*The main contributing authors to this blog are Evan Philip, Mario Dagrada and Vincent Elfving.*

The race to achieve quantum advantage on relevant problems is as hot as ever. On the hardware side, several competing architectures appeared in recent years whereas novel and improved near-term quantum algorithms emerge on an almost daily basis. Despite all the progress, quantum computing is still in its infancy and, at this stage, it is crucial to develop algorithms tailored for efficient execution on a specific hardware architecture. PASQAL is a full-stack quantum computing company and is, therefore, very well positioned to realize this vision. In this blog post, we will introduce the basics of PASQAL’s neutral atom quantum computing platform, and detail how this architecture is a perfect match with the promising near-term quantum machine-learning approach adopted by DQC (differentiable quantum circuit) algorithm, recently developed by PASQAL [1]. At PASQAL we believe that this combination may soon offer one of the first instances of narrow, industrially-relevant quantum advantage.

So, what is all the fuzz about and how does this architecture work? Let’s dive into it.

**1. Neutral-atom processors**

At PASQAL, we develop and commercialize second-generation quantum processing units (QPUs) based on ordered arrays of neutral atoms. In this architecture, each qubit is realized with a single neutral atom whose electronic energy levels represent the|0⟩ and |1⟩ states of the qubit. Upon the application of a carefully modulated laser, the atoms are trapped into the desired atomic registry using optical tweezers. Quantum computations can be achieved by shining fine-tuned laser pulses onto the atoms. Finally, the value of each qubit can be measured at the end of the computation by (literally) taking a camera image. This process is summarized in Fig. 1

Figure 1: Overview on how a neutral atom quantum processor works. Panel (a) shows the full process starting from the optical trap (SLM) for creating the atomic registry, to the moving tweezers for the quantum computation (in purple) and the CCD camera for measurement readout. Panel (b) is a photograph of the heart of the quantum processor: The chamber where the atoms are trapped. Adapted from Ref [6].

The neutral atom architecture has several advantages with respect to other competing platforms. The most salient one is their flexibility. On one hand, they are very scalable in terms of qubit count: At PASQAL, we currently run functional R&D prototypes with over 300 trapped atoms, and we plan on reaching 1000 qubits by the end of 2023. Thanks to nature, all these qubits are completely identical thus ensuring a lower overall noise and a coherence time of 10s of ms, an order of magnitude above competing architectures such as superconducting qubits and ion traps. On the other hand, the optical trap we develop allow us to arrange the atomic registry in arbitrary shapes, even in 3 dimensions, and realize almost any sort of 1-to-many and many-to-many qubit connectivity, thus reducing the needs for expensive SWAP gates to implement the most common quantum algorithms. This feature is also instrumental in achieving native implementation of multi-qubit gates such as Toffoli, another important differentiator from other platforms.

As just discussed, neutral atom QPUs can run in digital mode where circuit unitaries are decomposed into sets of compiled digital quantum gates. However, in contrast to most other platforms, they can also work in the so-called analog mode where the Hamiltonian of the full qubit system can be finely manipulated and used as a resource for computations. This mode paves the way for the development of novel quantum algorithms and the direct simulation of quantum systems on the QPU.

Given all these advantages, neutral atom processors have high chances of achieving quantum advantage in the future. However, at present there are still some hurdles to overcome towards this goal. For example, the fidelity, particularly on multi-qubit gates, needs to be improved. A similar increase is also required in the repetition rate, i.e. how many computations per second can be carried out by the processor, which is currently limited to the execution of ∼ 10 circuits per second. However, in both cases the main challenges do not stem from fundamental physics but from engineering limitations which can be 2 solved over time.

Let us now turn to the software side and see how the advantages of the neutral atom architectures can be leveraged to efficiently execute the DQC algorithm.

**2. Why neutral atoms processors are good for DQC**

In our recent blog post on the DQC algorithm [1], we showcased how to use physics-informed machine learning (PIML) [2] to solve differential equations directly on a qubit-based systems. This is achieved by a carefully crafted quantum neural network (QNN) as a universal function approximator to estimate the solution of the differential equation. In that blog post, we did not assume any specific hardware architecture to implement DQCs. Here we show how PASQAL’s neutral atom platform has certain salient features that make it an ideal platform to run the DQC algorithms.

In the previous section, we mentioned that the neutral atom platform can scale to hundreds or thousands of qubits already in the next couple of years; but how can we best make use of these qubits?

Within the DQC framework, this turns out to be quite advantageous. In particular, the maximum number of features that can be accessed by a QNN as a universal function approximator [1] depend exponentially on the number of qubits allocated to each data input dimension. With hundreds of qubits, a function f(x) on a single dimension x could potentially be decomposed by an enormous number of features, thus reaching an expressivity and accuracy not achievable by conventional methods. Perhaps, more interestingly, one could consider decomposing a multi-variate function f(x, y, z, t) of, say, three spatial dimensions and time over parallel registers and then entangle the entire register into a relatively shallow circuit. It is known from the conventional PIML methods that higher-dimensional problems are particularly interesting targets when compared to FEM methods [5]. Another use of a larger number of physical qubits, this time in common with other quantum algorithms, resides in error mitigation or correction protocols. In most of these techniques [4, 8], ancilla qubits are required to mitigate or stabilize the noisy circuit executions. While fully-error-corrected execution would require thousands of physical qubits per logical qubit, intermediate-scale mitigation could be done with smaller supports.

As discussed in the previous section, neutral atom QPUs offer the analog mode to directly manipulate the Hamiltonian of the qubit system. DQC can strongly benefit from this unique operating mode. On one hand, though DQC does not require universal entangling operations to work well, one can use carefully crafted pulses in the analog channel to entangle all the qubits; furthermore, these analog entangling layers may be interleaved with single-qubit rotations in a digital-analog approach. This allows a great reduction in the number of quantum gates needed for running the DQC algorithm efficiently, thus contributing to increased accuracy at attainable system sizes.

**2.1. Digital-Analog Mode**

The idea of digital-analog circuits sounds great, but how do you implement those on neutral atom systems? Intuitively, one might believe that one either runs the system in digital-mode or in analog-mode, but not side by side in one experiment.

In Figure 2 we schematically represent how individually-addressable digital gates may be implemented. We also show several strategies towards the implementation of a mixed digital-analog circuits on our neutral atom platform.

Figure 2: Different strategies for addressing individual qubits in a neutral atoms system and mix it with analog mode of operation. (a) with a single laser, multiple individual qubits can be addressed sequentially using an acousto-optic deflector (AOD). This can scale to a very large number of qubits without requiring equally many lasers. (b) Instead of redirecting the operational laser beam, the overall laser can be masked on some qubits by detuning those far away from the interaction frequency by using an additional SLM (see Ref. [9] where this technique was used for state preparation). In this way, effectively distinct rotations can be performed on distinct qubits. © digital-analog circuit diagram, consisting of both analog evolutions over the atomic qubit register, as well as digital gates which may target single or multiple qubits. (d) One strategy to mix digital and analog modes of operation, is by switching between the natural energy levels used to operate in, denoted as hyperfine (h), ground (g) and Rydberg (R) states of the atoms. In this scheme, the analog evolution is restricted to the {|1⟩, |r⟩} basis, while the digital (gate based) computations are done in the {|0⟩, |1⟩} basis. Switching between these is performed by applying laser pulses to transfer population.

**2.2. Two-qubit example**

Now that we’ve seen how digital-analog circuits can be implemented on neutral atoms processors, let’s take a look at a two-qubit example of solving a simple 1D ODE. In Figure 3(a) we show how a simple ODE is solved by a DQC-trained QNN.

In Figure 3(b) we show the digital-analog circuit which was used to obtain the solution. Digital single-qubit gates are combined with a ‘analog block’ implemented as a Hamiltonian evolution over the natural atom-atom interaction for a time t. Note that only a single variational ‘layer’ is used, but the degree of control is sufficient to independently tune all 4 Chebyshev polynomials in the 2 2 -dimensional Hilbert space. This is evidence of the entangling power of the analog block.

Figure 3: (a) 1D Ordinary Differential Equation in f(x) with boundary condition f(0) = 1, and which has a 4th order polynomial in x as a solution. The DQC-trained QNN output is plotted with filled circles. (b) Digital-Analog circuit used for the QNN for the solution shown in (a). Digital single-qubit gates are combined with a Hamiltonian evolution over the natural atom-atom interaction for a time t. Shifts s1, s2 and s3 denote the parameter shifts which may be used to compute derivatives of the function with respect to x and/or variational parameters θ and t.

While this serves as a simple example, in the following blog posts of the series we aim at expanding the applications of digital-analog strategies to the solution of more realistic scenarios.

**2.3. Challenges & Outlook**

Finally, let’s focus on two apparent obstacles and see how the combination of neutral atom QPUs and DQC can successfully cope with them. The first one is related to any quantum machine learning (QML) algorithm applied to classical input/data and it is commonly referred to as the data input-output problem: Even if the computation itself on the quantum hardware is efficient, the loading or extraction of data from or into the classical realm becomes a bottleneck for large data sets whereas this problem is easily handled by conventional computers. Secondly, we saw that current neutral atom processors are characterized by a low (but improving) repetition rate. With cycle times currently being several factors higher than other platforms, variational quantum algorithms with the same hyperparameters may run slower.

At a first sight, it seems we have a problem here. However, when combining the DQC algorithm with the neutral atom platform, these two issues can be successfully tackled:

- While datasets are often huge in standard machine learning tasks, posing a data input-output bottleneck for QML algorithms on current-day processors, in the particular case of PIML and DQC the number of data points can be orders of magnitude smaller; that is because the information used for training is not just included in the values of the training function itself, but also in the values of its derivatives and its potential correlations with other observables. This allows not only to handle the data input-output problem but also yields a significantly reduced number of total circuits to be evaluated when compared to other QML algorithms.
- In DQC, Hamiltonian averaging is used to represent function values [1, 7]. The accuracy of this estimate scales quadratically with the number of repetitions of the circuit (commonly referred to as shots). If we take the case of the famous variational quantum eigensolver for chemistry calculations, there is a particular target accuracy known as ‘chemical accuracy’, which is 1 kcal/mol. Even for small to medium sized systems, the fraction of this accuracy to the total spectral width of the Hamiltonian is huge, which implies a significant number of repetitions is required to attain this accuracy, often of the order of ∼ 106 . In the case of DQC, instead, we may allocate a cost-function spectrum of our choice, within a comfortable range of the solution-space, and thus require much fewer repetitions for a good relative accuracy. Furthermore, when gradient descent is used, the direction of the angle-update is much more important than accurately predicting how large the step in that direction should be. Hence, down-to 1 shot-per-loss derivative term may be sufficient in early stages of training [10] using the standard stochastic gradient descent. In this way, a low repetition rate does not represent a problem for the DQC algorithm.

In this post, we showed that the DQC algorithm constitutes a perfect candidate for efficient execution on neutral atom devices, thanks to the unique features offered by this platform. Be sure to check out our post about DQC and our first use-case example !

## Solving differential equations with machine learning

### Partial differential equations and finite elements

medium.com

## The power of differentiable quantum circuits

### The main contributing authors to this blog are Evan Philip, Mario Dagrada and Vincent Elfving.

medium.com

**References**

[1] Solving differential equations with machine learning

[2] The power of differentiable quantum circuits

[4] Suguru Endo et al. “Hybrid Quantum-Classical Algorithms and Quantum Error Mitigation”. In: Journal of the Physical Society of Japan 90.3 (2021), p. 032001. doi: 10.7566/JPSJ.90.032001. eprint: https://doi.org/ 10.7566/JPSJ.90.032001. url: https://doi.org/10.7566/JPSJ.90. 032001.

[5] Philipp Grohs et al. “A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations”. In: (2018). arXiv: 1809.02362 [math.NA].

[6] Loïc Henriet et al. “Quantum computing with neutral atoms”. In: Quantum 4 (Sept. 2020), p. 327. issn: 2521–327X. doi: 10.22331/q-2020–09–21- 327. url: https://doi.org/10.22331/q-2020-09-21-327.

[7] Oleksandr Kyriienko, Annie E. Paine, and Vincent E. Elfving. “Solving Nonlinear Differential Equations with Differentiable Quantum Circuits”. In: PRA 103.5, 052416 (May 2021), p. 052416. doi: 10.1103/PhysRevA. 103.052416. arXiv: 2011.10395 [quant-ph].

[8] Joschka Roffe. “Quantum error correction: an introductory guide”. In: Contemporary Physics 60.3 (July 2019), pp. 226–245. issn: 1366–5812. doi: 10.1080/00107514.2019.1667078. url: http://dx.doi.org/10.1080/ 00107514.2019.1667078.

[9] P. Scholl et al. “Microwave-engineering of programmable XXZ Hamiltonians in arrays of Rydberg atoms”. In: arXiv e-prints, arXiv:2107.14459 (July 2021), arXiv:2107.14459. arXiv: 2107.14459 [quant-ph].

[10] Ryan Sweke et al. “Stochastic gradient descent for hybrid quantum-classical optimization”. In: arXiv e-prints, arXiv:1910.01155 (Oct. 2019), arXiv:1910.01155. arXiv: 1910.01155 [quant-ph].