# QUANTUM COMPUTING WITH NEUTRAL ATOMS

Loïc Henriet  Pasqal, 2 avenue Augustin Fresnel, 91120 Palaiseau, France
Lucas Beguin  Pasqal, 2 avenue Augustin Fresnel, 91120 Palaiseau, France
Adrien Signoles  Pasqal, 2 avenue Augustin Fresnel, 91120 Palaiseau, France
Thierry Lahaye  Université Paris-Saclay, Institut d’Optique Graduate School,
CNRS, Laboratoire Charles Fabry, 91127 Palaiseau Cedex, France  Pasqal, 2
avenue Augustin Fresnel, 91120 Palaiseau, France  Antoine Browaeys  Université
Paris-Saclay, Institut d’Optique Graduate School, CNRS, Laboratoire Charles
Fabry, 91127 Palaiseau Cedex, France  Pasqal, 2 avenue Augustin Fresnel, 91120
Palaiseau, France  Georges-Olivier Reymond  Pasqal, 2 avenue Augustin Fresnel,
91120 Palaiseau, France  Christophe Jurczak  Pasqal, 2 avenue Augustin
Fresnel, 91120 Palaiseau, France  Quantonation, 58 rue d’Hauteville, 75010
Paris, France

###### Abstract

The manipulation of neutral atoms by light is at the heart of countless
scientific discoveries in the field of quantum physics in the last three
decades. The level of control that has been achieved at the single particle
level within arrays of optical traps, while preserving the fundamental
properties of quantum matter (coherence, entanglement, superposition), makes
these technologies prime candidates to implement disruptive computation
paradigms. In this paper, we review the main characteristics of these devices
from atoms / qubits to application interfaces, and propose a classification of
a wide variety of tasks that can already be addressed in a computationally
efficient manner in the Noisy Intermediate Scale Quantum[1] era we are in. We
illustrate how applications ranging from optimization challenges to simulation
of quantum systems can be explored either at the digital level (programming
gate-based circuits) or at the analog level (programming Hamiltonian
sequences). We give evidence of the intrinsic scalability of neutral atom
quantum processors in the 100-1,000 qubits range and introduce prospects for
universal fault tolerant quantum computing and applications beyond quantum
computing.

##  1 Introduction

One promising solution to the shortcomings of classical computing systems
consists in the utilization of specialized accelerators for dedicated
purposes. Examples of this type of computing systems include Graphics
Processing Units (GPU), originally designed to accelerate graphics tasks like
image rendering, or Field-Programmable Gate Arrays (FPGAs). Both technologies
now provide high performance pipelines for efficient and high-speed
computations, and are widely used as application-specific co-processors. Other
types of processors are emerging with a special focus on acceleration of
machine learning tasks.

In this respect, quantum co-processors, which manipulate the information at
the quantum level, hold great promise for the future. Unlike classical
information carried by digital bits 0 or 1, quantum information is commonly
encoded onto a collection of two-level quantum systems referred to as qubits.
A register containing $n$ qubits is described by a large complex vector of
dimension 2n. By taking advantage of the exponential amount of information
available in quantum registers through superposition and entanglement, several
quantum algorithms have been proposed that could outperform state-of-the-art
classical algorithms for specific computing tasks. The most prominent example
is Shor’s factoring algorithm. While factoring a large integer $N$ into the
product of its prime factors is a very hard task on a classical computer, it
can in theory be done quite efficiently on a quantum device. More
specifically, Shor’s algorithm runs in a time increasing as a polynomial of
$\log N$, whereas the best classical algorithm for this tasks requires sub-
exponential time.

In such quantum algorithms, computation is described as a sequential
application of quantum logic gates onto the qubits. This so-called digital
approach to quantum information processing presents several key advantages,
such as universality, as any operation can always be re-written with a finite
subset of basis gates, and cross-device compatibility. Although the actual
implementation of these quantum algorithms is still years away, as they would
require ideal digital quantum processors, active research is currently
exploring the capabilities of currently available quantum devices, which are
imperfect and comprise a relatively modest number of qubits. These devices
have recently shown that their computing capabilities can outperform classical
supercomputers for a specific computing task [2]. This opens up the so-called
Noisy Intermediate Scale Quantum (NISQ) computing era [1], where 50-1,000
qubits devices allow for the exploration of the entanglement frontier, beyond
what classical computers can do.

In order to build a viable quantum processor, a broad variety of physical
platforms are currently being investigated [3]. Among them, arrays of single
neutral atoms manipulated by light beams appear as a very powerful and
scalable technology to manipulate quantum registers with up to a few thousands
of qubits [4, 5]. In such quantum processors, each qubit is encoded into two
electronic states of an atom. Nature provides that all the qubits are strictly
identical when taken independently, as opposed to artifical atoms such as
superconducting circuits or Silicon spin qubits that need to be manufactured
with as little heterogeneity as possible. This feature is a remarkable
advantage for neutral atom quantum processors to achieve low error rates
during the computation. Atomic devices present other clear advantages with
respect to other platforms, such as a large connectivity and the ability to
natively realize multi-qubit gates.

In addition to the digital mode where the time evolution of qubits is
described by quantum gates, control over the device can be realized at the so-
called analog level, where the user can directly manipulate the mathematical
operator (the Hamiltonian) describing the evolution of the ensemble of atoms.
Not only does it allow for a finer level of control over pulses during the
application of gates, but it also makes it possible to directly use the
Hamiltonian of the system as a resource for computation. The fine level of
control allowed by this analog setting, together with the large number of
possible configurations, makes it a powerful tool for quantum processing.

Figure 1: Quantum stack and metrics to evaluate the performances of the
quantum processor. Applications (top block of the Quantum stack) can be
divided into two groups: Quantum Simulation problems that involve the study of
a quantum system, and standard computational problems. To solve these
problems, a quantum processor can either be used in a digital way or in an
analog way (middle block of the Quantum stack). The low-level part of the
stack corresponds to the physical quantum processor.

Fundamental research on quantum information processing platforms using neutral
atoms has been going on for years, and has led to impressive scientific
results such as the simulation of highly complex quantum systems well above
100 qubits, beyond the reach of classical high performance computers [6]. Only
recently did it become possible to contemplate manufacturing devices for
commercial use thanks to continuous progress in design and engineering. This
technology will be described in more detail in Section 2, with a particular
emphasis on the technological choices taken by the team of A. Browaeys and T.
Lahaye in an academic setting and at the company Pasqal.

Such a powerful machine can already bring value to several different fields,
which will be described in Section 3 (see top block of Fig. 1). The most
promising application is Quantum Simulation, where the quantum processor is
used to gain knowledge over a quantum system of interest. As Richard Feynman
already pointed out in the last century, it seems natural to use a quantum
system as a computational resource for quantum problems. Pure science
discovery will benefit from neutral atom quantum processors, and fields of
applications are numerous at the industrial level, including for example the
engineering of new materials for energy storage and transport, or chemistry
calculations for drug discovery. Beyond quantum simulation, neutral atom
quantum devices can also be used to solve hard computational problems, that
cannot be efficiently solved on classical devices. Applications of NISQ
computing notably encompass finding approximate solutions to hard
combinatorial optimization problems [7], or enhancing the performances of
Machine Learning procedures [8].

The multiple technologies that will be developed will in the longer term be
leveraged for the development of fault tolerant quantum computers, but also
quantum networking and metrology. These perspectives will be briefly touched
upon in Section 4.

##  2 Neutral atoms arrays

In this Section, we review the key technological blocks of neutral atom
quantum processors, based on configurable arrays of single neutral atoms. The
array can be seen as a register, where each single atom plays the role of a
qubit. One common choice is rubidium atoms, a very common species in atomic
physics that benefits from well-established technological solutions,
especially in terms of lasers. More precisely, two electronic levels of the
rubidium atoms are chosen to be the two qubit states, which we refer to as
$\ket{0}$ and $\ket{1}$. Since the number of electronic states in an atom is
infinite, there are various possible choices for implementing the qubit,
leading to a very rich variety of configurations.

Light is the main tool to control both the position and the quantum state of
the atoms. It is used to:

  * •

assemble and read-out registers made of hundreds of qubits (see Section 2.1),

  * •

perform fully programmable quantum processing (see Section 2.2).

For each dedicated task, a laser with a specific wavelength is required. In
addition, electronic controls are needed to tune the light properties, apply
instructions arising from the quantum software stack and extract information
through atomic detection, as illustrated in Fig. 2.

Figure 2: Schematic of the hardware components of a neutral atom quantum
device. The user sends, through the quantum software stack, instructions to
the register actuators, which initialize the quantum register, and to the
processing actuators, which perform the computation. Information in the
quantum register is extracted through detection of an image. It serves as an
input for real-time rearranging of the register and as an output of the
computation.

###  2.1 Operating an atomic qubit register

Unlike in solid quantum processors, e.g., superconducting, silicon or NV
qubits, the register in an atomic QPU is not permanently built but is
reconstructed after each processing. Hence, a typical computation cycle
consists in three phases: register preparation, quantum processing and
register readout. Although the processing stage is the one that receives most
of the attention because it is where entanglement plays a role, the two other
stages are of fundamental interest as well for developing a quantum processor.
Indeed, highly efficient preparation and readout of the register have a direct
impact on the performance of a quantum computing device.

####  2.1.1 Register loading

To prepare a register made of neutral atoms, one can use arrays of optical
tweezers. It uses the hardware components shown in Fig.3(a). As a starting
point, a dilute atomic vapor is formed inside an ultra-high vacuum system
operated at room temperature. With a first laser system (not shown), a cold
ensemble of about $10^{6}$ atoms and 1 mm3 volume is prepared inside a 3D
magneto-optical trap (3D MOT), leveraging numerous laser cooling and trapping
techniques [9]. Then, a second trapping laser system isolates individual atoms
within this ensemble. Using high numerical aperture lenses, the trapping beam
gets strongly focused down to multiple spots of about 1$\,\mu$m diameter: the
so-called optical tweezers [10]. Since the spots are only 10 mm away from the
lenses, the latter are placed inside the vacuum chamber (see Fig. 3(b)).
Within a trapping volume of a few $\mu$m3, each tweezer contains at most one
single atom at a time.

Figure 3: (a) Overview of the main hardware components constituting a quantum
processor. The trapping laser light (in red) is shaped by the spatial light
modulator (SLM) to produce multiple microtraps at the focal plane of the lens
(see inset). The moving tweezers (in purple), dedicated to rearranging the
atoms in the register, are controlled by a 2D acousto-optic laser beam
deflector (AOD) and superimposed on the main trapping beam with a polarizing
beam-splitter (PBS). The fluorescence light (in green) emitted by the atoms is
split from the trapping laser light by a dichroic mirror and collected onto a
camera. (b) Photography of the heart of a neutral-atom quantum co-processor.
The register is prepared at the center of this setup.

The number of tweezers and their arrangement in any arbitrary 1D, 2D or even
3D geometries is fully tailored by holographic methods [11]. Before passing
through the lens, the trapping beam is reflected onto a spatial light
modulator (SLM) that imprints an adjustable phase pattern on the light. In the
focal plane of the lens, the phase modulation gets converted into a desired
intensity pattern, as illustrated in Fig. 4. For this reason, neutral atom
platforms for quantum processing have a unique potential for scalability: the
size of the quantum register is only limited by the amount of trapping laser
power and by the performance of the optical system generating the optical
tweezers.

Figure 4: Examples of tweezer arrays in 2D or 3D geometries, extracted from
Ref [12].

Although each tweezer can host at most one atom, it happens only in about 50%
of the cases, the tweezer being empty otherwise. To detect which of the
tweezers are filled, the atoms are imaged by collecting their fluorescence
onto a sensitive camera (see Fig.3(a)). The atoms are then moved from site to
site in order to generate a pre-defined sub-register with unit filling. This
operation is done using programmable moving optical tweezers, with a success
rate above $99\%$. Elementary rearrangement steps are described in Fig.5(a).
From the analysis of the initial image, an algorithm computes on the fly a set
of moves to rearrange the initial configuration into the desired fully
assembled sub-register. To implement this active feedback within few tens of
milliseconds, the data are transferred through FPGA and the algorithm is run
on an external GPU. The moves are then communicated to the drivers of 2D
acousto-optic deflectors (AOD) that control the pointing and intensity of the
moving tweezers. After rearrangement is completed, an assembled image is
acquired to confirm the new positions of the atoms in the sub-register.

####  2.1.2 Register readout

Once the register is fully assembled, quantum processing can start. The next
section describes in detail how to realize the required quantum operations.
The processing by itself is extremely rapid since it happens in less than 100
$\mu$s, while the overall sequence, including loading and readout, lasts
approximately 200 ms.

Once processing is done, the atomic register is read out by taking a final
fluorescence image. It is performed such that each atom in qubit state
$\ket{0}$ will appear as bright, whereas atoms in qubit state $\ket{1}$ remain
dark, as illustrated in Fig.5(b). All the images are acquired on an electron-
multiplying charge-coupled-device (EMCCD) camera: it converts with a very high
sensitivity fluorescence photons into a bunch of electrons that produce a
detectable electronic signal. Efficiencies of more than $98.6\%$ have been
reported for this detection method [13], a value comparable to processors
based on superconducting qubits [14]. Given the probabilistic nature of each
possible outcome imposed by quantum mechanics, such computation cycles are
then repeated many times in order to reconstruct the relevant statistical
properties of the final quantum state produced by the algorithm.

Figure 5: (a) Moving a single atom from one site to another (both in red) in
the register. The moving optical tweezer (in purple) first takes the atom,
then transfers it and finally releases it into the other site. This operation
takes less than 1 ms. (b) Temporal sequence of one computation cycle. The
loading of the register being random, atoms are first rearranged to realize a
defect-free sub-register, on which the quantum processing is performed.

###  2.2 Quantum processing with atomic qubits

Neutral atom quantum processors (QPU) are able to implement both digital and
analog quantum processing tasks. In digital computing, quantum algorithms are
decomposed into a succession of quantum logic gates, described by a quantum
circuit as shown in Fig. 6(a). The quantum gates are realized by shining fine-
tuned laser pulses onto a chosen subset of individual atoms in the register.
In analog computing, lasers are applied to realize an Hamiltonian. The qubits
evolve in time according to the Schrödinger equation, as illustrated in Fig.
6(b). The final state of the system is probed by measuring the state of each
individual qubits. Implementation of digital and analog computing tasks are
discussed in this section.

Figure 6: Digital- vs analog processing. (a) In digital processing, a
succession of gates is applied to the qubits to implement a quantum algorithm.
Each gate is performed by addressing the qubits individually with laser beams.
(b) In analog processing the qubits evolve under a tailored Hamiltonian
$\mathcal{H}$, for instance by illuminating the whole register with a laser
beam. The wavefunction $\ket{\psi}$ of the system follows the Schrödinger
equation.

####  2.2.1 Digital quantum processing

Digital computing requires qubits robust against decoherence, i.e., weakly
coupled to their environment. Therefore one can use as a qubit the two
hyperfine ground states ($F=1$ and $F=2$) of the rubidium atom, because they
have very long lifetimes (of the order of tens of years) that prevent
radiative coupling to the electromagnetic environment. As discussed earlier,
gates are performed with laser beams. The spacing between atoms in the
register being typically several micrometers, specific qubits can be addressed
with high accuracy by strongly focusing the lasers (as illustrated in the
inset of the CNOT gate in Fig. 6(a)). Interestingly, one- and two-qubit gates
are all that is needed to constitute a universal gate set. For example, the
ability to realize arbitrary single qubit rotations and the well-known two-
qubit CNOT gate is sufficient for the realization of any quantum algorithm
[15].

Figure 7: Single- vs two-qubit gates. (a) Illustration of a NOT gate. When the
addressed spin is in the state $\ket{0}$ (top sphere), the laser field rotates
it by an angle of $\pi$ to end up in the state $\ket{1}$. When starting in
$\ket{1}$, it ends up in $\ket{0}$ (bottom sphere). (b) According to the state
of the control qubit (left column, in red), the target qubit (right column, in
blue) is rotating by $\pi$ or not (illustrated in the case of a target qubit
initially in $\ket{0}$.

One-qubit gates are specific unitary transformations described by 2-by-2
complex matrices transforming one qubit state into another. Notable examples
are the NOT-gate that changes the state $\ket{0}$ into $\ket{1}$ and vice-
versa, or the Hadamard (H) gate that generates superposition of both states
starting from a pure state. In the $\\{\ket{0},\ket{1}\\}$ basis, theses gates
read

$\displaystyle\text{NOT}=\begin{bmatrix}0&1\\\ 1&0\end{bmatrix}\hskip
113.81102pt\text{H}=\frac{1}{\sqrt{2}}\begin{bmatrix}1&1\\\
1&-1\end{bmatrix}.$

A nice way to illustrate how a quantum gate modifies the state of a qubit is
to use the Bloch sphere picture. In this picture, a qubit pure state is
represented by a 3D unitary vector, the Bloch vector, pointing somewhere on
the surface of a unit sphere, the Bloch sphere. The state $\ket{0}$ occupies
the North pole and the state $\ket{1}$ the South pole, and any point at the
surface of the sphere is associated with another unique qubit state, being a
coherent superposition of $\ket{0}$ and $\ket{1}$. As unitary transformations,
any single-qubit gate corresponds to a rotation of the Bloch vector onto the
Bloch sphere. In this respect, they can always be written as combinations of
the Pauli matrices $\sigma^{x},\,\sigma^{y}$ and $\sigma^{z}$. Regarding the
previous examples, the NOT-gate coincides with $\sigma^{x}$ and is a rotation
by a $\pi$ angle around the $x$ axis (see Fig. 7(a)), while the Hadamard-gate
writes $(\sigma^{x}+\sigma^{z})/\sqrt{2}$ and is a rotation of $\pi$ around
the $(x+z)$ axis.

Arbitrary rotations of the qubit state on the Bloch sphere can be performed by
driving the qubit transition with a control field. In our device the latter is
an optical laser field driving Raman transitions through an intermediate
atomic state [16, 17]. The atom-laser interaction is characterized by the Rabi
frequency $\Omega$ (its strength, proportional to the amplitude of the laser
field), the detuning $\delta$ (the difference between the qubit resonance and
the field frequencies) and their relative phase $\varphi$. Driving the control
field for a duration $\tau$ induces rotations around the $(x,y,z)$ axes with
angles $(\Omega\tau\cos{\varphi},\Omega\tau\sin{\varphi},\delta\tau)$. Hence,
any single-qubit gate can be implemented by tuning the pulse duration, the
laser intensity, the detuning and the phase of the laser. These parameters are
controlled using direct digital synthesizers (DDS) that drive acousto- and
electro-optic modulators (AOM and EOM) placed on the laser beams. As an
illustration, when driving the control field at resonance ($\delta=0$), the
qubit oscillates in time between the states $\ket{0}$ and $\ket{1}$. These so-
called Rabi oscillations are a general feature of quantum two-level systems,
and therefore serve as a test for controlling the efficiency of single-qubit
gates. In atomic registers they have been observed with an extremely high
contrast [17, 18], corresponding to single-qubit gate fidelities $\mathcal{F}$
higher than $99.5\%$. The fidelity $\mathcal{F}$ of a quantum operation is a
number between 0 and 1 that measures the closeness between the state actually
created by the operation and the theoretically expected state. A fidelity
$\mathcal{F}=99.5\%$ corresponds to a probability of $0.5\%$ of measuring an
undesired outcome after the operation. When stopped after half an oscillation,
i.e. for a pulse area equal to $\pi$, it exactly realizes a NOT gate. On the
other hand, the Hadamard gate requires a $\pi$ pulse at a detuning equal to
the Rabi frequency ($\delta=\Omega$) and the laser phase equal to 0.

Box 1 $|$ Implementation of the CNOT-gate

The CNOT gate flips a “target” qubit state if and only if a “control” qubit is
in the state $\ket{1}$. The corresponding matrix in the pair state basis
$\\{\ket{0_{c}0_{t}},\ket{0_{c}1_{t}},\ket{1_{c}0_{t}},\ket{1_{c}1_{t}}\\}$ is

$\displaystyle\text{CNOT}=\begin{bmatrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&0&1\\\
0&0&1&0\end{bmatrix}.$

There are several ways to realize this gate using dipolar Rydberg interaction
[19, 17]. Figure B1.1(a) illustrates the key mechanism with a simple sequence
of 3 pulses.

  * •

With initial state $\ket{1_{c}1_{t}}$, all the three pulses are off-resonant
and the state remains unchanged.

  * •

With initial state $\ket{0_{c}1_{t}}$, pulse 2 is off-resonant so the target
qubit state remains unchanged. Pulses 1 and 3 drive a 2$\pi$ rotation of the
control qubit. The pair state picks up a phase factor $e^{i\pi}=-1$.

  * •

With initial state $\ket{1_{c}0_{t}}$, pulses 1 and 3 are off-resonant so the
control qubit state remains unchanged. Pulse 2 drives a 2$\pi$ rotation of the
target qubit. The pair state picks up a phase factor $e^{i\pi}=-1$.

  * •

With initial state $\ket{0_{c}0_{t}}$, pulse 1 excites the control qubit to
state $\ket{r}$ with a $\pi$ rotation. Because of the Rydberg blockade, the
target qubit Rydberg state $\ket{r}$ is shifted out of resonance during pulse
2. The target qubit remains in state $\ket{0_{t}}$. Finally, pulse 3 brings
the control qubit back to state $\ket{0_{c}}$ with another $\pi$ rotation. The
pair state picks up a factor $e^{i\pi}=-1$.

This pulse sequence realizes, within a global $\pi$ phase, the controlled-Z
(CZ) gate

$\displaystyle\begin{bmatrix}-1&0&0&0\\\ 0&-1&0&0\\\ 0&0&-1&0\\\
0&0&0&1\end{bmatrix}=e^{i\pi}\begin{bmatrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&1&0\\\
0&0&0&-1\end{bmatrix}=e^{i\pi}\text{CZ}.$

The CZ gate can then be used to generate the CNOT gate by including Hadamard
gates on the target atom before and after the operation, as shown in
Fig.B1.1(b).

Figure B1.1 $|$ Implementation of the CNOT gate using Rydberg interactions.
(a) Principle of the controlled-Z gate based on dipolar Rydberg interaction.
First a $\pi$ pulse is applied on the control atom, then a 2$\pi$ pulse on the
target atom, and finally another $\pi$ pulse on the control one. (b)
Realization of a CNOT gate using a CZ gate and two Hadamard gates.

Two-qubit gates are unitary transformations described by 4-by-4 matrices that
transform one two-qubit state into another. They are the most basic but
crucial resource allowing the generation of entanglement in the register.
Physically, their implementation requires an interaction between the qubits.
However, neutral atoms in their electronic ground state can only interact
significantly via contact collisions. Therefore, single atoms –typically
separated by a few micrometers in the register– do not naturally feel each
other. In 2000, Jaksch et al. [20] proposed a scheme that takes advantage of
the dipole-dipole interaction between atoms when they are prepared to highly
excited electronic states. In these so-called Rydberg states, the atoms
exhibit a huge electric dipole moment that can be three orders of magnitude
bigger than in their ground state. Therefore, two Rydberg atoms –separated by
a few micrometers– will experience a dipole-dipole interaction strong enough
to shift significantly the energy of the doubly excited state, preventing the
excitation of two atoms at the same time. This effect is called Rydberg
blockade and is the basic mechanism to achieve a quantum logic: the excitation
of a first atom to a Rydberg state conditions the excitation of a second one.

Figure 8: Implementation of random quantum circuits on three distinct device
architectures with 48 qubits, illustrated on the left. The orange layout
displays nearest-neighbor connectivity in 2D, while the green layout has an
interaction radius (Rydberg blockade radius) of 2.3 units of lattice spacing.
The red layout corresponds to a 3D cubic lattice, with the same interaction
radius. These characteristics result in a large difference in terms of the
final number of gates necessary to realize quantum circuits. This is shown on
the right panel, where we plot the average gate count overhead with respect to
the number of two-qubit gates. The data points were obtained using Google Cirq
qubit routing compiler, and each point corresponds to an average over 64
random circuits.

Hence, Rydberg-mediated entanglement is particularly well suited to implement
the controlled-NOT (CNOT) gate (see Fig. 7(b)). Following the protocol
described in Box 1, the gate can be performed within 1 $\mu$s, leading to a 1
MHz clock rate for the QPU. Transient non-zero population of Rydberg states
may lead to errors in the process. Current imperfection levels in the
experiment allow us to perform more than a hundred gates within the typical
coherence time of the system. The experimentally achieved fidelities have been
measured to be $\mathcal{F}=94.1\%$ [21] and will be largely improved. In
particular, imperfections can be compensated by implementing optimal control
[22]. The goal is to affect the dynamics in a controlled way in order to
increase the probability of reaching the desired quantum state. It is
performed by shaping the optical pulses in time instead of applying basic
squared pulses. It calls specific algorithms that optimized the shapes to
maximize the gate fidelity. Practically it requires extremely fast arbitrary
waveform generators that are able to sample the pulses at the 100 ns
timescale.

The ability to achieve two-qubit gates between distant qubits has critical
consequences on the implementation of long quantum circuits. When a quantum
processor suffers from a limited connectivity, gates that are required for the
implementation of a given algorithm cannot be implemented natively.
Circumventing this problem requires the introduction of SWAP gates [15], that
exchange locally the quantum state between different qubits. This gate
addition comes at a high cost, as each additional gate introduces errors along
the computation. The relatively large connectivity of neutral atom devices
allows to mitigate this effect.

To illustrate this strong advantage, we use Cirq built-in qubit routing
compiler on a variety of random circuits to be implemented either on a device
with nearest-neighbor connectivity such as most transmon architectures for
superconducting qubits, or on a neutral atom device in 2D and 3D. Results are
displayed on Fig. 8, where we see that the overhead for neutral atom devices
(green points for the 2D device and red points for the 3D device) is 5 to 10
times lower than the one obtained with a device with nearest-neighbor
connectivity.

In addition to large connectivity, neutral atom devices present the advantage
of natively implementing multi-qubit gates involving more that 2 qubits [4, 5,
17, 23]. Such multi-qubit gates are instrumental for the efficient
implementation of several quantum algorithms, including for example Grover
search or the variational resolution of non-linear partial differential
equations [24]. In that respect, neutral atom quantum processors are for
example able to realize a three-qubit Toffoli gate, which corresponds to a NOT
operation applied on a target qubit conditioned on the states of two control
qubits, at a modest cost in terms of operations to be performed by the
processor. More specifically, a Toffoli gate can be achieved with a sequence
of only 7 pulses (2 Hadamard gates on both sides of a 5-pulse CCZ, as
illustrated on Fig. 9 (a) and (b)). On platforms that cannot natively
implement a Toffoli gate, one has to combine not less than six Controlled-NOT
gates and 9 single-qubit gates to achieve the same effect, leading to a very
large overhead as illustrated on Fig. 9.

Figure 9: (a) Sequence of 5 pulses realizing a CCZ gate. (b) Decomposition of
the Toffoli gate (CCNOT) into a CCZ surrounded by two single-qubit gates. (c)
The Toffoli gate can be decomposed into a minimum of six CNOTs, with
additional single qubits gates (Hadamard gates H and T ($\pi/8$) gates). This
last sequence if equivalent to a total of more than 30 laser pulses.

These properties of large connectivity and native multi-qubit gates, specific
to Rydberg-based gates, help reducing drastically the overall duration of the
processing. It is of fundamental interest since, for the current generation of
devices, the clock rate (relative to the computation, 1 MHz) and the
acquisition rate (relative to the measurement, 5 Hz) in a neutral atom QPU are
relatively small in comparison to other quantum computing platforms. Together
with the 2D and 3D capabilities, all these features contribute to making
neutral atom QPUs highly competitive. On top of that, what constitutes the
main advantages of neutral atom QPUs is their very high scalability: having
more qubits simply relies on the capability of increasing the number of
tweezers in the interference pattern of the laser, but does not involve
manufacturing new chips as it is the case with solid-state platforms where a
key challenge is the reproducibility of each single qubit over a large device.

####  2.2.2 Analog quantum processing

Neutral atoms arrays are also suitable to implement quantum Hamiltonians and
realize analog quantum processing. Rydberg atoms behaving as huge electric
dipoles, they experience dipole-dipole interactions that map into spin
Hamiltonians. Each qubit of the register then behaves as a spin whose states
are $\ket{\downarrow}=\ket{0}$ and $\ket{\uparrow}=\ket{1}$. Depending on the
Rydberg states that are involved in the process, the spins experience
different types of interactions that translate into different Hamiltonians
[6]. The most studied one is the Ising model, which is obtained when the
$\ket{\downarrow}$ state is one of the ground states and the $\ket{\uparrow}$
a Rydberg state [25, 26, 27, 28]. In this case the Hamiltonian writes

$\mathcal{H}(t)=\frac{\hbar}{2}\Omega(t)\sum_{j}\sigma_{j}^{x}-\hbar\delta(t)\sum_{j}n_{j}+\sum_{i\neq
j}\frac{C_{6}}{r_{ij}^{6}}n_{i}n_{j},$ (1)

with $n_{j}=(1+\sigma_{j}^{z})/2$ the Rydberg state occupancy and
$\sigma^{x,z}_{j}$ the Pauli matrices $\sigma^{x,z}$ of the spin $j$. The
first terms are induced by the laser that couples the qubit states and relate
to an effective magnetic field, with transverse and longitudinal components
$B_{\perp}\propto\Omega(t)$ and $B_{||}\propto-\delta(t)$. Physically, these
two terms describe how individual magnets would precess in the presence of an
external magnetic field. The parameters which control this precession are
$\Omega(t)$ and $\delta(t)$, respectively the Rabi frequency and detuning.
They can be varied by changing the intensity and frequency of the laser field.
The third term in (1) relates to the interactions between individual spins.
More specifically, it corresponds to an energy penalty that two qubits
experience if they are both in the Rydberg states at the same time. The
coupling between two spins $i$ and $j$ is of van der Waals type and depends on
the inverse of the distance between them $r_{ij}$ to the power of 6, and on a
coefficient $C_{6}$ relative to the Rydberg state (see Box 2).

The Ising Hamiltonian is a prototypical model that tackles many problems in
condensed matter. For example, this is a model that describes how quantum
magnets evolve at very low temperatures in material sciences. Interestingly,
it is also relevant for a broad variety of classical applications as it can
cast optimization problems that are computationally difficult to solve [29],
as further described in section 3. In neutral atom devices, it can be
implemented in 1D, 2D or 3D arrays containing few hundreds of atoms, far
beyond the computational capabilities of classical computers.

Box 2 $|$ Dipolar vs van der Waals interactions

Atoms in Rydberg states exhibit huge electric dipole moments. Two Rydberg
atoms thus interact via the electric dipole-dipole interactions, whose
Hamiltonian $V_{dd}$ is proportional to $1/R^{3}$, with $R$ the distance
between the two atoms [30].

At the first order of perturbation theory, two pair states $\ket{\psi}$ and
$\ket{\psi^{\prime}}$ are coupled only if the matrix element
$\bra{\psi}V_{dd}\ket{\psi^{\prime}}$ is not vanishing. It is the case when
considering for instance the states $\ket{\psi}=\ket{nS,n^{\prime}P}$ and
$\ket{\psi^{\prime}}=\ket{n^{\prime}P,nS}$, where $n$, $n^{\prime}$ are
principal quantum numbers, and S and P represent states with orbital angular
momentum $L$ respectively equals to 0 and 1. Since these two states have the
same energy, a pair of atoms initially prepared in the state
$\ket{nS,n^{\prime}P}$ will evolve into the state $\ket{n^{\prime}P,nS}$ and
back: the two atoms exchange energy corresponding to the difference between
the energy of the states $\ket{nS}$ and $\ket{n^{\prime}P}$. This exchange
mechanism, also known as "flip-flop", exhibits a $1/R^{3}$ scaling law and is
called dipolar interactions.

When pair states of same energy are not directly coupled to each other by
$V_{dd}$, they can interact via second-order processes. A common example is
the coupling of the pair state $\ket{nS,nS}$. It is dipole-coupled to states
with opposite parity, such as $\ket{n^{\prime}P,n^{\prime\prime}P}$, which are
separated in energy by $\Delta(n^{\prime},n^{\prime\prime})$. At the second
order of perturbation theory, each of those states induces an energy shift
proportional to
$|\bra{nS,nS}V_{dd}\ket{n^{\prime}P,n^{\prime\prime}P}|^{2}/\Delta(n^{\prime},n^{\prime\prime})$
[31]. In total, the pair state $\ket{nS,nS}$ experiences a van der Waals
energy shift $U_{vdW}=C_{6}(n)/R^{6}$, where $C_{6}(n)$ is a prefactor that
increases dramatically with $n$, the principal quantum number [32].

Another example of spin models that can be realized is the XY Hamiltonian

$\mathcal{H}(t)=\frac{\hbar}{2}\Omega(t)\sum_{j}\sigma_{j}^{x}-\frac{\hbar}{2}\delta(t)\sum_{j}\sigma_{j}^{z}+2\sum_{i\neq
j}\frac{C_{3}}{r_{ij}^{3}}\left(\sigma_{i}^{x}\sigma_{j}^{x}+\sigma_{i}^{y}\sigma_{j}^{y}\right).$
(2)

It naturally emerges when the spin states $\ket{\downarrow}$ and
$\ket{\uparrow}$ are two Rydberg states that are dipole-coupled, such as
$\ket{nS}$ and $\ket{nP}$ [33, 34]. Since the transitions are typically in the
tens of GHz range, the effective magnetic field (the two first terms in
equation (2)) are induced by a microwave field instead of an optical laser
field. The third term in equation (2) corresponds to a coherent exchange
between the spin states, transforming the pair state
$\ket{\downarrow\uparrow}$ into $\ket{\uparrow\downarrow}$. The dipolar
interaction that leads to the third term scales as the inverse to the distance
$r_{ij}$ to the power of 3 (see Box 2). The exchange interaction is very well
suited to study frustrated quantum magnets [35] or to investigate excitation
transport [36], especially in the context of photosynthesis to understand how
the light energy is carried towards the reaction center in light-harvesting
complexes [37]. Associated with controllable geometries of the qubits in the
register, it can also tackle problems of conductivity in topological materials
like organic polymers [38].

Combining various pairs of states and playing with the geometry of the spins
in the register, neutral atom QPU allows to implement a broad variety of spin
Hamiltonians, such as the XXZ model [39, 40] or Hamiltonians with non-spin
conserving terms [41]. As a very important tool for analog quantum processing,
our device also gives the possibility to apply simultaneously global and local
addressing beams. It can be used to introduce controllable disorder, which is
known to dramatically influence the behaviour of many-body systems [42, 43,
44, 45].

Among all the candidates for analog quantum processing, Rydberg atoms in
optical arrays are particularly well suited because they offer a very
favorable quality factor $\mathcal{Q}\sim 10^{2}$, given by the ratio of
coherent evolution rate over the decoherence rate. Indeed due to their huge
electric dipoles, Rydberg interaction energies are typically on the tens of
MHz range, at least two orders of magnitude larger than the energy scales
associated to decoherence. This gives an estimate of the number of quantum
operations that can be realized before the system loses its quantum properties
and becomes classical. This quality factor can even be further increased, for
instance by trapping the Rydberg atoms to reject the dephasing introduced by
their Doppler effect [46].

####  2.2.3 Prospects for improving the QPUs performances

The capabilities of a quantum processor are directly related to a number of
parameters such as the number of available qubits, the repetition rate of the
computation, the qubit connectivity in the register or the fidelities of
elementary operations, as illustrated in Fig. 10. An important goal is to
identify, study thoroughly and implement technologies to improve on each of
those features beyond current performances. We present here a brief overview
of the prospects in terms of hardware development.

First, the number of qubits available for computation is mostly limited by the
current trapping laser system. On the one hand, the development of new laser
systems delivering much higher optical power would enable the generation of
many more optical tweezers. Together with an improved imaging system based on
state-of-the-art microscope objectives, this would allow to scale the size of
the register up to a few thousands of qubits. At this point, the lifetime of
the atoms forming a sub-register configuration also becomes a limitation.
Because the latter is set by the residual pressure in the vacuum chamber, it
motivates another major future development: the design of QPUs in compact
cryogenic environments.

Figure 10: The three main axes of hardware developments to improve the
performances of the QPUs.

The other way to overcome the limitation set by the residual pressure is to
reduce the duration of each operation occurring in a computation cycle with
respect to the atoms lifetime (see Sec. 2.1). First, the development of high-
flux atomic sources would allow to reduce the time required to load the
registers. Then, a set of new hardware and techniques would be combined to
reduce the time required to assemble a defect-free sub-register of atoms, and
the time to acquire images. Overall, the repetition rate is expected to
improve by an order of magnitude. Even though the current acquisition
capabilities enable the realization of most experiments, the overall increase
of the repetition rate will facilitate the implementation of procedures that
require a very large number of repetitions, such as variational algorithms.

Finally, important efforts need to be focused on improving the manipulation of
the qubits and reducing the error rate of the quantum operations. In the next
generation of neutral atom processors, the electrostatic environment of the
atoms will have to be designed with special care to reduce even further the
decoherence induced by parasitic charges in the atoms surroundings. On the
other hand, because the optical pulses required for computing are generated by
radio-frequency pulses, the development of high-bandwidth low-noise
electronics would improve the quality of the QPU operations. In particular,
agile arbitrary waveform generators (AWGs) with high resolution and large
sampling rate would allow to tailor the pulses parameters for optimal control
which will boost even further the gate fidelities.

Beyond lowering the error rate of successive operations, another way to reduce
the effect of decoherence is to shorten the overall computation time. On the
one hand, our neutral atom device is well suited to develop tools allowing to
perform many operations at the same time (parallelization). On the other hand,
the number of operations for a given algorithm can also be reduced with a
larger connectivity between the qubits. Several techniques based on pulsing
either static or microwave electric fields can be developed to extend the
connectivity in the register.

##  3 Applications of a neutral atom QPU

The first natural application of a fully programmable neutral atom Quantum
Processing Unit (QPU) is to explore and solve complex quantum phenomena across
many areas of science, from the behavior of solid-state materials to chemical
and biochemical reaction dynamics. By piloting quantum entanglement and
superposition, one reproduces in the device the key elements that are thought
to be sufficient to explain such physical phenomena. In that sense, the
quantum device acts as a simulator of fundamental natural processes and can be
used to foster scientific discovery, at a very reduced computational cost. Due
to their quantum nature, the scientific problems explored in this Quantum
Simulation framework are not solvable easily on classical devices. At the
origin of the difficulty lies the exponential scaling of the size of the
Hilbert space with the number of interacting particles. In this regard, a
quantum advantage has already been reached in the study of quantum magnetism
[27, 47].

Beyond the simulation of scientific processes, neutral atom processors can be
used to solve hard computational problems, for which classical computers are
inefficient. Most of the quantum algorithms tackling computational problems,
such as Shor’s algorithm, have been designed to be executed on an ideal
quantum computer without errors. To reach such a perfect device in practice,
one needs to correct the errors that naturally occur at the hardware level
with dedicated error correction procedures. By partially measuring some parts
of the system, these procedures can detect if errors have occurred in the
course of the computation and actively correct them on the fly. Well-known
examples include the surface code [48, 49, 50]. Architectures that implement
error-correction procedures are called fault-tolerant; and they will not be
available in the near-future, due to the large overhead in the number of
qubits necessary to implement error correction. Even though fault-tolerant
quantum computing is still many years away, recent results from Google [2]
have shown that devices without error-correction and with a relatively modest
number of qubits can already outperform any classical supercomputer for
specific computing tasks. Applications in this Quantum Computing framework
notably include finding approximate solutions to hard combinatorial
optimization problems, solving non-linear partial differential equations or
enhancing the performances of Machine Learning procedures.

While the Quantum Simulation and Computing frameworks which will be presented
respectively in 3.1 and 3.2 are conceptually different, it is important to
realize that they are implemented on the exact same physical platform.

###  3.1 Quantum Simulation

Quantum Simulation consists in studying a synthetic quantum system with the
QPU, by implementing a model of interest for which no other way to solve it is
known. The model may be an approximate description of a real material or
molecule, but it can also be a purely abstract one. There are several
approaches to Quantum Simulation, which we describe below.

####  3.1.1 Quantum Simulation on neutral atom devices

As described in Section 2, neutral atom devices can natively realize quantum
spin Hamiltonians (see Eqs. (1) and (2)). This situation, where one sets the
Hamiltonian to exactly mimic the problem under consideration is called Analog
Quantum Simulation. This analog setting is not fully general, but corresponds
to relatively low requirements in terms of control and operation count, making
it the most competitive approach to date. In addition, continuous drive of the
system by an external source can extend the variety of phenomena that can be
simulated through Floquet engineering [51]. Even though neutral atom devices
can reproduce many models of interest, the range of applications of this
approach remains limited. To overcome this limitation, another framework has
recently emerged [52], consisting in using the QPU in tandem with a classical
computer. In this hybrid approach, a classical computer is used to optimize a
relevant cost function related to a target Hamiltonian, from measurements of
highly entangled states prepared on the QPU. In this procedure, the target
Hamiltonian never needs to be physically realized with the device. More
details about these hybrid classical-quantum procedures are given in Box 3.

While hybrid methods greatly extend the simulation capabilities, it is
believed that the only target Hamiltonians that can be efficiently simulated
are the ones which share the same symmetries as the resource Hamiltonians that
can be implemented in the quantum register [52]. Some models are then
intrinsically hard to simulate with neutral atom devices. For those models,
the solution consists in using quantum circuits made of gates [53]. This final
approach, called Digital Quantum Simulation, is known to be universal, in the
sense that any physical model of interest can be studied.

|  Analog Quantum Simulation |  Digital Quantum Simulation
---|---|---
Resource used for simulation |  Hamiltonians |  Gates
Key advantages |  Promising hybrid quantum-classical approaches |  Universal approach
Shortcomings |  Limited number of available configurations |  Requires a large number of gates
Status |  Quantum advantage already achieved |  Academic research

All of these approaches can be implemented on neutral atom QPUs, in various
fields of applications, as we describe below.

Box 3 $|$ Variational procedures with Quantum Processing Units

The general aim of these methods is to optimize for a given objective
function, and the key idea is to use both a quantum and a classical processor
in conjunction, with only a minimal number of operations realized on the
quantum processor. In this hybrid framework, the sole use of the quantum
processor is to prepare a given quantum state, parameterized by a set of
variational parameters. One then extracts from this state an estimate of the
objective function through repeated measurements. This estimate is in turn
used as the input into a classical optimization procedure, which returns new
variational parameters to prepare the quantum state for the next iteration.
This loop is repeated multiple times until some stopping criterion is
fulfilled, or the optimization procedure converges. Then, a final estimate of
the objective function is output.

In this framework, the quantum processor can be seen as a trainable quantum
algorithm which output is adapted variationally to best match the task under
consideration. Importantly, the variational nature of the procedure renders
these algorithms relatively resilient to errors [54], making them prime
candidates for an implementation on NISQ devices. The general principle of
these hybrid quantum-classical algorithms is illustrated in Fig. B3.1.

Figure B3.1 $|$ Principle of hybrid quantum-classical learning algorithms.
These algorithms are composed of both a quantum and a classical processor that
exchange information within a feedback loop. The quantum processor is used to
prepare and measure a $n$-qubit parameterized quantum state. The outcome of
the measurement is then used as the objective function in a standard classical
optimization procedure, that updates the parameter for the next iteration. The
operations on the quantum processor can be of various kinds: single-qubit
operations, two qubit operations, or global operations. These operations can
be indifferently expressed as digital quantum gates, or Hamiltonian time-
evolutions.

This approach might lead to speedups for certain tasks thanks to the “exotic”
correlations that can be encoded in the quantum state. Due to its hybrid
nature, this technique enhances the potential of the quantum processors
available today by drastically reducing the coherence time requirements for
the computation. The suitability of such procedures becomes evident when it
comes to learning a complex quantum system, such as a molecule’s energy levels
in chemistry.

####  3.1.2 Applications of Quantum Simulation

Quantum Simulation applications with arrays of neutral atoms include all of
many-body physics, the field that studies the behavior of ensembles of
interacting quantum particles. This is a very broad area encompassing almost
all condensed matter physics and quantum chemistry [55] but also nuclear and
high-energy physics.

##### Condensed-matter physics –

By allowing the simulation of quantum spin systems, neutral atom devices will
open up a variety of new opportunities in Condensed-matter physics. Spin
models have been extensively studied in the last 60 years in various contexts,
such as magnetism and excitation transport. However, many important open
questions remain the subject of active research, such as the nature of the
phase diagram when the spins are placed in arrays featuring geometrical
frustration, the dynamics of the system after a sudden variation of one
parameter of the Hamiltonian [56], the role of disorder in the couplings, or
their combination with situations where topology plays a role. In the past few
years, several important published studies [57, 58] have shown that neutral
atom devices developed at the Institut d’Optique Graduate School (IOGS) can
solve some of these scientific questions.

Beyond quantum spin systems, arrays of atoms can also bring new insight into
other solid-state systems of interest, like electronic systems. There indeed
exist explicit mappings between spin models, that can be implemented natively
with neutral atom devices, and electronic Hamiltonians, known as the Jordan-
Wigner [59, 60] and the Bravyi-Kitaev [61] transformations (see Box 4 for
details about the Jordan-Wigner transform). Future investigations along those
lines will allow for the study of new materials that will potentially offer
unprecedented functionalities for energy transport and storage, or exhibit
transformative properties such as high-temperature superconductivity.

Figure 11: Applications of neutral atom devices to study quantum matter and
biochemistry. The left panel shows the copper-oxide plane of a high
temperature superconductor [62]. One models such systems by many-body
electronic Hamiltonians (Fermi-Hubbard), that can be mapped on spin systems
through well-known transformations. For this application, numerical
simulations suggest that a NISQ device able to preserve its coherence for
circuit depths of a few hundreds could achieve a quantum advantage over the
best known classical methods [63]. The depth of a circuit corresponds to its
total duration, expressed in units of a single gate duration and assuming the
ability to carry out operations in parallel. The implementation of parallel
gates associated with a relatively modest increase of our gates’ fidelities
will allow neutral atom devices to go beyond this threshold. The right panel
shows the structure of Hemoglobin. Quantum Simulation of the active site of
the protein can bring new insight on its reactivity.

##### Quantum chemistry –

The ability to simulate electronic systems extends to quantum chemistry and
biochemistry problems. While classical computational mechanics is adequate for
describing most of the properties of these systems (e.g. with molecular
dynamics), the incorporation of quantum effects is instrumental in
understanding some physical processes at the microscopic level [64]. The
incorporation of many-body quantum effects then allows us to refine models and
better understand the reactivity of some molecules, by providing a more
complete modelling of the electronic degrees of freedom of the molecules’
active sites. This kind of study generally amounts to characterizing the low
lying eigenstates of a very large electronic Hamiltonian. Quantum approaches
to find eigenvalues have previously relied on the quantum phase estimation
(QPE) algorithm [15], which offers an exponential speedup over classical
methods,but remains unpractical for NISQ quantum devices without error-
correction. An alternative approach was proposed, leveraging the capabilities
of quantum hardware through a variational procedure [65, 66].

Box 4 $|$ Jordan-Wigner transform

We illustrate here how to map 1D chain of electrons (fermionic particles) onto
a 1D chain of qubits (spin-1/2 particles described by standard Pauli
operators, i.e. hard-core bosonic particles).

Computing the anti-commutator of the two spin operators $\sigma^{+}_{j}$ and
$\sigma^{-}_{j}$ at the same spatial site $j$, we find
$\\{\sigma^{+}_{j},\sigma^{-}_{j}\\}=1$, similarly to electronic creation and
annihilation operators. However, for different sites, spin operators commuter
$[\sigma^{+}_{j},\sigma^{-}_{i}]=0$ in contrast to electron operators which
anti-commute.

Jordan and Wigner introduced in Ref. [59] a transformation that recovers the
true fermionic commutation relations from spin-1/2 operators. More
specifically, this transformation reads

$\displaystyle\sigma_{j}^{\dagger}=e^{\left(-i\pi\sum_{k=1}^{j-1}f_{k}^{\dagger}f_{k}\right)}\cdot
f_{j}^{\dagger}$
$\displaystyle\sigma_{j}=e^{\left(+i\pi\sum_{k=1}^{j-1}f_{k}^{\dagger}f_{k}\right)}\cdot
f_{j}$ $\displaystyle\sigma_{j}^{\dagger}\sigma_{j}=f_{j}^{\dagger}f_{j}.$

The additional factor above, also named the Jordan-Wigner string, introduces a
phase factor at each site, which depends on the density of excited particles
located on its left. While we presented its 1D-version, extensions to this
transformation also exist in higher dimensions.

The mapping above can be used to map any electronic Hamiltonian from
condensed-matter physics or quantum chemistry onto a spin (qubit) Hamiltonian.
This is the first step of all Quantum Simulation procedures of electronic
systems with neutral atom processors. In 2D, this mapping comes with an extra
difficulty as simple local fermionic Hamiltonians are mapped to non-local spin
Hamiltonians and vice versa.

##### High-energy physics, nuclear physics and cosmology –

The great control over individual particles allowed by modern quantum
simulators enables to realize lattice gauge theory models in practice. Gauge
theories are important in particle physics, and include the prevailing
theories of elementary particles: quantum electrodynamics, quantum
chromodynamics and the standard model of particle physics. Lattice gauge
theory is the study of gauge theories on a spacetime that has been discretized
into a lattice, which applications also extend to Condensed-matter and nuclear
physics [67]. In that respect, one pioneering experiment studied the Lattice
Schwinger model with trapped ions [52] using a variational approach. Such
studies could be implemented on neutral atom devices, which also provide the
ability to explore the behavior of 2D lattice models, as noted for example in
Ref. [68]. As such, neutral atom devices constitute an appealing platform for
testing fundamental theories of physics in the high-energy regime, with a
development cost four orders of magnitude lower than a particle collider.

We have seen above some of the applications of neutral atom quantum devices
for quantum simulation.

###  3.2 Quantum Computing tasks in the NISQ era

Beyond Quantum Simulation problems, variational procedures with parameterized
quantum circuits described in Box 3 can also be applied to more general
computational tasks, as was first proposed in Ref. [7] in the context of
optimization problems.

####  3.2.1 Variational algorithms for computing purposes

Even though no clear proof of a speed-up with respect to the best classical
algorithms exists, several key points have been identified to understand and
enhance performances, and the route to quantum advantage using variational
procedures for computational problems seems within reach.

One particularly promising opportunity for improvement is the implementation
of hardware-efficient variational algorithms, that maximally exploit the
strengths of available quantum computing devices. Indeed, not all quantum
hardware is equivalent, as the set of operations that can be natively
implemented depends on the chosen qubit technology (superconducting circuits,
neutral atoms, ions, photons, …). On a neutral atom platform, one important
example is the native resolution of a well-known graph problem, the Maximum
Independent Set (MIS) problem (see details in Box 5). This problem, which has
various direct applications in network design [69] or finance [70], becomes
hard to solve on a classical computer when the size of the graph grows [71].
More generally, solving efficiently the MIS problem would provide a way to
solve any interval scheduling problem, with applications in many fields
(telecommunication, computing tasks allocation in HPC, or even satellite
photography to name a few).

Box 5 $|$ Hardware-efficient implementation of a variational algorithm using
neutral atom devices

Considering an undirected graph composed of a set of vertices connected by
unweighted edges, an independent set of this graph is a subset of vertices
where no pair is connected by an edge. The objective of the MIS problem is to
find the largest of such subsets.

The MIS problem on unit disks graphs can be tackled by using an ensemble of
interacting cold neutral atoms as a quantum resource, where each atom
represents a vertex of the graph under study. As with any quantum system, the
dynamics of the atoms are governed by the Schrödinger equation, involving a
Hamiltonian (energy functional) depending on the atomic positions, the
electronic energy levels and their interactions. Interestingly, the physical
interactions encoded in the Hamiltonian constrain the dynamics to only explore
independent sets of the graph under study, then leading to an efficient search
in the set of possible solutions [72, 54], as illustrated in Fig. B5.1. This
example underlines the prime importance of targeting the trainable quantum
network to the specific task at hand, so that the class of trial quantum
states that are generated represents good candidates for solving the problem
under consideration.

Figure B5.1 $|$ From a spatial configuration of Rydberg atoms to a unit-disk
MIS graph problem. The positions of the atoms, that have two internal energy
levels, are chosen to match directly the graph under consideration. The levels
of two Rydberg atoms strongly interact if the distance between the atoms is
smaller than a typical distance ($r_{b}$, see left part), resulting in the
impossibility for the two atoms to be both in the same state at the same time
(phenomenon of Rydberg blockade, as described in Section 2.2.1). This
naturally corresponds to the independent set constraint in the graph defined
by the atoms, with edges linking atoms that sit at a distance closer than
$r_{b}$. We show on the right the corresponding graph, with the red vertices
forming one independent set.

As such, well known notions in the field of Quantum Information Processing
such as gate fidelity or quantum volume are not always the best criteria for
assessing the performances of a given quantum hardware on a specific task.
Exploring the same phase space by using only standard digital quantum gates in
a nearest-neighbor architecture would be far more demanding than with atoms. A
recent preprint from the Google group supported this statement, by finding
that the best performance of a variational algorithm for optimization was
achieved on a problem that maps directly to the native hardware layout of the
device [73].

By solving the MIS problem for several hundreds sites, neutral atom devices
could outperform the best existing classical algorithms for this task [72,
74]. Beyond this particular example, progress in the hardware will lead to
efficient implementation of variational algorithms on Pneutral atom devices
for a variety of other tasks, as we describe below.

####  3.2.2 Applications of NISQ Quantum Computing

Almost all the numerical tasks that are computationally-intensive for standard
computing resources are candidates for quantum acceleration using neutral atom
devices. This notably encompasses hard optimization problems, the resolution
of non-linear partial differential equations or Machine Learning routines.

##### Combinatorial optimization problems –

The first natural example corresponds to solving NP-hard combinatorial
optimization problems, as initially proposed by Farhi et al. [7]. Finding the
exact best solution to these problems is not always the main concern, and
coming up with an approximate solution in a decent amount of time is often
sufficient for application purposes. This fact encouraged the early
development of approximate variational algorithms working on noisy devices, as
exactness is not strictly required. These algorithms are often referred to in
the literature as Quantum Approximate Optimization Algorithms (QAOA).

Solving combinatorial optimization problems remains a chief concern in many
industrial areas, such as logistics, supply-chain optimization, or network
design. In addition, neutral atoms represent a natural architecture for
solving optimization problems with multiple-body terms using the Lechner-
Hauke-Zoller (LHZ) architecture [75, 76].

##### Non-linear differential equations –

Non-linear differential equations are ubiquitous in all fields of science and
engineering. Examples notably include large-scale simulations for reliable
weather forecasts, computational fluid dynamics in aeronautics, or even
finance. An algorithm was recently proposed to solve non-linear partial
differential equations [24] using a quantum accelerator within a variational
procedure. The efficient implementation proposed in that paper makes concrete
the prospect of utilizing intermediate-scale quantum processors for solving
non-linear problems on grid sizes beyond the scope of conventional computers.

##### Machine Learning tasks –

Machine Learning aims at automatically identifying structures and patterns in
large data sets. In order to identify these patterns, algorithms often resort
to standard linear algebra routines such as matrix inversion or eigenvalue
decomposition. For example, support vector machines, one of the most
successful traditional machine learning approaches for classification, can be
cast to a linear system of equations, and then be solved using matrix
inversion. The large dimensionality of the vector spaces involved in these
operations makes their implementation at large scale very resource intensive,
thus motivating the development of innovative methods to lower their
computational cost. Researchers at MIT already proposed by the end of the 90s
the use of a quantum processor to perform these linear algebra routines more
efficiently (see review [77]). The discovery of these algorithms triggered a
lot of research efforts at the intersection of machine learning and quantum
information processing, but the resource requirements for their implementation
in term of numbers of qubits and gate fidelities is unfortunately prohibitive
for devices that are and will be available in the short term.

Motivated by these developments, people tried to find other ways to leverage
Machine Learning methods on available NISQ quantum processors, albeit for
other purposes than data intensive applications such as the ones mentioned
above. New quantum applications for Machine learning are today rapidly
developing [8, 78], and several classifiers have been elaborated [79, 80] as
well as sampling methods of classically inaccessible systems. Another
promising direction concerns the use of Quantum machine learning for
experimental quantum data, opening new prospects in enhanced quantum sensing
or metrology.

As exemplified in the preceding Section, the high controllability and
versatility of neutral atom quantum devices can bring instantaneous value to
researchers and practitioners in Quantum Simulation and Quantum Computing in
various fields.

##  4 Perspectives

The emergence of quantum devices with a few hundred physical qubits (i.e. not
error-corrected) opens many exciting perspectives in quantum computing and
quantum simulation towards quantum advantage with respect to supercomputers.
Among other platforms, fully programmable neutral atom devices display unique
characteristics. By controlling quantum entanglement and superposition, either
with quantum gates in the digital configuration or with Hamiltonians in the
analog configuration, they represent a powerful tool to tackle scientific
problems and computing challenges. We explicitly illustrated in the preceding
sections how a wide variety of fields will be impacted by the advent of
quantum-accelerated computing. We also showed the prospects for future
developments at the hardware level, that will allow us to reach the 100 -
1,000 qubits range, and even beyond.

In the longer term, neutral atom platforms exhibit many interesting features
to push further their computational power and develop new applications. One
first appealing direction will be to couple together several processors with
optical interconnects. This multi-core architecture would then allow us to
substantially scale up the number of qubits available for computation. Not
only will the increase of the number of physical qubits allow for the
exploration of more industry-relevant usecases, but it will also make possible
the implementation of quantum error correcting codes such as the surface code
[48, 49, 81] on large system sizes. This is one first possible path for the
construction of a general-purpose fault-tolerant quantum computing device with
neutral atoms.

Coupling together distinct processors will require to build coherent
interfaces between our atomic qubits and single photons, which in itself
constitutes a great experimental challenge. But developing such an efficient
interfacing between individual processors and the outer world will also unlock
other development avenues.

One application of reliable light-matter interfacing consists in using the
atomic ensemble as a quantum memory for a photonic qubit. In that framework,
quantum information encoded in an incoming photon can be stored in the atomic
medium using the phenomenon of electromagnetically-induced transparency (EIT)
[82, 83]. Under EIT conditions, an atomic ensemble becomes transparent to
light and a single photon can propagate inside it without losses under the
form of a mixed light-matter excitation called a polariton. The polariton
velocity is greatly reduced as compared to the speed of light in vacuum, and
can even be temporarily set to zero [84], transforming then the atomic
ensemble into a quantum memory. Such a quantum memory could be used for the
distribution of quantum information over large distances, which represents a
great challenge due to photon loss. Several so-called quantum repeater
protocols [85, 86] based on atomic ensemble memories have been proposed to
overcome this challenge, and experiments are progressing rapidly [87, 88].

Figure 12: Diagram showing the impact of future technological improvements on
(i) the quality of our processors in the NISQ era, and (ii) the development of
our future atom-based quantum technologies. Diagram adapted from Ref. [89].

In addition, light-matter interfaces could be harnessed to engineer non-
linearities between single photons at the single-particle level. This can be
done by combining the EIT phenomenon with the strong dipole-dipole
interactions in Ryberg media. Under EIT conditions, a single photon can
propagate without losses through the atomic medium. When a second photon
penetrates inside the medium, the strong dipole-dipole interaction between the
atomic components of the two polaritons will result in an effective
interaction between the two photons [90, 91]. Such processes constitute a
powerful resource for photonic quantum information processing, where quantum
logic is applied between photonic quantum bits (see for example Refs. [92, 93,
94, 95]). In this area of Rydberg non-linear optics [96, 97], the Rydberg
atoms are no longer the support of the quantum information, but rather act as
a source of non-linearity for photonic qubits.

An interesting recent development about photonic quantum computing is the
ability to engineer relatively simple error-correction procedures. The
infinite dimensional Hilbert space of photons can be used to redundantly
encode quantum information, without having to dramatically increase the number
of quantum units. Such realizations have also been proposed with
superconducting qubits that are used as resources (ancilla) to perform
photonic gates in the microwave frequency range [98, 99, 100]. The
implementation of such photonic (or bosonic) codes would correspond to a
second path towards a fault-tolerant architecture.

As briefly illustrated above, and summarized in Fig. 12, cold neutral atom
devices hold great potential for the development of multiple key technologies
in the second quantum revolution.

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