Miles Stoudenmire: breaking everything apart and putting it back together again with tensors

Imagine a giant Lego ball made of billions of multicoloured blocks. With so many pieces attached to each other in every which way, it’s a big, complicated mess. You can’t see all the blocks and colours inside the ball, so it is difficult to understand the makeup of this object, or know what you can do with it.

The most logical solution is to break it apart into smaller component blocks, so you can see the pieces more clearly, and then you can start putting parts of it back together.

That’s what working with tensor networks is like.

Miles Stoudenmire, a former Perimeter Institute postdoctoral researcher who is now a research scientist at the Flatiron Institute Center for Computational Quantum Physics (CCQ) is a tensor network expert.

Tensor networks are a mathematical way of representing and manipulating extremely large, complex objects—especially quantum states—by breaking them into smaller pieces and then reconnecting them together again in a structured pattern.

As another analogy, imagine a computer that contains so much information that the data cannot be seen all at once because there is too much of it, Stoudenmire says. “But you could sip the data, and when you put some of it together, you have a panel of the data to work with. When you are done with that, you can compress it and open up another panel.”

Stoudenmire, who recently visited Perimeter Institute to give an informal talk about his work, said his interest in tensors began when he was a graduate student at the University of California Santa Barbara. 

Back then, he was working with what is known in physics as the “Monte Carlo method.” That is a computational technique that uses repeated random sampling to obtain numerical results, effectively using randomness to solve complex problems. 

But while this Monte Carlo method gives you “glimpses, like snapshots” of data, Stoudenmire was looking for a way to make it quicker and more precise. One of his advisors suggested that he go to a conference about tensors, and that opened his eyes to a better way of tackling large amounts of data.

“One way to think about tensors is to think about a spreadsheet, like an Excel spreadsheet, that we can treat like a matrix. One thing you can do with a matrix is to break it apart to detect redundancies in the data and find patterns. So you might have a matrix of a million by million, but you may find that it is really only made out of four columns that are just repeating over and over in different combinations,” he says. 

Although tensor networks began as a branch of theoretical quantum physics, they have recently found their way into much more everyday applications, like predicting like motion of air over an airplane wing, or the motion of oil through a pipe, Stoudenmire adds. 

One of the applications is the development of quantum computing “simulators,” which can be thought of as software that runs on a regular computer, but mimics what a quantum computer processor can accomplish when it is small or running a more limited type of program. 

The original idea was to run a simulator across a few steps, and after that, when the data became too complicated, you would turn to an actual quantum computer to do the rest. As it turned out, these tensor-driven quantum simulators were, in some instances, better than researchers had imagined, Stoudenmire said. “Sometimes the simulator could just keep going.” 

One can also think of programming a quantum computer as using tensors, he adds. “People in the quantum computing field don't often think of it that way, but you could view quantum computer operations, or quantum gates, as tensors.”

But the area that has been the bread and butter of tensor network research is condensed matter physics. 

“We can store all the probabilities of where you might find particles of a gas or a solid and discover what would happen if two atoms move closer together. We can compute properties, like how good of a conductor is this metal, or excitations of some exotic gas,” Stoudenmire says.

Tensor research can provide useful predictive value for scientists working in condensed matter labs. “It could tell them, for example, that if you synthesize a crystal in this way, it will have this level of conductivity. It can give them more knowledge up-front.”

More recently, Stoudenmire has been applying his tensor expertise to a technique called “sketching.” In this method, “you can do random rotations of your data, switch them down to lower dimensions, get sketch of your data and recover what you need to know about it. It speeds everything up.”

He came to Perimeter to do postdoctoral research in 2013, drawn to work with Guifre Vidal, who at the time was straddling the fields of quantum information and quantum matter at Perimeter as a leading expert in tensor networks. Vidal is now with Google Quantum AI and a distinguished visiting research chair at Perimeter.

Stoudenmire also worked with Roger Melko, a Perimeter research associate faculty with an interest in developing efficient state-of-the-art algorithms for strongly interacting quantum many-body systems in quantum matter and in error-correcting phases of quantum computers, areas where tensor networks are important.

Ultimately, what Stoudenmire loves most about working with tensors is that “it seems to be a very thorough and honest way to do physics, because you can’t leave out any details.”