Quantum computing’s edge looked closer after a hard physics problem seemed beyond classical machines. But a new result shows compressed math and smarter algorithms can match or beat that benchmark, raising fresh questions about where true quantum advantage really begins.

For years, quantum computers have carried a bold promise. They could solve problems so complex that even the world’s best classical computers would fail. That promise fueled a global race among scientists and technology companies to prove “quantum advantage,” the point where quantum machines outperform traditional computing systems.

Now, physicists at the Center for Computational Quantum Physics at the Simons Foundation’s Flatiron Institute and collaborators at Boston University have shaken that narrative. Using advanced mathematics, tensor networks and clever coding, the team solved a difficult quantum physics problem that another group had claimed only a quantum computer could handle.

The breakthrough shows that classical computers may still have far more power than many researchers expected. In fact, some of the calculations were completed on a personal laptop.

“Whenever we see these kinds of claims, we’re always a bit skeptical,” said Joseph Tindall, associate research scientist at the Center for Computational Quantum Physics and first author of the study. “Like, ‘Did you try this? Did you try that?’”

The work challenges growing assumptions about the limits of conventional computing. It also opens new paths for studying quantum materials, optimization problems and the strange behavior of entangled particles.

The story began earlier this year when another research team reported that a quantum computer had simulated a highly complicated system of interacting qubits. The researchers argued that classical computers could not realistically match the calculation.

Qubits are the building blocks of quantum computing. Unlike ordinary computer bits, which exist as either 0 or 1, qubits can exist in several states at once through a quantum property called superposition. They can also become entangled, meaning their behavior remains linked even when separated by distance.

That entanglement creates a major problem for classical simulation. As more qubits interact, the mathematical description of the system grows exponentially larger. Very quickly, storing the full quantum state becomes impossible for normal computers.

“When you have lots of particles that interact by quantum physics, you have this wave function that describes the state of the system,” Tindall said. “It’s this huge object that rapidly gets bigger and bigger the more particles there are.”

The original quantum experiment focused on Ising spin glasses, disordered quantum systems arranged on square, cubic and diamond-shaped lattices. These systems are notoriously difficult to model because their particles interact in complex and unpredictable ways.

Researchers used a quantum annealer to study how these systems evolved over time. Earlier classical methods struggled to keep up as system sizes increased. That led many to believe the quantum machine had crossed a major milestone.

Instead of accepting defeat, the Flatiron team revisited the problem with a different approach. They relied on tensor networks, mathematical structures that compress enormous quantum wave functions into manageable pieces.

Tindall compared tensor networks to a compressed computer file.

“It’s this very powerful compression that can be very effective,” he said. “You’ve taken all this information, and you’ve compressed it into this mathematical data structure.”

Each qubit becomes part of an interconnected web of smaller mathematical objects called tensors. These tensors capture the relationships between particles while avoiding the need to store every possible quantum state directly.

The researchers also adapted an older algorithm called belief propagation, originally developed in the 1980s. Though more approximate than some advanced methods, it required far fewer computing resources.

“It’s way cheaper, and we can run it much more directly on lots of harder problems,” said study co-author Miles Stoudenmire, research scientist at the Center for Computational Quantum Physics.

This combination of tensor networks and belief propagation allowed the team to simulate systems containing hundreds of qubits. Remarkably, the computational cost scaled roughly linearly with system size instead of exploding exponentially.

That meant doubling the number of qubits only roughly doubled the workload.

Many of the calculations ran on modest hardware. Tindall performed initial simulations on a personal laptop using ITensor, a tensor network software library developed at the Flatiron Institute.

The software handled three-dimensional tensor networks, one of the most difficult frontiers in computational physics.

“This really is a bit of a frontier,” Tindall said. “Especially in three dimensions.”

The team simulated the same lattice systems studied in the earlier quantum experiment. In cylindrical and diamond lattice geometries, their classical simulations produced errors smaller than those from the quantum annealer. In dimerized cubic lattices, the classical and quantum results were comparable.

The researchers also verified their accuracy using smaller systems where exact benchmark answers were known. Their simulations matched theoretical predictions and successfully reproduced the results reported by the quantum computing team.

Importantly, the researchers did not rely on noisy measurement sampling. Instead, they directly calculated correlations between qubits using efficient message-passing techniques.

One of the biggest challenges in quantum simulation involves managing entanglement growth during time evolution. As quantum systems evolve, particle relationships become increasingly complicated.

The researchers handled this using a second-order Trotterization method combined with belief propagation updates. Their approach compressed the tensor networks repeatedly while preserving critical quantum information.

The simulations also reproduced important physical behavior tied to phase transitions. The team studied a phenomenon known as Kibble-Zurek physics, which describes how systems behave when pushed across critical transitions too quickly.

In cylindrical lattices, they observed correlation patterns consistent with theoretical predictions. In diamond lattices containing up to 900 qubits, the simulations showed scaling behavior expected from genuine nonequilibrium quantum physics.

That result mattered because it demonstrated that classical methods were not merely approximating small pieces of the problem. They were capturing deep physical behavior across large systems.

The findings do not mean quantum computing has failed. Instead, they reveal how quickly classical algorithms continue to improve.

“The good side of the classical versus quantum computing debate is that there’s a lot of synergy,” Tindall said.

Quantum devices still hold long-term promise for certain problems. But studies like this show that the line separating classical and quantum capability keeps shifting.

Researchers in both fields often inspire each other. Quantum experiments provide difficult targets for classical simulations. Classical simulations, meanwhile, help verify whether quantum devices truly achieve an advantage.

The Flatiron team now plans to push even further. Their next goal involves simulating systems where electrons move freely between lattice sites, a much harder challenge connected directly to superconductors and quantum materials.

“They’re really, quantitatively, a lot harder problems,” Stoudenmire said. “So that’s one of our next big bars that we want to clear.”

This breakthrough could significantly expand what scientists can accomplish with existing computers. By squeezing more performance from classical hardware, researchers may reduce the need for expensive quantum machines in certain areas of physics and materials science.

The methods could help scientists study superconductors, magnetic materials and other quantum systems that are difficult to analyze experimentally. Better simulations may accelerate the search for new materials used in energy, electronics and advanced computing technologies.

The approach may also improve optimization methods. Many real-world problems involve searching through enormous numbers of possible solutions. Tensor network techniques and belief propagation could eventually help tackle logistics, scheduling and machine learning challenges more efficiently.

Perhaps most importantly, the research reminds scientists that computational progress rarely follows a straight line. As algorithms improve, problems once labeled impossible can suddenly become solvable.

Research findings are available online in the journal Science.