Physicists have used ordinary computers, advanced mathematics, and specialized software to solve difficult problems in quantum physics that were once said to be beyond the reach of classical machines.
The study was conducted by researchers at the Simons Foundation Flatiron Institute’s Center for Computational Quantum Physics (CCQ) and collaborators at Boston University. Their method proved efficient enough that some calculations could be performed on personal laptops.
This approach has the potential to expand the range of quantum mechanics problems that scientists can study by extracting more computational power from conventional hardware. It may also provide a useful strategy for optimization problems where researchers need to identify the best answer among many possible solutions.
The research results were published in a magazine science.
Simulation of hundreds of interacting qubits
The challenge involved modeling hundreds of interacting “qubits,” which are equivalent to the quanta of bits used in traditional computers. The qubits were arranged in a square, cubic, or diamond-shaped lattice.
Traditional bits store 0 or 1. However, qubits can exist in multiple superpositions. This feature gives quantum systems unusual capabilities, but at the same time makes it extremely difficult to reproduce their behavior with classical computers.
Articles from March 2025 (also included) scienceanother research team reported using quantum computers to calculate the dynamics of particularly complex qubit systems. The research team claimed that no classical computer could match this achievement.
“Whenever we (at CCQ) see these kinds of claims, we always get a little skeptical,” says Joseph Tyndall, an associate research scientist at CCQ and lead author of the new Science paper. “Like, ‘Did you try this? Did you try that?'”
For CCQ researchers, this argument provided a compelling way to test the limits of their own technology.
Study co-author Miles Staudenmayer, a research scientist at CCQ, said the issue was an opportunity to “take the tool for a test drive.” “We could have chosen a more arbitrary target,” Staudenmayer said. “But I was like, ‘Why not choose this one that has a big statement?'”
The challenge of quantum entanglement
One of the biggest obstacles was quantum entanglement. When qubits are entangled, their properties remain connected, even if the qubits are far apart. As a result, researchers are unable to model each qubit independently.
Instead, sophisticated algorithms are required to describe the entire system.
“Quantum physics gives us a wave function that describes the state of a system when you have a lot of interacting particles,” Tyndall says. “This huge object grows rapidly as more particles are added.”
The wave function contains the information needed to describe a quantum system, but its size increases exponentially as particles are added.
Because the size of the wavefunction grows exponentially, “you can’t store it directly in a computer,” he says. Dealing with such huge wave functions is a recurring problem in quantum physics. However, these calculations are essential for predicting the behavior of quantum materials, including superconductors.
Compressing huge quantum systems
Researchers have overcome this barrier by developing and applying new tools based on tensor networks. These mathematical structures compress the information contained in the wave function, allowing it to be processed more efficiently.
Tindall likens this approach to “taking a zip file of wavefunctions with all the information and compressing them into a mathematical data structure filled with small tables of interconnected numbers.”
This compression made the simulation manageable on classical computers. Tindall completed many of the first calculations on his laptop using ITensor, a high-performance tensor network software library created at CCQ.
New simulations show how the ITensor team is adapting tensor technology to new types of problems. In this case, the researchers used a 3D tensor network to model three-dimensional quantum mechanics.
“This very powerful compression is very effective, but it’s a very complex mathematical object,” Tyndall says. “This is really a bit of a new frontier, because working with these objects, especially in three dimensions, is unprecedented. Working with these objects requires sophisticated code and algorithms, which is a software engineering challenge in itself.”
Old algorithms find new uses
Many of the simulations required only relatively modest computing resources. In early calculations, Tyndall used belief propagation. This is an algorithm developed in the 1980s, and researchers have recently adapted it to quantum systems.
“It’s a little more approximate than some of the other methods, but it’s much cheaper and more straightforward to perform for many difficult problems,” Staudenmayer says.
He contrasts this with “more sophisticated methods in the past in our field,” saying, “Some of these three-dimensional problems would have been so large that we wouldn’t even have been able to address them.”
Although the hardware was modest, the results reached state-of-the-art levels of accuracy. The simulations produced solutions consistent with theoretical predictions, with good results for small problems where the correct answer could be verified.
Most importantly, the results matched those previously obtained using quantum computers. The difference is that the new calculations don’t require quantum hardware.
Classical computing and quantum computing can work together
This discovery further deepens the debate about where classical computing ends and quantum dominance begins. But Tyndall and Staudenmayer emphasize that the two fields are not simply in competition with each other.
Classical simulations help researchers understand what quantum computers can do, but advances in quantum hardware may give rise to new classical techniques.
“The nice thing about the classical versus quantum computing debate is that there are a lot of synergies between the kinds of simulations we’re interested in and the code we write and what quantum computers can achieve,” Tyndall says. “It helps guide us, and it helps guide quantum computing researchers, because it’s clear that the barrier to entry for us to simulate certain things is much easier than they are, because we don’t have to build a quantum computer. We can just write code and press ‘run’ on a personal computer.”
Challenge to the next quantum simulation
Researchers are now developing methods that go beyond systems made entirely of qubits. Their next goal is to model electrons that can move between different sites.
Although these systems are very difficult to simulate, they are also directly relevant to understanding real quantum materials.
“Those are actually much more difficult problems quantitatively,” Staudenmayer said. “So that’s one of the next big hurdles we want to clear.”
