Building New Bridges in Science

To understand is to create new connections, to find links between something we do not know and what we know well. Understanding many-body quantum systems – collections of many quantum particles -- holds the promise to bring us a step closer to building quantum computers, extremely fast and efficient machines that will help us solve problems that classical ones cannot handle. The peculiar nature of quantum interactions, however, makes this an arduous enterprise.

Researchers in the ICTP’s Condensed Matter and Statistical Physics (CMSP) section, including ICTP scientist Marcello Dalmonte, and Tiago Mendes-Santos, a researcher at the Paris-based quantum computer startup company PASQAL, have recently found an important connection between these intricate systems and network theory. This could help us tackle the complexity of many-body quantum systems, and therefore help us build more efficient quantum computers. Their results were published in 2024 in the prestigious journal Physical Review X.

Unlike a classical particle, which follows a precise trajectory in space and time, a quantum particle is described by a wave function. This function represents the probability of finding the particle in a certain state at a given moment. A system consisting of many quantum particles is described by a wave function that results from the combination of the wave functions of all its constituents. And because of the strange nature of quantum mechanics, the total wave function depends on all the possible states that each particle can be found in.

New measuring techniques give us accurate snapshots of the wave function of many-body quantum systems. Efficiently extracting and processing information from such observations, however, remains challenging. Researchers in ICTP’s CMSP section and their collaborators working in research centres located in Germany and in France have recently found a smart way to use the rich information provided by modern measurements of the wave function of many- body quantum systems that relies on network theory.

Mendes-Santos started working on these questions in 2020, while a postdoctoral researcher at ICTP, in Dalmonte’s group. Eager to tackle the complexity of many-body quantum systems in new ways, the two started collaborating with Alexander Rodriguez, a professor at the University of Trieste, to apply his expertise in Artificial Intelligence to quantum many-body systems and unveil some of their properties.

“We wanted to extract the maximum information from the measurements made available by the most recent real experiments with programmable quantum simulators,” Mendes-Santos explains. “We showed that these datasets, made out of series of snapshots, can exhibit interesting properties when mapped onto a network,” he continues.

“We obtained the network by introducing the notion of “distance” between each pair of snapshots. The more different two snapshots are, the more distant they are from each other. Each snapshot will correspond to a vertex of the network and two points will be connected only when the snapshots are similar enough. We can then use network theory to characterize the graph that we obtain in this way,” Mendes-Santos explains.

“A fundamental property of networks is their degree distribution, which describes the probability that a node in the network has a certain number of connections. By looking at the degree distribution, you can tell if the network is random, or if it has a structure,” Mendes-Santos says, and continues, “What we observed is that in certain regimes, and particularly close to a phase transition, the degree distribution of our networks follows a power law. In the jargon of network theory, these networks are called “scale-free”.”

Scale-free networks are some of the most-studied classes of networks and they have many interesting properties. For example, they are characterized by a significant number of large nodes that we call “hubs”, which have many connections. Many networks in real life are scale-free -- think for example of the network of airports, where Paris Charles de Gaulle, London Heathrow, or JFK in New York play the role of hubs. Another example are social networks, where the hubs are represented by the so-called influencers.

By applying advanced data-mining techniques, Mendes-Santos and his collaborators have established a crucial two-way bridge between network theory and quantum physics. This connection has the potential to reveal significant new insights into both fields. In particular, network theory could help us characterize large-scale quantum devices, and thus bring us a step closer towards building more effective quantum computers. These results also exemplify how sophisticated data analysis and artificial intelligence algorithms can aid scientists in exploring frontier research topics, in particular within the context of quantum science and technology.

Read the full paper: https://journals.aps.org/prx/pdf/10.1103/PhysRevX.14.021029