The Mathematical Tools Trailblazing the Quantum Future

Quantum physicists are constantly running up against comically large numbers. Just storing the mathematical description of a simple quantum system of only 100 atoms would require far more storage space than that available on every computer hard drive on Earth combined.

This exponentially daunting challenge has led some to think that the only way to solve some of the thorniest quantum problems is with quantum computers. But a team at the Flatiron Institute’s Center for Computational Quantum Physics (CCQ) is using mathematics to show otherwise.

A key test for the CCQ’s method came in March 2025, when a group of quantum computing researchers reported that they had used a quantum computer to complete a task that they claimed would take a classical computer tens of thousands of years to figure out.

“I’m always a bit skeptical of these types of claims,” says Joseph Tindall, an associate research fellow at the CCQ. “But this one in particular, I realized there were some approaches they had overlooked.”

Tindall and his colleagues at the CCQ are experts at solving complex problems in quantum physics using conventional computers and cutting-edge techniques. He and his collaborators set out to complete the task without a quantum computer and, within a few months, had refined a method to solve the problem. Their approach was so efficient that they could solve one of the tasks at hand on a consumer desktop computer in just 30 minutes.

The secret to the team’s success was a mathematical object known as a tensor network. Tensor networks use mathematical tricks to simplify systems that can be represented by immense numerical tables of numbers. This allows them to compress huge amounts of data, such as those in quantum systems, into manageable, interconnected tables known as tensors.

Beyond testing claims of quantum supremacy, tensor networks are enabling scientists to tackle many other difficult problems in areas ranging from computer science and mathematics to condensed matter and quantum physics. Foundational work being done at the CCQ is helping lay the groundwork for future breakthroughs, from understanding high-temperature superconductors to developing leaner machine learning approaches.

Research into tensor networks began in the 1970s, led by physicist and mathematician Roger Penrose, who later shared a Nobel Prize in physics for his mathematical description of black holes. Penrose’s notation for tensor networks is still used, but the applications of these mathematical systems have expanded far beyond the problems for which he introduced them.

In 1992, Steven R. White devised a powerful algorithm for understanding quantum systems — the density matrix renormalization group — which is now understood (thanks in part to the work of Ulrich Schöllwock) to inherit its efficiency from an underlying tensor network structure. Over the last 15 years, the use of tensor networks has exploded, and they are now found in fields from mathematics to chemistry.

“There are a lot of problems in math and science where you have huge amounts of data — data so big it can’t fit on your computer,” Tindall says. “A tensor network offers a way of compressing that data, kind of like a zip file.” These techniques come from a branch of mathematics called linear algebra that deals with tables of numbers known as matrices. Linear algebra is typically used to describe systems in low dimensions such as those involving a single matrix. But by linking matrices together, researchers realized that linear algebra could be used to describe problems with more dimensions.

Using networks of these linked matrices — that is, tensor networks — researchers can take very complex systems found in quantum dynamics and simplify them enough that they can be handled by classical computers. Some of the mathematical methods used in this simplification were developed recently, while others are hundreds of years old. Some of those techniques, though known to mathematicians, are new to quantum dynamics research.

“One of our mathematicians recently introduced us to this ancient technique widespread in mathematics, which is new to us as physicists,” says Miles Stoudenmire, a research scientist at the CCQ. “It’s blowing our minds because it lets us solve certain problems exponentially faster.”

These problems include challenges in quantum physics in which a system might change quickly early on but settle down over time. The ancient mathematical method, which uses unconditionally stable differential equation solvers, enables the system to be simulated more quickly over time so the entire simulation can be completed much faster.

Today, tensor networks are used to study a range of problems from chemistry to disease transmission. However, their primary application remains studying quantum systems. Some of this research is allowing scientists, including those at the CCQ, to better understand superconductors — materials through which electricity flows without resistance. Superconductors are important in many technologies, from magnetic resonance imaging (MRI) to quantum computing. The work could ultimately lead to the development of better superconductor technologies.

To help expand the use of tensor networks, a group at the CCQ headed by Matthew Fishman is developing ITensor, an open-access software library that helps researchers around the world develop better and more efficient tensor network software. By providing a standardized way to record system information, ITensor simplifies running complex simulations such as those involving quantum systems.

“ITensor has been very popular with physics professors all around the world,” Stoudenmire says. “It’s a tool that lets their students really quickly and reliably describe a quantum system in a lab.”

An additional software library built on top of ITensor, the Tensor Network Quantum Simulator, is also being used by quantum computing companies to help benchmark their quantum claims. Researchers at the CCQ are continually working to improve their tensor network libraries to better handle complex systems and incorporate new techniques under development.

“Tensor networks are a very nascent field still, and it’s important to have state-of-the-art codes available so that people can use them to solve new problems,” Tindall says. “The aim of what we’re doing with ITensor and other projects at the CCQ is to really make things more accessible so we can push the boundaries of what’s possible with tensor networks.”