Quantum computing with neutral atoms

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 2D and 3D 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 discuss the technological strategy of Pasqal, a startup developing such quantum processors to tackle some of the world's most challenging problems. We present the full quantum stack, 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 State Quantum 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 Pasqal's technology in the 100-1,000 qubits range and introduce prospects for universal fault tolerant quantum computing and applications beyond quantum computing.


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 2 n . 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]. 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.
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 [5]. Only recently did it become possible to contemplate manufacturing devices for commercial use thanks to continuous progress in design and engineering. The company Pasqal has been created in 2019 to leverage the technology developed at Institut d'Optique by the team of A. Browaeys and T. Lahaye in an academic setting precisely for the purpose of building fully programmable Quantum Processing Units (QPUs) for practical quantum advantage for customers. This technology will be described in more detail in Section 2.
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 Pasqal 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, Pasqal 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 [6], or enhancing the performances of Machine Learning procedures [7].
Prospective users will have two distinct ways to explore these applications with Pasqal devices, either at the digital level (programming gate-based circuits) or at the analog level (programming Hamiltonian sequences), as illustrated in the middle block of Fig. 1. For the digital control, we will provide access through all the major frameworks for digital quantum information processing (Qiskit, Cirq, Braket, Myqlm, QuTip ...), allowing the users to leverage existing hardware-agnostic software tools for applications. For the analog control, we will additionally

Applications Quantum Simulation
Chemistry, Material science, Drug discovery …

Computational problems
Optimization, Machine Learning, partial differential equations, … Laser controls

Digital control
Sequence of gates in a Quantum circuit

Sequences of Quantum Hamiltonians
Quantum software

Quantum Hardware
Detection system

Metrics Quantum stack
Quality, speed or reliability of the procedure in comparison with state-of-the art classical methods Complexity of the procedure, and/ or gate count Number of atoms, gate fidelity quality factor , and repetition rate of the QPU Quantum register ℱ Figure 1: Pasqal 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, the processor developed at Pasqal can 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. build a web-based graphical user interface for easy and interactive programming of experiments. In both cases, users will be provided with a precise control over the device dynamics. In the NISQ era, software and hardware development come by pair, therefore some specific software tasks will be developed internally, allowing us to unlock the full potential of Pasqal devices. Details about the access to Pasqal quantum devices are provided in Section 4.
Pasqal's vision is to become the leading player in the field of quantum-enhanced applications based on the manipulation of individual neutral atoms. 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 5.

Neutral atoms arrays
The technology at the heart of Pasqal is 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. At Pasqal we use 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 |0 and |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 the Pasqal 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.

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. 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 beamsplitter (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 the Pasqal co-processor. The register is prepared at the center of this setup.

Register loading
To prepare a register made of neutral atoms, we use arrays of optical tweezers, a technology developed over the last years by the group of A. Browaeys and T. Lahaye at Institut d'Optique (IOGS), co-founders of Pasqal. 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 mm 3 volume is prepared inside a 3D magneto-optical trap (3D MOT), leveraging numerous laser cooling and trapping techniques [8]. 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 µm diameter: the so-called optical tweezers [9]. 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 µm 3 , each tweezer contains at most one single atom at a time.
The number of tweezers and their arrangement in any arbitrary 1D, 2D or even 3D geometries is fully tailored by holographic methods [10]. 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, the neutral atom platform for quantum processing has 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.
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 Figure 4: Examples of tweezer arrays in 2D or 3D geometries, extracted from Ref [11].
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.

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 µs, 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 |0 will appear as bright, whereas atoms in qubit state |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 [12], a value comparable to processors based on superconducting qubits [13]. 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.

Quantum processing with atomic qubits
The Pasqal Quantum Processing Unit (QPU) is 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 in the Pasqal QPU are discussed in this section.

Digital quantum processing
Digital computing requires qubits robust against decoherence, i.e., weakly coupled to their environment. Therefore we use as a qubit the two hyperfine ground states (F = 1 and F = 2) of the rubidium atom, because they have infinite lifetimes 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 [14]. 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 |0 into |1 and vice-versa, or the Hadamard (H) gate that generates superposition of both states starting from a pure state. In the {|0 , |1 } basis, theses gates read 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 |0 occupies the North pole and the state |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 |0 and |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 σ x , σ y and σ z . Regarding the previous examples, the NOT-gate coincides with σ x and is a rotation by a π angle around the x axis (see Fig. 7(a)), while the Hadamard-gate writes (σ x + σ z )/ √ 2 and is a rotation of π 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 the Pasqal device the later is an optical laser field driving Raman transitions through an intermediate atomic state [15,16]. The atom-laser interaction is characterized by the Rabi frequency Ω (its strength, proportional to the amplitude of the laser field), the detuning δ (the difference between the qubit resonance and the field frequencies) and their relative phase ϕ. Driving the control field for a duration τ induces rotations around the (x, y, z) axes with angles (Ωτ cos ϕ, Ωτ sin ϕ, δτ ). 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 synthetizers (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 (δ = 0), the qubit oscillates in time between the states |0 and |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 [16,17], corresponding to single-qubit gate fidelities F higher than 99.5%. The fidelity 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 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 π, it exactly realizes a NOT gate. On the other hand, the Hadamard gate requires a π pulse at a detuning equal to the Rabi frequency (δ = Ω) 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 |1 . The corresponding matrix in the pair state basis There are several ways to realize this gate using dipolar Rydberg interaction [16,18]. Figure  B1.1(a) illustrates the key mechanism with a simple sequence of 3 pulses.
• With initial state |1 c 1 t , all the three pulses are off-resonant and the state remains unchanged.
• With initial state |0 c 1 t , pulse 2 is off-resonant so the target qubit state remains unchanged. Pulses 1 and 3 drive a 2π rotation of the control qubit. The pair state picks up a phase factor e iπ = −1.
• With initial state |1 c 0 t , pulses 1 and 3 are off-resonant so the control qubit state remains unchanged. Pulse 2 drives a 2π rotation of the target qubit. The pair state picks up a phase factor e iπ = −1.
• With initial state |0 c 0 t , pulse 1 excites the control qubit to state |r with a π rotation. Because of the Rydberg blockade, the target qubit Rydberg state |r is shifted out of resonance during pulse 2. The target qubit remains in state |0 t . Finally, pulse 3 brings the control qubit back to state |0 c with another π rotation. The pair state picks up a factor e iπ = −1.
This pulse sequence realizes, within a global π phase, the controlled-Z (CZ) gate 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  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. [19] 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.
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 µs, leading to a 1 MHz clock rate for the QPU. Transient non-zero popu-lation 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 F = 94.1% [20] and will be largely improved [21]. 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 [14], 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 Pasqal 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 Pasqal device in 2D and 3D. Results are displayed on Fig. 8, where we see that the overhead for Pasqal devices (green points 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.
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 such as the one developed at Pasqal present the advantage of natively implementing multi-qubit gates involving more that 2 qubits [4,16]. 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 [23]. In that respect, Pasqal 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.
control target p p 2p These properties of large connectivity and native multi-qubit gates, specific to Rydbergbased 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, 50 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 the Pasqal QPUs highly competitive. On top of that, what constitutes the main advantages of Pasqal 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.

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 dipoledipole interactions that map into spin Hamiltonians. Each qubit of the register then behaves as a spin whose states are |↓ = |0 and |↑ = |1 . Depending on the Rydberg states that are involved in the process, the spins experience different types of interactions that translate into different Hamiltonians [5]. The most studied one is the Ising model, which is obtained when the |↓ state is one of the ground states and the |↑ a Rydberg state [24][25][26][27]. In this case the Hamiltonian writes with n j = (1 + σ z j )/2 the Rydberg state occupancy and σ x,z j the Pauli matrices σ 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 ⊥ ∝ Ω(t) and B || ∝ −δ(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 Ω(t) and δ(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 the to 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 [28], as further described in section 3. In Pasqal devices, we will have the ability to implement it 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 [29].
At the first order of perturbation theory, two pair states |ψ and |ψ are coupled only if the matrix element ψ| V dd |ψ is not vanishing. It is the case when considering for instance the states |ψ = |nS, n P and |ψ = |n P, nS , where n, n 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 |nS, n P will evolve into the state |n P, nS and back: the two atoms exchange energy corresponding to the difference between the energy of the states |nS and |n 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 |nS, nS .
It is dipole-coupled to states with opposite parity, such as |n P, n P , which are separated in energy by ∆(n , n ). At the second order of perturbation theory, each of those states induces an energy shift proportional to | nS, nS| V dd |n P, n P | 2 /∆(n , n ) [30]. In total, the pair state |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 [31].
Another example of spin models that can be realized in the Pasqal register is the XY Hamiltonian It naturally emerges when the spin states |↓ and |↑ are two Rydberg states that are dipolecoupled, such as |nS and |nP [32,33]. 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 |↓↑ into |↑↓ . 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 [34] or to investigate excitation transport [35], especially in the context of photosynthesis to understand how the light energy is carried towards the reaction center in light-harvesting complexes [36]. Associated with controllable geometries of the qubits in the register, it can also tackle problems of conductivity in topological materials like organic polymers [37].
Combining various pairs of states and playing with the geometry of the spins in the register, Pasqal QPU allows to implement a broad variety of spin Hamiltonians, such as the XXZ model [38,39] or Hamiltonians with non-spin conserving terms [40]. 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 [41][42][43][44].
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 Q ∼ 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 [45].

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 for Pasqal is to identify, study thoroughly and implement technologies to improve on each of those features beyond current performances. The prospective in terms of hardware development are described in detail in [21]. Here, we limit ourselves to presenting only a brief overview.
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 will enable the generation of many more optical tweezers. Together with an improved imaging system based on state-of-the-art microscope objectives, this will 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.

Number of qubits
Computation repetition rate Improved processing 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 will allow to reduce the time required to load the registers. Then, a set of new hardware and techniques will 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.
Finally, important efforts will be focused on improving the manipulation of the qubits and reducing the error rate of the quantum operations. In the next generation of QPUs, the electrostatic environment of the atoms will 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 will improve the quality of the QPU operations. In particular, agile arbitrary waveform generators (AWGs) with high resolution and large sampling rate will allow to tailor the pulses parameters for optimal control which will boost even further the gate fidelities. To confirm the validity of our approaches aimed at decreasing the error rates of the quantum operations, we will set up benchmarking procedures both in the digital and in the analog mode [46][47][48]. This is the object of a collaboration with the team of Prof. Elham Kashefi at the School of Informatics at the University of Edinburgh (UK), and at CNRS at the Paris Sorbonne University (France).
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 will be developed to extend the connectivity in the register.

Applications of Pasqal QPU
The first natural application of our fully programmable 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 with the neutral atom device developed at the IOGS academic group, in the study of quantum magnetism [49,50] (see also Ref. [26] from Harvard University).
Beyond the simulation of scientific processes, Pasqal 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 [51][52][53]. 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 at Pasqal on the exact same physical platform.

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.

Quantum Simulation on Pasqal devices
As described in Section 2, Pasqal 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 [54]. Even though Pasqal 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 [55], 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 [55]. Some models are then intrinsically hard to simulate with Pasqal devices. For those models, the solution consists in using quantum circuits made of gates [56]. This final approach, called Digital Quantum Simulation, is known to be universal, in the sense that any physical model of interest can be studied. All of these approaches can be implemented on Pasqal 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 [57], 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 molecules energy levels in chemistry.

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 [58] but also nuclear and high-energy physics.
Condensed-matter physics -By allowing the simulation of quantum spin systems, Pasqal 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 [59], 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 [49,60] have shown that neutral atom devices developed at IOGS can solve some of these scientific questions.
Beyond quantum spin systems, arrays of atoms can also bring new insight into other solidstate systems of interest, like electronic systems. There indeed exist explicit mappings between spin models, that can be implemented natively with Pasqal devices, and electronic Hamiltonians, known as the Jordan-Wigner [61,62] and the Bravyi-Kitaev [63] 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 Pasqal devices to study quantum matter and biochemistry. The left panel shows the copper-oxide plane of a high temperature superconductor [64]. 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 [65]. 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 Pasqal 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 [66]. 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 [14], 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 [67,68]. On this subject, Pasqal is the lead of the Innov'up research project QUACHA (Quantum Chemistry with Atom arrays), together with the London-based Quantum software startup Rahko (see Section 3.1.3). Another research project involves the Paris-based startup Qubit Pharmaceuticals to explore the integration of Pasqal's QPUs into High Performance Computing workflows that are already used extensively to model very large molecules and complex processes [69], such as, among other things, the interaction of the SARS-CoV-2 Spike surface protein and the ACE2 human cell receptor [70].

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 σ + j and σ − j at the same spatial site j, we find {σ + j , σ − j } = 1, similarly to electronic creation and annihilation operators. However, for different sites, spin operators commuter [σ + j , σ − i ] = 0 in contrast to electron operators which anti-commute.
Jordan and Wigner introduced in Ref. [61] a transformation that recovers the true fermionic commutation relations from spin-1/2 operators. More specifically, this transformation reads 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 Pasqal 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 [71]. In that respect, one pioneering experiment studied the Lattice Schwinger model with trapped ions [55] using a variational approach. Such studies could be implemented on Pasqal devices, which also provide the ability to explore the behavior of 2D lattice models, as noted for example in Ref. [72]. 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 Pasqal's QPU for quantum simulation. Next, we present more details on a current Quantum Chemistry project for applications in drug-discovery.

Concrete usecase: quantum chemistry calculations for drug-discovery
The QUACHA (Quantum Chemistry with Atom arrays) project, involving Pasqal and the quantum software company Rahko, aims at using variational algorithms to study the interactions between metalloproteins, which are enzymatic proteins containing metal ions, and their substrate. About one third of the enzymes known today are metalloproteins. They are involved in many biological mechanisms, and their mode of action is always the same. By binding to a particular substrate in a specific way, they allow certain chemical reactions to take place by lowering the activation energy of these reactions. The initial substrate is then transformed into the final product.
Understanding the docking properties of a given metalloprotein on its substrate plays a major role in drug discovery. Comprehension of this anchoring mechanism, and how to inhibit or accelerate it through the action of other molecules, would entail the modification of targeted biological processes. This is why many metalloproteins have been considered as drug targets for diseases such as cancer, diabetes, anemia, AIDS, malaria, bacterial infections, fibrosis and neurodegenerative diseases.
Traditional methods such as Density Functionality (DFT) do not calculate these energies precisely enough because of quantum correlations that are usually omitted in the calculations. On the other hand, laboratory experiments are very expensive and slow. Hemoglobin for example is well known for binding to oxygen, i.e. the binding energies are low enough to allow the blood to transport oxygen in the body. It has been shown [73,74] that DFT is insufficient to describe this binding process and results in binding energies that are too high. On the other hand, Dynamic Mean Field Theory [75] (DMFT) or the methods of the self-consistent multiconfigurational field (CASSCF/CASPT2) give good results, but at very high computational costs. These methods therefore prove impractical for larger proteins or industrial applications. Moreover, their use on supercomputers generates significant energy consumption.
In the QUACHA project, we propose the development and implementation on Pasqal devices of a new variational quantum algorithm to address this problem. More precisely, we will develop a hybrid algorithm adapting the CASSF/CASPT2 method to Pasqal processors. By taking advantage of the quantum properties of the processor, this new algorithm could faith-fully reproduce the precise structure of metalloprotein binding mechanisms at a relatively low computational cost compared to traditional classical algorithms.

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. [6] in the context of optimization problems.

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 hardwareefficient 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 the neutral atom platform developped by Pasqal, 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 [76] or finance [77], becomes hard to solve on a classical computer when the size of the graph grows [78]. 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 Pasqal 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 Schrdinger 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 [57,79], 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. r b 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 [80].
By solving the MIS problem for several hundreds sites, Pasqal devices could soon outperform the best existing classical algorithms for this task. Beyond this particular example, progress in the hardware will lead to efficient implementation of variational algorithms on Pasqal devices for a variety of other tasks, as we describe below.

Applications of NISQ Quantum Computing
Almost all the numerical tasks that are computationally-intensive for standard computing resources are candidates for quantum acceleration using Pasqal 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. [6]. 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. Pasqal is currently exploring the resolution of one such problem within a contract with the French leading electricity provider EDF, as we will describe in Section 3.2.3. In addition, neutral atoms represent a natural architecture for solving optimization problems with multiple-body terms using the Lechner-Hauke-Zoller (LHZ) architecture [81,82].
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 [23] 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 nonlinear 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 [83]). 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 [7,84], and several classifiers have been elaborated [85,86] 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.
We have seen here numerous applications of NISQ devices for computational purposes. In the following, we describe in more detail one particular example related to an optimization problem.

Concrete use-case : smart-charging of a fleet of electric vehicles
With the development of publicly accessible electric vehicles in cities, one faces many scheduling problems related to their availability and charging constraints. One such problem consists in minimizing the total charging time of N vehicles on k distinct charging stations. Under some constraints, this problem amounts to solving a combinatorial optimization problem called Maxk-Cut. The objective of Max-k-Cut is to partition the N vertices of a graph into k ensembles, so that the cut is maximal (the cut is the sum of all the weights of edges between nodes that  Figure 12: (a) Mapping for seven vehicles on three charging stations. The scheduling and association of each vehicle to a given station correspond to solving a Max-3-cut problem on a 7-nodes graph. The affiliation of a node to a charging station is illustrated by a color-code. The edges contributing to the cost function are illustrated by full black lines, while the ones that are discarded are dashed. The computational complexity of this problem grows exponentially fast with the system size. (b) Optimization landscape for a Max-k-cut problem using QAOA on an emulated Pasqal processor as a function of two variational parameters α and β. One shows the minimum search using both a gradient-based optimizer and a genetic algorithm.
are not in the same group). An example is illustrated on Fig. 12 (a).
An efficient resolution of Max-k-Cut with a variational procedure involving N ln k qubits has recently been devised by Pasqal's research team, thus unlocking the potential of Quantum Processors for real-world operational problems. One illustrates in Fig. 12 (b) the optimization landscape for one instance of a Max-k-cut problem. As can be seen in the plot, the optimization landscape is very irregular, requiring the design of efficient software methods in the classical optimization loop of the procedure (cf Fig. B3.1 in Box 3). Focusing on the performances of the quantum procedure on this particular use-case enables to develop general tools and knowhow for solving other industrial problems, beyond the specific application of smart-charging of electric vehicles.
As exemplified in the preceding Section, the quantum device developed at Pasqal constitutes a general-purpose programmable Quantum Processing Unit. Its high controllability and versatility, illustrated previously in Section 2, can bring instantaneous value to researchers and practitioners in Quantum Simulation and Quantum Computing in various fields. The next Section details how users will be able to access Pasqal devices, as well as specific software tools that are being developed internally.

The quantum software stack
As a hardware provider, Pasqal does not have the ambition to create a stand-alone software suite for accessing its computing resources: • regarding digital quantum information processing, Pasqal QPUs will benefit from an integration to well-known existing frameworks such as Qiskit (IBM), Cirq (Google), QuTiP (Python toolbox) or Myqlm (Atos), • regarding specific Hamiltonian design, a dedicated Pasqal web portal will be developed, on top of interfaces to standard frameworks that enable Hamiltonian tuning.

Digital framework
The general workflow for sending digital quantum circuits to Pasqal back-ends is summarized in Fig. 13.
Front-end (Cirq, Qiskit, QuTiP, Myqlm, …) Pasqal system software (ii) Abstract quantum circuit (i) Implementable quantum circuit, obtained directly from the user, or from a third-party compiler PasQasm -Processing of user-submitted quantum circuits at an abstract level -Transformation of circuits for optimal execution on one of Pasqal's quantum processors

PasQasm
Pasqal quantum compiler Implementable quantum circuit Figure 13: Pasqal software pipeline for digital (gate-based) quantum processing. We will provide an interface to the main existing frameworks for quantum circuit creation. Users can choose to send (i) quantum circuits that are readily implementable on the device, or (ii) abstract unconstrained quantum circuits that will be processed by our compiler before execution. Besides the real QPU, users will also have the possibility to run their circuits on a classical emulator of our device, with a limited number of qubits. This emulator, which will incorporate realistic noise models of the device, will allow users to calibrate the real experiments they want to run on the QPU.
As indicated in the top block of Fig.13, Pasqal QPUs will be accessible from multiple frontends. One particularly interesting feature of standard digital frameworks is the compatibility with hardware-agnostic tools for circuit optimization and libraries for applications.
An interface with Cirq has already been developed, with a classical emulator of the device running as a backend, and we now illustrate how to use it for running a quantum circuit.

Concrete example: sending circuits to a Pasqal QPU with Cirq
We believe that users should have the ability to fully control the circuits that are executed on our devices. As such, we will give them the ability to specify an exact gate sequence to be executed on the hardware, an implementable quantum circuit, as illustrated in Fig. 13 (i). That level of control will enable users to benchmark the performance of specific sequences of operations. We illustrate here one such example.
After importing Cirq, the first step is to create qubits. We consider here 36 qubits arranged on a square grid in 2D.
Then, one can instantiate a device, that takes as an input the qubit list defined above. Another parameter that the user can control is the Rydberg blockade radius, or control radius, that controls the maximum distance between two qubits for a two-qubit gate to be directly realized (see Section 2.2.1). We set it here to be 3.1, in units of the lattice spacing. Now that we have qubits on a Pasqal device, we can start creating a circuit. This is done by first instantiating a Circuit object, taking our recently created Pasqal device as device parameter. Then, one can add gates to the circuit by using the append method. As you can see here, Cirq knows that controlled-NOT gates are not native in Pasqal devices. As such, it automatically decomposes it into a succession of three gates.
When the distance between the two qubits involved in the gate is greater than the control radius, as shown for example in the following where one tries to add a two-qubit gate between the first and last qubit of our register, Cirq will raise an error.
After adding gates to his/her circuit, the user can either run it locally on his/her laptop or send it to our remote Pasqal server, to be executed on the QPU or the emulator. To send a circuit to our remote server, one has to instantiate a PasqalSampler object, specifying our url and a password. The final step is to call the run method.
The compatibility of Pasqal software system with Cirq enables the user to leverage libraries like TensorFlow-Quantum for quantum machine learning, or Openfermion for Quantum Chemistry applications. For example, one can solve the optimization problem of Section 3.2.3 using TensorFlow-Quantum. This is done in Ref. [87].
A user interested in applications might not be too familiar with the hardware characteristics, therefore leading to poor performances in the circuit design. That is why Pasqal Application Programming Interface (API) also provides the opportunity to send unconstrained quantum circuits directly to our system (see Fig. 13 (ii)). This will free the application developer to think about how his/her algorithm is implemented in practice. In that case, a modification of the quantum circuit will be done to optimize its performance on our specific hardware, during a so-called compilation stage.

Compilation of quantum circuits with PasQasm
The specificity of Pasqal devices in terms of gates, connectivity and spatial configurations makes it imperative to develop our own compilation solution, that we call PasQasm (Pasqal Quantum Assembly Model). The role of the compiler is to get the maximum performance out of the specific resources available. Below, we describe the main steps of the compilation procedure.
The first consideration, after the translation from a given front-end, is to process a highlevel abstract circuit to see if it can be re-ordered and simplified. At that optimization stage, general simplifications to the original quantum circuit are made, reducing as much as possible the number or the complexity of the circuit operations.
Then comes the crucial transpilation part, during which the quantum circuit is modified and optimized for execution on a specific device. As important as this optimization is, one needs to ensure that the time necessary to run this pre-processing stage remains contained, and that the procedure is scalable to larger circuit sizes. As such, one focuses on approaches that run at most in a time that grows as a polynomial of the number of qubits and operations in the quantum circuit. The close association between the hardware and the transpiler will allow for the use of device-specific calibration data in the compilation process. This stage is meant to take turns with the previous one, as the outcome of one might still be further improved by the other. This transpilation part can be divided into several key steps: • Mapping qubits to positions in a physical layout: each qubit is assigned a position in a layout that tries to maximize its connectivity to qubits with whom it shares the largest number of multi-qubit operations.
• Enabling operations between physically disconnected qubits: Whenever the chosen distribution of qubits in a layout does not allow for all multi-qubit operations to happen naturally, the circuit needs to be modified for the operation to be realized. This is done by introducing SWAP gates that exchange the quantum state of connected qubits.
• Converting all non-native gates into native gates: Ultimately, a device is only capable of performing operations in a small (although universal) set of quantum gates. All the gates not contained in this set are thus decomposed into equivalent sequences of gates native to the device.
• Scheduling gate execution: Since (i) the amount of operations a device can do simultaneously is limited and (ii) distinct operations have different durations, the ordering of operations influences the total duration of a quantum circuit. Here, the operations are sequenced considering the devices parallelization capabilities and gate duration to minimize the total execution time.
A final step of assembly converts the entire quantum circuit into a sequence of hardware instructions to be executed on the chosen backend.
A working version of our compiler has already been developed, and several options for improvements are currently being explored, including for example error adaptive compiling. Incorporation of third-party compilers within Pasqal system software will also be possible.

Hamiltonian control over the device
As described in Section 2.2, control over Pasqal QPUs can either be achieved via quantum gates, or by using Hamiltonians (analog control). In this latter case, the user should be able to tune the values of the physical parameters of a Hamiltonian chosen among a finite set of allowed configurations. Some python frameworks allow for this type of Hamiltonian control, such as QuTip, Atos myqlm, or Openpulse in Qiskit. A common API to these frameworks will enable users to devise custom experiments with analog control over the device.
On top of this standard access, we will develop a dedicated web-based graphical-user interface (GUI). There, users will be able to directly program the parameters of the experiment he/she wants to perform. A sketch of this interface is shown in Fig. 14. As illustrated in this Figure 14: Mockup showing a possible GUI for analog control over Pasqal devices. The interface will allow for a full control over the experiment to be realized, within the range of experimentally accessible parameters. figure, the user will have full control over many aspects: • The device on which he/she wants to perform an experiment, • The spatial arrangement of the atoms, which can be a standard lattice or a custom configuration chosen by the user.
• The type of Hamiltonian configuration, to be chosen among a finite list of allowed configurations.
• The type of experiment to be performed, i.e. standard analog quantum simulation, or variational program.
After the specification of these general characteristics, precise control over the parameters used by the machine will be possible by uploading user-defined profiles for the chosen parameters.

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 Pasqal 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 [51,52,88] 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 Pasqal 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 Pasqal 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) [89,90]. 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 [91], 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 [92,93] based on atomic ensemble memories have been proposed to overcome this challenge, and experiments are progressing rapidly [94,95].
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 [97,98]. Such processes constitute a powerful resource for photonic quantum information processing, where quantum logic is applied between photonic quantum bits (see for example Refs. [99][100][101][102]). In this area of Rydberg non-linear optics [103,104], 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 [105][106][107]. 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. 15, cold neutral atom devices hold great potential for the development of multiple key technologies in the second quantum revolution. Among them, metrology and sensing applications have already benefited from a dynamic industrial development, as illustrated by the activities of the french company muquans, an industrial partner of Pasqal. Pushing forward the capabilities of neutral atom quantum technologies, Pasqal will first offer to practitioners in Quantum Computing and Quantum Simulation the ability to control a powerful NISQ quantum processor. In the longer term, skills and technologies will be developed to establish a leading actor of quantum-enhanced devices based on atoms.