nearest-neighbor and onsite coupling (dashed pink). Additional
infidelity due to residual coupling to motional modes (dashed light-blue)
exhibits a similar behavior. Bottom: The required total drive coupling, given
in units of the lowest transverse mode frequency (bottom). The crosstalk
compensated solution (dark-blue) only incurs a small, $\sim 10\%$, overhead on
the initial LSF solution (orange).

We study the efficacy of our crosstalk mitigation method by considering a
segmented ion-crystal system used for a 3d Ising model simulation, similar to
that shown in Fig. 2 above, with $S=5$, $d=3$ and $B_{A}=B_{B}=3$. We design
the control pulses that implement the required qubit-qubit couplings, for
various optical trapping frequencies. The resulting operation infidelity is
evaluated, to leading order in the qubit-qubit coupling deviations and in
residual ion displacements, as,

$I=I_{\text{short}}+I_{\text{long}}+I_{\text{motion}}$ (7)

such that $I_{\text{short}}$ approximates the infidelity due an erroneous
qubit-qubit coupling within each cell and crosstalk to nearest-neighbor cells,
$I_{\text{long}}$ is the infidelity due to qubit-qubit coupling between cells
that are not nearest-neighbors and $I_{\text{motion}}$ approximates the
infidelity due to residual coupling between the qubits and the phonon modes of
motion.

Figure 7(top) presents the infidelity, $I$, of the initial LSF solution, based
on the typical cell (orange) and the crosstalk corrected solution (dark-blue).
It exhibits an apparent cross-over between two regimes at
$\omega_{\text{o.t.p}}=\omega_{\text{rad}}$, at which point the transverse
mode band structure is manifested. At low optical confinement the performance
of the uncorrected, typical-cell solution (orange) is low, such that our
leading-order estimation breaks-down. At strong optical confinement this
solution generates an infidelity of $\sim 10^{-2}$, as expected from our
analysis above. However the crosstalk-corrected solution (dark-blue)
successfully generates solutions that have an infidelity that is better
(lower) than $10^{-4}$ for the entire multi-qubit operation, throughout the
strong optical confinement regime.

We account for the different contributions to the crosstalk-corrected
solution, given by Eq. (7). We note that $I_{\text{short}}$ (dashed pink), has
a negligible contribution, regardless of optical confinement, while
$I_{\text{long}}$ (dashed purple) converges to low values only at the strong
confinement regime. This behavior is expected as our optimization actively
corrects for on-site and nearest-neighbor terms, and ignores other terms. This
implies that strong optical confinement successfully decouples next nearest-
neighbor (and further) cells, removing the need to take the entire ion-crystal
into account, and enabling scalability. Similarly, due to a well-formed band
structure, the motional infidelity, $I_{\text{motion}}$ (dashed light-blue),
becomes small at strong optical confinement, and in general dominates the
overall infidelity.

Figure 7(bottom) shows the total drive coupling, i.e. Rabi frequency, required
to drive the initial LSF solution (orange) and the crosstalk corrected
solution (dark-blue), given in terms of the frequency of the lowest radial
mode, $\nu_{1}$, as a characteristic scale. The compensated solution incurs a
small overhead of $\sim 10\%$ on the initial solution.

We remark that our architecture and crosstalk compensation method can also be
implemented by global driving of entire cells using similar techniques, such
as in Refs. [12, 14].

##  VII Mid-circuit measurements

The ability to perform mid-circuit projective measurements, and apply coherent
feedback based on the measured results, is at the heart of QEC, as well as
additional central quantum computational tools [67, 68, 69]. In trapped-ions
based systems, projective state detection is typically performed by state-
dependent florescence. This poses a technical challenge, as the scattered
photons usually heat-up the ion-crystal due to photon recoil. Furthermore the
photon may be resonant with neighboring ions and scattered by them, resulting
in decoherence of the quantum state of the entire system. Thus, so far, mid-
circuit measurements have been implemented in trapped ions either in small
ion-crystals, which are well separated from other parts of the quantum
register, by ‘shelving’ all computational qubits to non-resonant atomic
states, or by using two different atomic species spectraly separating logical
and incoherent operations [70, 71, 72, 73].

In our optically segmented ion-crystal scheme, optically confined ions are
well-suited for mid-circuit measurements, as their motion is separated to
independent bands (top part of the spectra in Figs. 4 and 5), thus photon
scattering heating remains local and does not heat the other crystal modes.
Furthermore, the optical fields generating the trapping also substantially
light-shift the atomic transition lines of the ions, specifically the photon
emission lines, such that photon scattering is largely suppressed. Thus no
physical shuttling of the ions into dedicated measurement regions is required,
nor the use of multiple ion-species.

Figure 8: Top: schematic of mid-circuit measurement protocol, showing three
types of involved ions, computational qubit ions, C (green), barrier ion to be
measured, B (light-green) and optically confined barrier ions, A (purple).
Operations are ordered from left to right. Data is encoded on ion C (vertical
line), and one of the qubit states of the ion is ‘shelved’ to a non-
fluorescing state using local control. Next an intermediate optical
segmentation configuration is used, confining both ions A and B, and
separating their motion from the computational qubits. The state of the
barrier ions is measured (‘Detect’) and then reset (‘Prepare’). Lastly the
optical segmentation configuration is switched. Bottom: Relevant atomic levels
of ${}^{40}\text{Ca}^{+}$ ions, showing $4S_{1/2}$ the ground state manifold,
the metastable $4D_{5/2}$ manifold and the short-lived $4P_{1/2}$ manifold.
The $S\leftrightarrow P$ transition varies between the no-optical confinement
(left) and optically confined (right) cases, allowing to separate the two
cases spectrally (see main text). The main optical field used in the protocol
are shown, namely at $729$ nm (red), coupling the $S\leftrightarrow D$ levels,
at $400$ nm (dark-blue), generating Raman transition between qubit states as
well as additional local light-shifts, and at $397$ nm (light-blue) coupling
the $S\leftrightarrow P$ levels.

We describe the protocol used to measure the qubit state of some target ions,
without incurring decoherence on computational qubits, or heating of bulk
modes. After the measurement is completed, we cool and re-initialize the
measured ions. Figure 8 (top) illustrates these general steps involved in
performing mid-circuit measurements. For simplicity we consider the required
operations on three types of ions in the ion-crystal, i.e. computational qubit
ions, ‘C’ (green), barrier ions to be measured, ‘B’ (light-green) and
optically confined barrier ions, ‘A’ (purple). Crucially, our measurement
method can be performed in parallel throughout the entire ion-crystal on all
required ions. In our architecture ions may be optically confined specifically
for the purpose of mid-circuit measurements, without having an essential role
in segmentation of the ion-crystal, as showcased in the application of a QEC
surface code, above.

For concreteness we consider a specific realization, in trapped
${}^{40}\text{Ca}^{+}$ ions. Furthermore we consider a local control over a
single wavelength, at $\sim 400$ nm, which couples the two qubit states
defined in the $4S_{1/2}$ ground state manifold via Raman transitions mediated
by the short lived $4P$ manifolds. This local control, acting independently on
all ions, is primarily used for generating single qubit rotations and multi-
qubit programmable gates, however here it is utilized as a means to ‘localize’
global control fields using light-shifts, detailed below. While additional
local controls, that can be used in our scheme, will act to simplify the
resulting mid-circuit measurement protocol, we show that a single local
control remarkably suffices to facilitate all requirements.

The main atomic levels used for standard qubit operation are presented in Fig.
8 (bottom) in the case of no optical confinement (left) and with optical
confinement (right). The ground state manifold is coupled to the narrow meta-
stable $4D_{5/2}$ manifold via a $729$ nm field (red arrow). In the case of no
optical confinement, this transition can be localized by light-shifting the
$4S_{1/2}$ manifold of a target ion by using the locally controlled $400$ nm
field (dark-blue arrow). For these purposes a small, e.g. few MHz light-shift
suffices. On the other hand, optical confinement strongly light-shifts the
$4S_{1/2}\leftrightarrow 4P_{1/2}$ level (right), making the $729$ nm coupling
off resonant. In principle, this strong shift can be taken into account,
however since the $S\leftrightarrow D$ transition is narrow it imposes
stringent requirements on the intensity stabilization of the optical
confinement. Here we simply do not make use of the $S\leftrightarrow D$
transition for optically confined ions relaxing the requirements for intensity
stabilization.

However, Raman transitions within the qubits subspace, mediated by the $400$
nm field, are negligibly affected by optical confinement, and exclusively in
terms of their Rabi frequency.

An additional field resonant with the $S\leftrightarrow P$ transition at $397$
nm is used (light-blue). Its frequency does require adjustment to the light-
shift due to optical confinement, however the $P$ level is broad, thus the
required stability of the optically confining fields is reasonable. The
combination of local $400$ nm and $397$ nm fields, together with standard
repump fields (not shown), adjusted as well for light-shifts, accommodates for
all the necessary steps, i.e., state detection, various cooling techniques and
state preparation.

The mid-circuit protocol shown in Fig. 8 is carried out by first encoding some
information on ion B via the entanglement operations detailed above (vertical
line connecting ions C and B), e.g. parity information of a plaquette shown in
Fig. 3. Then, one of the qubit states of ion B is ‘shelved’ to the D manifold
via a combination of $400$ nm light-shift and $729$ nm coupling, making this
operation local on the ion.

Next, the segmentation configuration is changed to an intermediate setting
where all barrier ions are illuminated. The purpose of optically confining ion
B is to prepare it for being measured, by separating its motion from the
computational qubit’s motion and to light-shift its $S\leftrightarrow P$
transition. However the confinement of ion A cannot be relaxed since ion B
might fully populate the $D$ level, in which case it is not optically
confined. State detection is performed using the light-shifted $397$ nm field,
followed by qubit reset and preparation using a combination of the $397$ nm
and $400$ nm fields.

During the detection and preparation steps all optically confined ions scatter
$397$ nm photons, such that all of them are required to be cooled down in the
‘prepare’ step, which can be done in parallel. The scattered photons are
substantially detuned, by approximately $2$ GHz, from the absorption lines of
the computational qubit ions, having a linewidth of few tens of MHz, thus
negligibly influencing their state. Nevertheless the state of neighboring
computational qubits, located few micrometers apart, may be further protected
by shelving both of their qubit states to the $D$ level (at the shelving
step). Alternatively they may simply remain unutilized.

Lastly the segmentation configuration can now be changed, exchanging the roles
of ions ‘A’ and ‘B’. The cooled ‘A’ ions are returned to the bulk of the cells
and cool-down the bulk modes via sympathetic cooling.

##  VIII Summary

Here we propose a scalable architecture for quantum computing; it is based on
a large register of trapped-ion qubits together with dynamically operated
optical potentials. Our proposed architecture circumvents the two most
prominent challenges in working with ever-larger ion-crystals – prohibitively
high heating rates and spectral crowding of the ions’ motional modes. It does
so by effectively segmenting an arbitrarily large trapped ion-crystal into
several independent segments of a manageable size. Despite the local structure
that results, connectivity across the full trapped ion-crystal is enabled by
rapidly reconfiguring the optical potentials. The optical potentials further
enable mid-circuit measurements of the confined ions, followed by classical
feedback.

The utility of this architecture is emphasized when combined with a method for
programmable multi-qubit entangling gates, such as that proposed in Ref. [14].
We have used this method to numerically study the application of independent
multi-qubit unitaries on each cell in parallel and in a scalable manner.
Moreover we have extended this method to enable arbitrarily good compensation
of crosstalk errors that arise between adjacent segments.

Our architecture requires modest hardware resources and makes use off well-
established experimental techniques; it is thus an ideal platform for quantum
simulation, quantum computation, as well as ultimately for fault tolerant
quantum computing.

###### Acknowledgements.

This work was supported by the Israel Science Foundation Quantum Science and
Technology (Grants 2074/19, 1376/19 and 3457/21).

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##  IX Supplemental material

###  IX.1 Mode structure calculations

Mode structure, i.e. motional mode frequencies and mode participation matrices
are generated by diagonalizing a classical Hamiltonian. Specifically we
consider an isospaced RF trapped ion-crystal, of ions with mass $m$ and charge
$e$. Optical trapping potentials are taken into account by incorporating an
additional on-site potential term for illuminated ions. In all of our examples
in the main text and below we have considered equally-spaced trapped
${}^{40}\text{Ca}^{+}$ ions, with an inter-ion distance of $5\text{
}\mu\text{m}$ and a $400\text{ nm}$ optical field that generates the
additional confinement and segmentation.

We assume the ion-crystal forms a stable linear chain, such that axial and
transverse modes can be considered independently. Thus, following Ref. [54],
for axial modes we write the secular matrix,

$V_{n,m}^{\text{axial}}=\begin{cases}-2\frac{eE_{\text{n.n}}}{d\left|n-m\right|^{3}}&n\neq
m\\\
4\zeta\left(3\right)\frac{eE_{\text{n.n}}}{d}+\frac{1}{2}m\omega_{\text{o.t.p}}^{2}b_{n}&n=m\end{cases},$
(8)

with $b_{n}=1$ ($b_{n}=0$) if ion $n$ is (not) illuminated by an optical
potential, $d$ the distance between ions, $\zeta\left(3\right)\approx 1.202$
is Apéry’s constant, $E_{\text{n.n}}=\frac{e}{4\pi\epsilon_{0}d^{2}}$ is the
electric field strength created by nearest-neighbor ions and $\epsilon_{0}$ is
the vacuum permittivity. The eigenvalues of $V^{\text{axial}}$ are
$\frac{1}{2}m\omega_{\text{a}}^{2}$, with $\omega_{\text{a}}$ the axial
motional mode frequencies. The eigenvectors are normal-modes of motion
associated with the corresponding frequency, designated in the main text as
the mode-matrix $A$.

Similarly, transverse (radial) modes are calculated by diagonalizing the
matrix,

$V_{n,m}^{\text{radial}}=\begin{cases}\frac{eE_{\text{n.n}}}{d\left|n-m\right|^{3}}&n\neq
m\\\
-2\zeta\left(3\right)\frac{eE_{\text{n.n}}}{d}-\frac{1}{2}m\left(\omega_{\text{rad}}^{2}+\omega_{\text{o.t.p}}^{2}b_{n}\right)&n=m\end{cases},$
(9)

with $\frac{1}{2}m\omega_{\text{rad}}^{2}$ the radial trapping potential
generated by the RF trap, which in the analysis presented here is fixed at
$\omega_{\text{rad}}=3$ MHz. Figures 4 and 5 of the main text are generated by
computing and diagonalizing the matrices in Eqs. (8) and (9).

The resulting transverse mode-matrix $R$, is the orthogonal matrix that
diagonalizes the secular matrix in Eq. (9) and is used in the main text to
estimate crosstalk between segments. This is realized by considering the mode-
dependent qubit-qubit coupling (Eq. (5) of the main text),

$J_{b}^{\left(c,s\right),(c^{\prime},s^{\prime}}=\sum_{m=1}^{S}R_{\left(b,m\right)}^{\left(c,s\right)}R_{\left(b,m\right)}^{(c^{\prime},s^{\prime})},$
(10)

Figure 9 highlights the resulting structure of the $J_{b}$’s for several
bands, with the horizontal and vertical axes representing ion indices within
the ion-crystal (excluding $B_{A}$ barrier ions). Specifically the ‘zig-zag’,
$b=0$, band (left) the COM, $b=34$, band (right) and an intermediate, $b=17$,
band (middle). Couplings are shown in a symlog scale (colors represent a
logarithmic scale in both the positive and negative directions from the
origin, with an interval between $\pm 10^{-6}$ that is linearly scaled). All
of the presented coupling maps show considerable inter-cell coupling and
negligible crosstalk, i.e. coupling between ions in different segments, with
nearest-neighbor segments constituting the most relevant correction.
Furthermore long-wavelength, i.e. high $b$, modes generate more crosstalk, as
is also shown in Fig. 5 of the main text.

Figure 9: Band-dependent qubit-qubit coupling, $J_{b}$. Horizontal and
vertical axes are indices of qubits within the entire ion-crystal (excluding
$B_{A}$ barrier ions). Couplings are shown in a symlog scale (colors represent
a logarithmic scale in both the positive and negative directions from the
origin, with an interval between $\pm 10^{-6}$ that is linearly scaled). The
optical segmentation generates a block-diagonal structure, that signifies
strong coupling within each cell and negligible coupling between qubits in
different cells. Couplings due to three bands are shown, the ‘zig-zag’ $b=0$
band (left), the COM $b=34$ band (right) and an intermediate band (center).
Clearly crosstalk, seen as coupling outside of the diagonal blocks, is more
relevant for high $b$, long wavelength modes.

###  IX.2 Excitation rate calculations

In order to evaluate the excitation rate of a given axial mode of motion we
recall Eq. (3) of the main text, i.e. in presence of some electric field
noise, $\delta E(t,r)$, the excitation rate for a given motional mode is [11],

$\Gamma^{(k)}=\frac{e^{2}}{4m\hbar\omega_{k}}S_{E}^{(k)}(\omega_{k}),$ (11)

with $\omega_{k}$ the frequency of mode $k$ and
$S_{E}^{\left(k\right)}\left(\omega_{k}\right)$ the spectral density of the
electric field noise, given by,

$S_{E}^{\left(k\right)}\left(\omega_{k}\right)=2\sum\limits_{i,j}\int
d\tau\langle\delta E_{k}^{i}\left(\tau\right)\delta
E_{k}^{j}\left(0\right)\rangle e^{-i\omega_{k}\tau}.$ (12)

Here $\delta E_{k}^{i}\left(\tau\right)$ is the projection of $\delta E$ on
the $k$th mode of motion at the position of ion $i$, i.e. $\delta
E_{k}^{i}\left(t\right)=\delta E(t,r_{i})A_{k}^{\left(i\right)}$ and $r_{i}$
is the position of the $i$th ion.

We assume that the leading order contribution to $\delta E$ is spatially
uniform and neglect all other contributions. Furthermore, motivated by
theoretical and experimental results we assume that
$S_{E}^{\left(k\right)}\propto\omega_{k}^{-\alpha}$, with $\alpha$ here set to
1.

This considerably simplifies $\Gamma^{\left(k\right)}$ and enables us to
obtain the total axial excitation rate shown in Eq. (4) of the main text, up
to a proportionality factor,

$\Gamma\propto\sum\limits_{k}\Gamma^{\left(k\right)}\propto\sum\limits_{k}\frac{e^{2}}{4m\hbar\omega_{k}^{2}}\sum\limits_{i,j}A_{k}^{\left(i\right)}A_{k}^{\left(j\right)}.$
(13)

Crucially, the effect of optical segmentation comes about spectrally in the
formation of bands, such that the summation on $k$ in Eq. (13) can be
decomposed as,

$\Gamma\approx\sum\limits_{b=1}^{B+C}\frac{e^{2}}{4m\hbar\omega_{b}}\sum\limits_{s=1}^{S}\sum\limits_{i,j}A_{b,s}^{\left(i\right)}A_{b,s}^{\left(j\right)}+\mathcal{O}\left(\frac{\text{BW}_{b}}{\Delta\omega_{b}}\right),$
(14)

with $A_{b,s}^{\left(i\right)}$ a re-arrangement of $A$ according to the band
structure and the error term due to the bandwidth of the bands,
$\text{BW}_{b}$ and the gap between bands, $\Delta\omega_{b}$, which is
assumed to be small. From a mode-structure point of view, the optical
segmentation comes about as an independent motion between adjacent cells
(shown in Fig. 9), thus the sum on $s$ restricts $i$ and $j$ to the same cell,
leading to a linear scaling with $S$.

###  IX.3 Crosstalk mitigation method

We provide further details on our approach to mitigating crosstalk between
cells in the optically-segmented ion-crystal. In order to do so we recall some
basic facts about the operation of the LSF method, which is discussed in
detail in Ref. [14]. We assume the ions are driven by local fields that, in
the spectral domain, all have the same $M$ tone pairs, but can vary in the
amplitude of each pair. Thus the drive is described by a set of vectors
$\textbf{r}_{n}$ such that $\left(\textbf{r}_{n}\right)_{m}$ is the amplitude
of the $m$th tone pair that drives the $n$th ion. This choice of degrees of
freedom allows to generate a bi-partite multi-qubit entanglement operation of
the form,

$U=\exp\left(\sum\limits_{n,m=1}^{N}\varphi_{n,m}\sigma_{x}^{\left(n\right)}\sigma_{x}^{\left(m\right)}\right),$
(15)

with $\varphi_{n,m}$ a completely controlled ‘target’ operation and
$\sigma_{x}^{\left(n\right)}$ a Pauli-$x$ operator acting on the $n$th ion.

Indeed, generating $U$ can be mapped to finding solutions (of the
$\textbf{r}_{n}$’s) to the quadratic constraints,

$\textbf{r}_{n}^{T}A_{n,m}\textbf{r}_{m}=\varphi_{n,m}\quad 1\leq n<m\leq N,$
(16)

with $A_{n,m}$ a set of real-valued $M\times M$ symmetric matrices that
quantify the coupling between ions $n$ and $m$ due to the drive’s spectral
components. Specifically these coupling matrices are given by,

$A_{n,m}=\sum\limits_{j=1}^{N}R_{j}^{\left(n\right)}R_{j}^{\left(m\right)}A_{j},$
(17)

with $A_{j}$ a set of real-valued $M\times M$ symmetric matrices that quantify
the coupling between the drive’s spectral components and mode $j$ [12].
Typically we will be interested in solutions to small norm $\textbf{r}_{n}$’s
that satisfy Eq. (16), which constitutes a NP-hard optimization problem.

In LSF we make use of a given non-trivial solution,
$\textbf{r}_{n}=\textbf{z}$ for all $n=1,..,N$, that generates the target
$\varphi_{n,m}^{\left(0\right)}=0$ for all $n$ and $m$, coined the zero-phase
solution (ZPS). Then, solutions to other ‘full’ arbitrary targets can be
efficiently converted from $\bm{z}$. Crucially the ZPS does not depend on the
ion index, $n$, and thus can be found directly by considering the $A_{j}$’s
(instead of the $A_{n,m}$’s), reducing the number of quadratic constraints
that are required to be satisfied.

We utilize these concepts and apply them to the optically-segmented ion-
crystal. Generally we consider a ‘typical’ effective cell as an independent
ion-crystal and find a ZPS for it. We then convert the ZPS to solutions to
full targets, which reflect the required operations on the different cells,
i.e. $U_{s,n}^{\left(A,B\right)}$ in Fig. 1 of the main text. By simply
setting these solutions as the drive of the optically segmented ion-crystal we
obtain entanglement gates of the form of Eq. (15), up to $\sim 10^{-2}$
infidelity, which arises mainly due to crosstalk between adjacent cells, but
also due to using a typical cell system (e.g. cells at the edge of the ion-
crystal are slightly different than in the bulk).

To construct a typical cell system of the ion-crystal, we consider a
fictitious system that has $C+B_{B}$ equally spaced ions with corresponding
transverse motional modes. The mode frequencies are constructed from the
average frequency of each band in the motional spectrum band structure of the
segmented ion-crystal. The mode participation matrix is constructed by
similarly averaging the participation of each ion in each mode, partitioned to
cells. The segmented ion-crystal has $S$ modes per band (and not a single
mode), which needs to be taken in to account by the typical cell system.
Indeed we do so by writing our drive degrees of freedom in a way that is
resilient to small inaccuracies in the motional frequency [74, 66, 75].

This construction, along with other fixed system parameters such as the
entanglement gate time, ion type etc., is sufficient in order to construct
$A_{j}$’s and $A_{n,m}$’s of a typical cell, and to generate ZPS of it. We
remark that we do not make use of tones that lie within bands since these will
generate a differential coupling between the different modes within a band,
and are not well approximated by the typical cell system.

We use LSF to convert the typical cell ZPS to different target cell solutions
for the various cells, and obtain a set of amplitudes,
$\bm{r}_{s,c}^{\left(0\right)}$ corresponding to the drive of qubit $c$ in
cell $s$. The qubit-qubit coupling between ions $c$ ($c^{\prime}$) in cell $s$
($s^{\prime}$), ‘inherited’ from the typical LSF solution, can be evaluated
as,
$\varphi_{\left(s,c\right),\left(s^{\prime},c^{\prime}\right)}^{\left(i\right)}=\left(\textbf{r}_{s,c}^{\left(i\right)}\right)^{T}A_{\left(s,c\right),\left(s^{\prime},c^{\prime}\right)}\bm{r}_{s^{\prime},c^{\prime}}^{\left(i\right)}$,
with designating the LSF solution as $i=0$.

The resulting couplings are compared to the intended target,
$\Delta\varphi_{\left(s,c\right),\left(s^{\prime},c^{\prime}\right)}^{\left(i\right)}=\varphi_{\left(s,c\right),\left(s^{\prime},c^{\prime}\right)}^{\left(i\right)}-\varphi_{\left(s,c\right),\left(s^{\prime},c^{\prime}\right)}^{t}$,
with the ‘t’ inscription referring to the required ideal target dictated by
the intended unitary operator, such that,
$\varphi_{\left(s,c\right),\left(s^{\prime}\neq s,c^{\prime}\right)}^{t}=0$.
Nearest-neighbor cell crosstalk errors are given by,
$\Delta\varphi_{\left(s,c\right),\left(s\pm
1,c^{\prime}\right)}^{\left(i\right)}$, and target inaccuracies are given by,
$\Delta\varphi_{\left(s,c\right),\left(s,c^{\prime}\right)}^{\left(i\right)}$.

Since the total infidelity is small, crosstalk and target inaccuracies can be
mitigated by linearizing the quadratic constraints in Eq. (16) and iteratively
improving the resulting fidelity. Specifically we consider an iteration of the
form
$\bm{r}_{s,c}^{\left(i+1\right)}=\bm{r}_{s,c}^{\left(i\right)}+\bm{\delta}_{s,c}^{\left(i+1\right)}$,
starting from $i=0$. We construct a set of linear equations for the correction
$\bm{\delta}^{\left(i+1\right)}$. In order to make this technique scalable we
want to avoid considering all $N$ ions in the ion-crystal in the same set of
linear equations. This is made possible by that fact that crosstalk is
dominated by coupling to nearest-neighbor cells (see Fig. 5 in the main text
and Fig. 9).

Specifically we focus on two adjacent ‘target’ cells, $s$ and $s+1$, and
derive linear equations for them. These are,

$\displaystyle\left(\bm{\delta}_{s,c}^{\left(i+1\right)}\right)^{T}A_{\left(s,c\right),\left(s,c^{\prime}\right)}\bm{r}_{s,c^{\prime}}^{\left(i\right)}+\left(\bm{\delta}_{s,c^{\prime}}^{\left(i+1\right)}\right)^{T}A_{\left(s,c\right),\left(s,c^{\prime}\right)}\bm{r}_{s,c}^{\left(i\right)}$
$\displaystyle=\Delta\varphi_{\left(s,c\right),\left(s,c^{\prime}\right)}^{\left(i\right)}$
(18)
$\displaystyle\left(\bm{\delta}_{s+1,c}^{\left(i+1\right)}\right)^{T}A_{\left(s+1,c\right),\left(s+1,c^{\prime}\right)}\bm{r}_{s+1,c^{\prime}}^{\left(i\right)}+\left(\bm{\delta}_{s+1,c^{\prime}}^{\left(i+1\right)}\right)^{T}A_{\left(s+1,c\right),\left(s+1,c^{\prime}\right)}\bm{r}_{s+1,c}^{\left(i\right)}$
$\displaystyle=\Delta\varphi_{\left(s+1,c\right),\left(s+1,c^{\prime}\right)}^{\left(i\right)}$
(19)
$\displaystyle\left(\bm{\delta}_{s,c}^{\left(i+1\right)}\right)^{T}A_{\left(s,c\right),\left(s+1,c^{\prime}\right)}\bm{r}_{s+1,c^{\prime}}^{\left(i\right)}+\left(\bm{\delta}_{s+1,c^{\prime}}^{\left(i+1\right)}\right)^{T}A_{\left(s,c\right),\left(s+1,c^{\prime}\right)}\bm{r}_{s,c}^{\left(i\right)}$
$\displaystyle=\Delta\varphi_{\left(s,c\right),\left(s+1,c^{\prime}\right)}^{\left(i\right)}$
(20)
$\displaystyle\left(\bm{\delta}_{s,c}^{\left(i+1\right)}\right)^{T}A_{\left(s,c\right),\left(s-1,c^{\prime}\right)}\bm{r}_{s-1,c^{\prime}}^{\left(i\right)}$
$\displaystyle=\Delta\varphi_{\left(s,c\right),\left(s-1,c^{\prime}\right)}^{\left(i\right)}$
(21)
$\displaystyle\left(\bm{\delta}_{s+1,c}^{\left(i+1\right)}\right)^{T}A_{\left(s+1,c\right),\left(s+2,c^{\prime}\right)}\bm{r}_{s+2,c^{\prime}}^{\left(i\right)}$
$\displaystyle=\Delta\varphi_{\left(s+1,c\right),\left(s+2,c^{\prime}\right)}^{\left(i\right)}.$
(22)

with $c,c^{\prime}=1,..,C$. The first (second) row accounts for inaccuracies
of the target implemented on cell $s$ ($s+1$), the third row accounts for
crosstalk between cells $s$ and $s+1$, and the forth (fifth) row accounts for
crosstalk between cell $s$ ($s+1$) and cell $s-1$ ($s+2$). The choice to
correct the drive of two adjacent cells simultaneously is since this
construction allows crosstalk between cells $s$ and $s+1$ to be mitigated
simultaneously by both $\bm{\delta}_{s,c}$ and $\bm{\delta}_{s+1,c^{\prime}}$,
which has been found to be more efficient than the correction of the crosstalk
with control over a single qubit drive, e.g. the terms in the fourth and fifth
rows.

Crucially all of these terms result in $C\left(C-1\right)+3C^{2}$ linear
constraints, and are independent of the total number of segments. Moreover
iterative optimizations can be performed by considering independent blocks of
4 adjacent cells, that can be efficiently parallelized and interlaced. We
remark that an additional linear constraint can be added in order to minimize
the amplitude of the $\bm{r}_{n}$’s [14].

We perform $f$ optimization iterations, until convergence of the solution, or
until meeting a fidelity criteria. We use the resulting solutions and
benchmark the expected fidelity of the corresponding entanglement operations.
The resulting infidelity is evaluated, in leading order, as,

$\displaystyle I$
$\displaystyle=I_{\text{short}}+I_{\text{long}}+I_{\text{motion}}$ (23)
$\displaystyle I_{\text{short}}$
$\displaystyle=2\sum_{s=1}^{S}\sum_{c,c^{\prime}=1}^{C+B_{B}}\left(\left|\Delta\varphi_{\left(s,c\right),\left(s,c^{\prime}\right)}^{\left(f\right)}\right|^{2}+\left|\Delta\varphi_{\left(s,c\right),\left(s+1,c^{\prime}\right)}^{\left(f\right)}\right|^{2}+\left|\Delta\varphi_{\left(s,c\right),\left(s-1,c^{\prime}\right)}^{\left(f\right)}\right|^{2}\right)$
(24) $\displaystyle I_{\text{long}}$
$\displaystyle=2\sum_{s=1}^{S}\sum_{s^{\prime}\neq s,s\pm
1}^{S}\sum_{c,c^{\prime}=1}^{C+B_{B}}\left|\Delta\varphi_{\left(s,c\right),\left(s^{\prime},c^{\prime}\right)}^{\left(f\right)}\right|^{2}$
(25) $\displaystyle I_{\text{motion}}$
$\displaystyle=\sum_{j=1}^{N}\sum_{n=1}^{N}\left|\alpha_{j}^{\left(n\right)}\right|,$
(26)

with $I_{\text{short}}$ representing the residual unwanted coupling between
qubits that is accounted for by our mitigation iterations, $I_{\text{long}}$
representing the residual unwanted coupling between qubits that is not
accounted for by our mitigation, and $I_{\text{motion}}$ accounting for
unwanted residual displacement of the $j=1,...,N$ motional modes. Here
$\alpha_{j}^{\left(n\right)}$ is the displacement of mode $j$ due to ion $n$
and is thus linearly related to the $\bm{r}_{n}$s and can be easily evaluated
[12, 14]. While we are not directly controlling and minimizing the latter term
in our optimization iterations, the resilience to motional frequency errors of
the typical cell, discussed above, ensures that it remains small.

###  IX.4 Crosstalk and power analysis

We analyze the dependence of the crosstalk, in its mode-structure dependent
formulation, $\varepsilon_{\text{J},b}$ in Eq. (6) of the main text, on the
choice of the number of barrier ions per cell, $B_{A}$, and on the optical
confining power used per barrier ion,
$\omega_{\text{o.t.p}}/\omega_{\text{rad}}$. The comparison to the radial RF
trapping frequency, $\omega_{\text{rad}}$, is a natural scale for analyzing
crosstalk in entanglement operations that utilize transverse modes of motion.

Figure 10: Crosstalk dependence on $\omega_{\text{o.t.p}}$ and $B$. Left: Mean
crosstalk (colored regions) for varying number of barrier ions (horizontal
axis) and varying optical confinement, $\omega_{\text{o.t.p}}$, per barrier
ions, given in units of the RF radial trapping, $\omega_{\text{rad}}$
(vertical axis). We observe that a perturbative crosstalk, set for values
below 0.1 (area above white dashed line), is generated by setting
$\omega_{\text{o.t.p}}\geq 0.75\omega_{\text{rad}}$, almost regardless of $B$.
Nevertheless increasing $B$ is helpful in reducing the crosstalk. The total
optical power per cell is proportional to
$P=B\omega^{2}_{\text{o.t.p}}/\omega^{2}_{\text{rad}}$ (colored lines, with
the values of $P$ in the legend). Right: Estimation of band-dependent
crosstalk, $\varepsilon_{J,b}$, based on mode structure analysis in an ion-
crystal with cell size of $C=32$ qubits and $S=4$ segments, and a varying
number of barrier ions, $B_{A}=B_{B}$ (color). In general the addition of
barrier ions decreases the overall crosstalk in the system. Limiting cases are
found for $B_{A}=1,2$ in which some of the collective modes do not involve the
barrier ions, resulting in a non-typical crosstalk structure. Clearly for
$B_{A}\geq 3$ this behavior is suppressed. The mean crosstalk of the curves
with of $1\leq B\leq 5$ at $\omega_{\text{o.t.p}}=\omega_{\text{rad}}$ are
correspondingly shown in the left.

Figure 10 (left) shows the mean crosstalk averaged over all bulk-bands,
$\langle\varepsilon_{\text{J},b}\rangle_{b}$ (color), for a segmented ion-
crystal made of $S=4$ segments, each containing $C=32$ computational qubits
and $B$ (horizontal axis) barrier ions, such that each barrier ion is
illuminated by an optical confining potential with a trapping frequency
$\omega_{\text{o.t.p}}$ (vertical axis). Clearly, as the optical trapping
frequency decreases the mean crosstalk increases, and passes the perturbative
limit, set here to $0.1$ (white dashed). We also present constant lines of the
expression, $P=B\omega_{\text{o.t.p}}^{2}/\omega_{\text{rad}}^{2}$, as it is
proportional to the total required optical power per-cell (colored lines, see
value of $P$ in the legend).

Figure 10 (right) shows $\varepsilon_{J,b}$ for $C=32$, $S=4$ and various
values of $B_{A}$, between 1 and 14 (color), such that for all $B_{A}$ barrier
ions $\omega_{\text{o.t.p}}/\omega_{\text{rad}}=1$. The curves show expected
crosstalk due to $b=1,..,C+B_{B}$ bands, with $B_{B}=B_{A}$, similarly to the
inset of Fig. 5. The high-index cell bands are typically the main contributors
to the crosstalk as they correspond to long-wavelength excitations of the
cells, analogues to low-order oscillating multipoles, thus having a stronger
coupling to adjacent cells. Clearly, a larger barrier reduces the overall
crosstalk. Nevertheless, the curves do not show a monotonic behavior, which is
especially obvious with $B_{A}=1$ (blue) and $B_{A}=2$ (orange), which exhibit
resonant-like features. These resonances are likely due to the modes of motion
in which the barrier-ions only weakly participate and therefore cannot isolate
the motion of ions within a single cell. For $B_{A}\geq 3$ this effect is
generally suppressed, leading to only a residual distance-dependent Coulomb
interaction between the cells. Furthermore, since our crosstalk mitigation
technique relies on linearization, we have to work in the perturbative
crosstalk regime, which is already shown to apply at $B_{A}\geq 2$, motivating
our choices in the main text.