The Mathematical Tools Trailblazing the Quantum Future

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.”