## The dawn of the quantum internet

We are at the beginning of a new era of technology — an era in which we can harness the power of quantum mechanics to revolutionize the ways we compute, sense, communicate, and secure data. Central to this revolution lies one vision: the quantum internet.

The quantum internet is a global network of *quantum* computers linked by *quantum* communication. In the long term, the quantum internet will enable distributed quantum computing alongside a slew of operational advantages in network security, distributed sensing, and communication throughput. In the near term, however, the storage and processing capabilities of quantum network hardware are severely limited due to the presence of noise. These limitations hinder our ability to realize the full advantage of the quantum internet.

Nevertheless, we have already begun to build rudimentary quantum communication networks. Within these networks, quantum states are prepared at specified devices—called sources—and sent using quantum communication to detector devices, which measure the received quantum states. Nascent quantum communication networks are naturally referred to as *prepare-and-measure networks* and are able to implement simple applications such as secure key distribution, certified random number generation, and entanglement swapping.

*Predicted stages of development for quantum networks. In each stage, novel functionality is introduced and scaled. Over time, this incremental development leads to the quantum internet [1].*

While new functionality enables new applications, it is not required for scaling. A quantum network can be scaled by manufacturing many devices and linking them together into a network. In this way, it is reasonable to expect that prepare-and-measure networks will be implemented at scale relatively soon.

To design quantum network applications at scale we will undoubtedly rely upon simulation and optimization tools, which are limited by a key technical challenge. That is, large quantum systems are notoriously difficult to simulate and optimize using classical methods. At some point, the quantum networks we build will exceed our capacity to simulate and optimize them on a classical computer.

How can we design quantum networks if our classical approaches fail?

Perhaps quantum networks should be designed using quantum methods? Recently, hybrid quantum-classical algorithms have shown promise of outperforming classical simulation and optimization methods, even on the noisy quantum computers of the near term.

When applied to quantum networks, the hybrid algorithm’s advantage is derived from the ability of quantum computers to simulate quantum physics efficiently. Then, using classical software to perform variational optimization, the simulation can be tuned in a procedure that resembles training a neural network. These so-called *variational quantum optimization* algorithms may play a key role in assisting the design and development of quantum networks. Remarkably, these hybrid methods can even be applied on quantum network hardware to train protocols directly against the hardware’s noise.

## Turning theory into action

In a collaboration between Xanadu and the University of Illinois at Urbana-Champaign (UIUC), we have taken a step towards these theorized hybrid quantum network design tools.

**Project Contributors from left to right**: Thomas Bromley (Theoretical Physicist @ Xanadu), Brian Doolittle (Physics PhD Candidate @ UIUC), Nathan Killoran (Head of Software @ Xanadu), and Eric Chitambar (Associate Professor of Electrical and Computer Engineering @ UIUC).

Our first step was to develop a hybrid framework for simulating and optimizing quantum networks. Our open-source software is publicly available as a Python package called qNetVO: The Quantum Network Variational Optimizer. The goal of qNetVO is to extend PennyLane‘s trainable quantum circuits to quantum networks, making it easy to set up and optimize quantum network simulations on both classical and quantum hardware.

## Simulating quantum networks

qNetVO can simulate noisy prepare-and-measure networks by mapping a description of the network into a quantum circuit that simulates the network. Using PennyLane, the network simulation can be run on a wide range of quantum processors or classical simulators. As an example, we’ll consider a network that has two independent sources $ \Lambda_1 $ and $ \Lambda_2 $ that each prepares an entangled state. The entangled states are then sent through a noisy communication link and measured by three detectors. Such networks are an important element for entanglement swapping and, when scaled, they can be used for entanglement distribution and long-distance communication using quantum repeaters.

*A quantum network and its corresponding simulation circuit [2].*

## Optimizing quantum networks

PennyLane’s framework for automatic differentiation of quantum circuits can then be used to optimize the quantum network simulation. This is done by first defining a cost function $ \text{Cost}(\mathbf{P}^{\text{Net}}(\Theta)) $ that evaluates some important quantity of the network using the probability distribution obtained from the network simulation $ \mathbf{P}^{\text{Net}}(\Theta) = {P(\vec{a}|\vec{x},\Theta_{\vec{x}})}_{\vec{a},\vec{x}} $. For example, the cost function could be an error probability, entropic quantity, or some distance measure.

The goal in our optimization is to minimize the cost by tuning the simulation parameters $ Θ $
. Our goal can be achieved using an optimization procedure called *gradient descent*, in which a function is optimized by incrementally following the path of steepest descent. PennyLane allows us to compute the gradient of the cost function on either classical or quantum hardware.

## Demonstrating the value of variational quantum optimization

Our next step in this process was to justify that our variational quantum optimization software could, indeed, provide value as a quantum network design tool. For this demonstration, we turned to a key network property: nonlocality.

## Quantum nonlocality

Colloquially, nonlocality is the quantum phenomenon deemed “spooky” by Einstein. The spookiness is illustrated by a scenario in which a source shares an entangled state between two independent detectors, $ A $ and $ B $. If each detector performs certain measurements, then stronger-than-classical correlations can form between the two locations. These nonlocal correlations cannot be obtained using classical physics, making them an ideal test to prove that a system is quantum. Thus, establishing nonlocal correlations is important to the development of quantum technology.

*(Left) Test procedure for generating nonlocal correlations. (Right) Quantum (orange) versus classical (blue) correlations in measurement results [3].*

Quantum nonlocality can be extended to networks that have many sources and detectors. Similarly to the single source case, the classical correlations in networks are bound by inequalities, which can be violated by quantum correlations. However, the noise in near-term quantum network hardware is detrimental to the formation of these nonlocal correlations. As such, if a noisy quantum network can’t demonstrate nonlocality, it’ll have a tough time convincing anyone that it is truly quantum.

Thus, it is important to understand how these nonlocal correlations deteriorate in the presence of noise. This is a problem to which we applied the optimization and simulation capabilities of qNetVO.

## Variational quantum optimization of nonlocality in noisy quantum networks

In our recent work, “Variational Quantum Optimization of Nonlocality in Noisy Quantum Networks” [2], we use qNetVO to optimize a network’s state preparations and measurements for maximal nonlocality with respect to a fixed noise model. The main idea behind our work is to take an inequality $ S_{\text{Net}}\leq \beta $ that bounds the classical correlations, and apply it as a cost function $ \text{Cost}(\Theta) = - S_{\text{Net}}(\Theta) $ , parameterized by the settings of the network simulation. Then we use variational quantum optimization to minimize the cost, thereby maximizing the quantum violation to the classical bound.

Using PennyLane’s classical simulators on a laptop computer, we simulated noisy networks with up to 8 qubits. We found not only that our optimization results were consistent with the theoretical results derived in literature, but that we can identify examples of nonlocal correlations stronger than those reported in the literature.

We then turned to IBM’s 7-qubit quantum processor to show that our optimizations also work on hardware. For all considered networks, variational quantum optimization was able to find maximal violations when run on noisy quantum hardware, meaning our optimization converged on average to the expected theoretical maximum.

*Using IBM’s quantum hardware to maximize nonlocality in different quantum networks (see [2] for details).*

## Conclusion

Quantum networks will revolutionize our communications infrastructure. However, there is much to learn about the advantages that these networks can provide. To design applications that realize these advantages, we will need tools to simulate and optimize network applications. We argue these tools can be built using hybrid quantum-classical computation.

To argue this point, we developed the qNetVO software and applied it in our recent work [2] to maximize nonlocality in noisy quantum networks. We showed that variational quantum optimization can reproduce known results and even be used to gain interesting and novel insights. As such, our software has shown its value to current research.

It will be exciting to see how hybrid algorithms can be used to support the design of future quantum networks. However, the most exciting path forward is, perhaps, the direct application of our techniques on quantum network hardware.

*Variational quantum optimization can be used to train a quantum network like a neural network. This may allow us to optimize protocols against the noise inherent to quantum networking devices.*

In conclusion, variational quantum optimization will play an important role in moving us forward as dawn breaks over the quantum internet.

## Acknowledgements

The collaboration between UIUC and Xanadu was made possible through the Quantum Information Science and Engineering Network (QISE-Net) fellowship awarded to Brian Doolittle and the support of the Hybrid Quantum Architectures and Networks (HQAN) Quantum Leap Challenge Institute.

## References

[1] Wehner, Stephanie, David Elkouss, and Ronald Hanson. “Quantum internet: A vision for the road ahead.” Science 362.6412 (2018).

[2] Doolittle, Brian, Thomas Bromley, Nathan Killoran, and Eric Chitambar. “Variational Quantum Optimization of Nonlocality in Noisy Quantum Networks.” arXiv preprint arXiv:2205.02891 (2022).

[3] Brunner, Nicolas, Daniel Cavalcanti, Stefano Pironio, Valerio Scarani, and Stephanie Wehner. “Bell nonlocality.” Reviews of Modern Physics 86, no. 2 (2014): 419.