In 2025 a team of quantum-computing researchers demonstrated that they could calculate the behaviour of a particularly complex system of qubits using a real quantum computer — and argued that no ordinary computer could ever reproduce the result. A year later, physicists at the Flatiron Institute solved that very same problem on a laptop.
The difficulty comes from quantum entanglement. In a system of hundreds of interacting qubits arranged on a square, cubic or diamond-shaped grid, the particles cannot be treated as separate objects — even when they are far apart. The mathematical object that describes the whole system, called the wave function, balloons in size the more particles you add. As one of the researchers put it: with enough qubits, the wave function is simply too big to store in memory. That explosion of complexity is what gave quantum advocates the claim that only a quantum machine could solve it.
The Flatiron team sidestepped the problem by using something called a tensor network — a mathematical structure that connects small tables of numbers to represent a much larger object. One researcher compares it to a zip file for the wave function: all the information is compressed into an efficient, interconnected data structure that a classical computer can actually hold and manipulate.
The calculations were carried out using ITensor, a high-performance tensor-network software library built at the Flatiron Institute, plus an algorithm called belief propagation from the 1980s — recently repackaged for quantum systems. It is more approximate than the most sophisticated methods, but far cheaper to run, and much more direct on hard three-dimensional problems. "Why not pick a problem that has a big claim attached to it?" a co-author noted, when choosing this as a test case.
The simulations matched both the theoretical predictions and the results previously obtained with a quantum computer — on modest hardware. The message is not that quantum computers are useless, but that clever algorithms can stretch the reach of ordinary machines further than we once thought. Tensor-network methods open a practical route for studying quantum materials such as superconductors, and may offer a fresh approach to tough optimisation problems too. Sometimes, the right piece of mathematics beats a bigger machine.