Many things in life become easier to understand once you've found he right words to talk about them. That also goes for topics in maths and physics — in particular, it goes for the famously tricky theory of quantum mechanics. Frank Verstraete, a physicist at the University of Cambridge, has helped to create a new language to talk about the quantum world. It doesn't make the theory less strange — nothing will ever do that — but it makes the calculations involved in appliying it a whole lot simpler. In a recent interview Verstraete gave us a gist of what the new language is all about.

The root of the problem

The fundamental particles of nature are small, but we ourselves are big. So are all the other creatures and objects we come into contact with every day. This means that the number of particles that make up all we see is absolutely vast. For example, just 1g of iron contains over 10,000,000,000,000,000,000,000 atoms. And each of these atoms is itself made up of even smaller particles, such as quarks and electrons.

It took humanity quite a long time to figure out how all matter is made up of tiny fundamental particles. It's only around a hundred years ago that quantum mechanics, the theory of physics that describes the world at very small scales, was formulated. Most of the credit for this goes to Werner Heisenberg and Erwin Schrödinger. (See here for a very brief introduction to quantum mechanics.)

The theory was a breakthrough. "Quantum mechanics was the biggest disruption in physics ever," says Verstraete. "Before quantum mechanics, we did not really understand anything about matter."

Thanks to quantum theory we gained, for the first time ever, a complete understanding of simple atoms such as hydrogen. We also gained the tools to tackle more complicated atoms and larger systems of particles, such as lumps of iron. The problem is that to this day we don't know how to properly employ these tools to obtain useful answers for all the questions we have about the quantum world.

The exponential wall

The problem stems from the fact that, to describe the quantum world, you need an awful lot of information. To illustrate this, first think of an ordinary, non-quantum, billiard ball moving around on a table. A single piece of information describes what this system looks like at a given moment in time: the location of the ball. If there are two balls on the table, you need two pieces of information: the locations of both balls. Generally, if there are n balls, you need n pieces of information. As you would expect, the number of pieces of information needed grows with the number of balls, but it grows in proportion. The growth is what's called linear.

If, instead of a system of billiard balls, you have a system of quantum particles, things become a lot more complicated. A quantum particle can be in a state of superposition where it is at many locations at the same time. It is only when you measure its position that quantum reality collapses to a single one of the many possibilities. That's why we never observe superposition in real life — once you look, it disappears.

The mathematics that describes quantum mechanics also allows something called entanglement: two particles, even if they are far apart from each other, can be related so that when the position of one of them is decided through a measurement, the position of the other is immediately decided as well. Like superposition, the concept of entanglement is counter-intuitive. But entanglement exists as clear as day within the mathematics of quantum mechanics and has been established in experiments.

A consequence of this strange state of affairs is that a quantum system can exist in many more configurations than a system made up of macroscopic objects such as billiard balls — there's a multitude of entangled states that billiard balls can't occupy. As a result, the number of pieces of information you need to describe a system of quantum particles grows exponentially with the number of particles involved. Exponential growth is a whole lot faster than the linear growth in our system of billiard balls.

The galloping complexity means that, although quantum mechanics provides the equations that describe many-particle quantum systems, solving these equations is extremely hard. It's usually impossible when you're looking at a system of more than a few particles. This is often called the exponential wall problem of many-particle quantum systems.

Approximating the world

Given this problem, it seems a miracle that physics is possible at all. The fact that we're able to describe and even predict the behaviour of so many physical systems is partly down to the fact that quantum effects disappear in the macroscopic world. You don't need quantum physics to describe billiard balls. When it comes to the quantum world, descriptions are possible because we have found good ways of approximating the solutions to the equations involved.

"The whole history of physics is basically a continuation of approximations," says Verstraete. "A history of approximating problems that are actually exponentially hard." Such approximations are surprisingly successful. They underlie much of chemistry and material science and have given us nuclear energy, lasers, transistors, and MRI scanners to name just a few applications.

But there are also many systems for which approximations won't do. An interesting example are superconductors which produce the high-intensity magnetic fields that are needed in MRI scanners, maglev trains, and particle accelerators. Current superconductors only function in extremely low temperatures that are costly to produce, so scientists are looking for high-temperature alternatives. These, however, require more than mere approximations of many-particle quantum systems.

Road maps of entanglement

This is where tensor networks come in. Visually, tensor networks are diagrams that look, unsurprisingly, like networks. Mathematically, they are pictorial devices that help you keep track of the complicated calculations (involving mathematical objects called tensors) of quantum mechanics. In terms of physics, tensor networks focus on one of the main sources of complexity: entanglement.

The networks were first developed in the 1990s in the context of quantum computing. Quantum computers use quantum particles called qubits instead of the ordinary 0-or-1 bits familiar from ordinary computation. Qubits can be in a superposition of both 0 and 1, and they can become entangled. The idea is to exploit their entanglement to achieve vast speed-ups compared to ordinary computation. In computing, more complexity means more computational power. (See here for a brief introduction to quantum computing.)

"Tensor networks formed a whole new language that was created to describe how a quantum computer would work in terms of entanglement." says Verstraete. "It's all about how particles interact and finding new ways of describing quantum correlations between them."

What Verstraete and his collaborator Ignacio Cirac realised in the early 2000s is that tensor networks are useful, not just to describe quantum computers, but also to describe other many-particle quantum systems. While the ordinary formulation of quantum mechanics (in terms of something called wave functions) takes into account the entire big picture — all that is going on in a system like, say, a superconductor— tensor networks identify routes along which entanglement propagates from particle to particle. They provide entanglement road maps.

"Tensor networks lead to a huge simplification," says Verstraete. "This way of viewing the system has lots of structure and symmetries and all the usual tools that have been brought to physics seem to re-apply."

Leaping over the exponential wall

A particularly interesting result comes from the fact that we are most interested in many-particle systems that are in a so-called ground state, a state of lowest energy. This is because the properties of many-particle systems are largely determined by the properties of the electrons inside them, and electrons find the temperatures we tend to be interested in, room level temperatures, extremely cold. This makes them sluggy, so the overall energy of the system is low.

"It has been discovered that there's much less entanglement in low energy states than in high energy states," says Verstraete. In particular, entanglement at low temperatures is heavily constrained: it can only happen between particles that are close to each other. "If you look at some region within your system and the correlation to its surrounding, you find that all the correlations are in some sense concentrated on the boundary of the region." (Technically, the correlations between the region and its surroundings can be described by an effective theory that is formulated on the lower-dimensional boundary of the region.)

"Tensor networks have been constructed exactly to describe such situations," says Verstraete. By targeting only the places where correlations are actually happening, they present a much simpler picture than the ordinary formulation of quantum mechanics. "This has led to an exponential reduction of complexity. Tensor networks give a new way of breaking down the exponential wall."

In his work Verstraete, together with his colleagues, is devising novel computational methods for optimising tensor networks, and applies these methods to a range of areas: from condensed matter physics, quantum field theory, and atomic physics to statistical physics and quantum computing.

Finding words

Tensor networks have come a long way since their conception 30 years ago. Verstraete admits that when he first started advocating their wider use, people thought he was a "complete crackpot". "They thought there is no way that this new simplistic way of looking at things in terms of pictures would work."

"But things have changed enormously in the last fifteen years. We have a new language so we can ask new questions. Plenty of new problems have emerged and there's a new way of understanding what it means to have many-body wave functions."

Verstraete has also spread the word about quantum mechanics more generally. When he met his now wife, Céline Broeckaert, he was frustrated by how difficult it is to explain the theory to non-experts, without recourse to the language of mathematics. Broeckaert, a theatre director, novelist and social-impact producer, proceeded to ask just the questions that were needed to develop an accessible perspective on the theory.

Two years later the two were married and published a book about quantum mechanics that became a bestseller in Belgium. The English version, called Why nobody understands quantum physics, will be published in the UK in June 2025 by Pan Macmillan. It goes to show — if you can find a language that suits both your audience and your ideas, even the trickiest of theories becomes accessible.

About this article

Frank Verstraete is Leigh Trapnell Professor of Quantum Physics at the Department of Applied Mathematics and Theoretical Physics at the University of Cambridge.

Marianne Freiberger, Editor of Plus, interviewed Verstraete in March 2025.