This article explores tensor networks — a novel conceptual approach to quantum physics that significantly streamlines the complex calculations required to apply quantum theory effectively.

Understanding often follows language — and that applies just as much to physics and mathematics as it does to daily life. Quantum mechanics, notorious for its perplexities, is no exception. Frank Verstraete, a physicist at the University of Cambridge, has been instrumental in shaping a fresh framework for discussing the quantum realm. While this new perspective doesn’t diminish the inherent strangeness of quantum mechanics, it does make working with its mathematics far more manageable. In a recent discussion, Verstraete shared key insights into this innovative framework.

Where the challenge begins

At the heart of the issue lies the size discrepancy between us and the building blocks of the universe. Humans and the objects we interact with are composed of an immense number of fundamental particles. Consider that a mere gram of iron contains more than 10²² atoms, each comprised of even smaller constituents like electrons and quarks.

It took decades of scientific progress to recognize that all matter consists of these tiny, elementary units. Only about a century ago did the framework of quantum mechanics emerge — thanks largely to pioneers like Werner Heisenberg and Erwin Schrödinger — offering a means to describe the world at the smallest scales.

This theory marked a revolutionary shift. “Quantum mechanics was the most profound upheaval in physics,” Verstraete explains. “Before it, our grasp of the material world was extremely limited.”

Quantum mechanics opened the door to understanding basic atoms like hydrogen and allowed scientists to begin analyzing more complex atoms and large collections of particles. However, despite its power, applying quantum theory to practical problems remains incredibly challenging — and, in many cases, unsolved.

The exponential challenge

The core difficulty arises from the sheer volume of data needed to model quantum systems accurately. Imagine a simple scenario: a billiard ball rolling across a table. To describe its state at any given time, you only need to know its position — a single data point. With two balls, you require two points of information, and for n balls, you need n values. This linear growth is straightforward and predictable.

Now, contrast this with quantum particles. These can exist in superposition — occupying multiple locations at once. Only upon measurement does this ambiguity resolve into a single observable outcome, which is why we don’t witness superposition directly. Moreover, quantum particles can be entangled: measuring one particle instantaneously affects the state of another, regardless of the distance between them. This deeply non-intuitive property has been consistently confirmed through experiments.

Because of these phenomena, quantum systems can adopt vastly more configurations than classical systems. This means the amount of information needed to describe such systems scales not linearly, but exponentially with the number of particles involved. And exponential growth quickly becomes unmanageable.

Although quantum mechanics provides the equations needed to describe multi-particle systems, solving those equations becomes computationally infeasible as the system size increases — a dilemma often referred to as the exponential wall.

Approximation as a survival strategy

Given these hurdles, it’s astonishing that physics has been able to make the progress it has. Part of the reason lies in the fact that quantum effects typically fade in the macroscopic world. You don’t need quantum theory to understand billiard balls. And within the quantum domain, scientists have developed sophisticated approximation methods that make modeling tractable.

“The entire evolution of physics could be seen as a chain of ever-improving approximations,” Verstraete says. “We’ve learned how to tackle problems that are otherwise computationally intractable.” These approximations underpin vast areas of modern science, from chemistry and material science to nuclear energy, lasers, semiconductors, and MRI technology.

However, some systems resist approximation. Superconductors — which power MRI machines, high-speed trains, and particle accelerators — are one example. The search for superconductors that operate at higher, more practical temperatures is ongoing, but approximations alone won’t suffice to understand or develop them.

Mapping entanglement: the role of tensor networks

This is where tensor networks enter the picture. These visual structures resemble actual networks and serve as schematic representations that simplify the mathematical manipulations involving tensors — multi-dimensional data arrays central to quantum physics. Tensor networks specifically address the complexity introduced by entanglement.

Originally conceptualized in the 1990s within quantum computing, tensor networks were designed to depict systems using qubits — quantum analogs of classical bits that can exist in superpositions and become entangled. Quantum computing aims to harness this complexity to perform calculations that would be impossible for classical computers.

“Tensor networks gave us an entirely new language to describe how a quantum computer functions in terms of entanglement,” Verstraete notes. “They help us understand the intricate interactions and correlations between quantum particles.”

In the early 2000s, Verstraete and Ignacio Cirac recognized that these tools weren’t limited to quantum computing. They could also simplify the analysis of any many-body quantum system. While traditional quantum mechanics uses wave functions to describe the total state of a system, tensor networks chart the pathways along which entanglement spreads, providing structured “maps” of these correlations.

“With tensor networks, the picture becomes much clearer,” Verstraete says. “They reveal patterns, symmetries, and structures in the system that other methods overlook — reintroducing familiar physics techniques in new contexts.”

This illustration shows a tensor network (depicted as an interwoven red and blue grid) that outlines entanglement patterns in a complex quantum system (represented by the upper layer). It’s like a musical score revealing the hidden architecture of a composition.

Overcoming exponential barriers

One particularly powerful insight is tied to systems in their ground state — the state of lowest possible energy. These states are especially relevant because electron behavior largely determines a system’s properties, and electrons at room temperature are effectively in a very cold, low-energy environment.

“It turns out that low-energy states exhibit much less entanglement than high-energy ones,” Verstraete explains. Crucially, in low-temperature conditions, entanglement is limited to nearby particles. “If you consider a particular region and examine its relationship to the surrounding environment, the correlations tend to be concentrated around the boundary,” he adds. (More technically, this boundary-centric entanglement can be captured using an effective theory defined on a lower-dimensional surface enclosing the region.)

“Tensor networks were specifically developed to tackle scenarios like these,” explains Verstraete. By focusing solely on regions where quantum correlations are actually present, they offer a far more streamlined representation compared to traditional quantum mechanics. “This approach has drastically cut down the complexity. Tensor networks provide an entirely new framework for overcoming the exponential barrier.”

Verstraete, in collaboration with his team, is pioneering advanced computational techniques aimed at refining tensor network models. These innovations are being applied across diverse fields, including condensed matter physics, quantum field theory, atomic interactions, statistical mechanics, and the realm of quantum computing.

Shaping a new vocabulary

Since their origin three decades ago, tensor networks have evolved significantly. Verstraete recalls that when he initially began promoting their broader application, many dismissed his ideas as absurd. “People were convinced that this visual and seemingly simplistic interpretation couldn’t possibly be effective,” he says.

Why nobody understands quantum theory, co-authored by Broeckaert and Verstraete.

“However, the landscape has shifted dramatically over the past fifteen years. We now possess a fresh conceptual language, enabling us to pose different questions. Countless novel challenges have arisen, and with them, a deeper comprehension of the nature of many-body wave functions.”

Beyond his academic pursuits, Verstraete has also taken steps to demystify quantum mechanics for the broader public. When he first encountered Céline Broeckaert — now his wife — he was grappling with the difficulty of conveying the essence of quantum theory to those without a background in mathematics. Broeckaert, a theatre director, author, and producer with a focus on social change, posed precisely the kind of probing questions that helped frame the theory in more relatable terms.

Two years after their initial conversations, the pair got married and co-authored a book on quantum physics that went on to become a bestseller in Belgium. The English edition, titled Why nobody understands quantum physics, is set for release in the UK in June 2025 through Pan Macmillan. Their experience illustrates that with the right language — one that bridges the gap between abstract ideas and everyday understanding — even the most complex scientific theories can resonate with a wide audience.