D-Wave said classical computers could not match its quantum chip. A laptop just did.

A personal laptop running open-source software has matched the outputs of D-Wave Systems' 5,000-qubit Advantage2 quantum annealer on a class of quantum dynamics problems the company stated in March 2025 were beyond the reach of any classical computer. Researchers at the Center for Computational Quantum Physics at the Simons Foundation's Flatiron Institute, working with Boston University collaborators, published their findings in Science, showing that a mathematical compression framework called tensor networks — combined with a message-passing inference algorithm first developed in 1982 — can reproduce D-Wave's results without quantum hardware. D-Wave has issued a formal rebuttal disputing that the classical method reproduces its full result, arguing that the hardest problem instances and highest-order physical measurements remain beyond what the classical approach can reach.

The result adds a new episode to a running pattern in quantum computing: claims of beyond-classical performance, followed by classical simulation advances that close the gap — or dispute where the gap lies.

D-Wave Claimed Quantum Annealing Could Simulate What Frontier Could Not

In March 2025, a large collaboration anchored at D-Wave published a paper in Science reporting that the company's Advantage2 processor — a quantum annealing machine operating at near absolute zero — had simulated the non-equilibrium dynamics of a magnetic spin glass system faster and more accurately than Oak Ridge National Laboratory's Frontier supercomputer could. The system being simulated was a transverse-field Ising model: a standard condensed matter physics problem in which hundreds of qubits interact in disordered lattice geometries shaped like squares, cubes, and diamonds. D-Wave's chief scientist Mohammad Amin said at the time that the quantum machine could "simulate some problems that cannot be solved classically," a claim he suggested would "always survive."

That claim was immediately contested. IEEE Spectrum and Scientific American both reported that experts in classical quantum simulation pushed back from the day of publication. The dispute mattered because D-Wave's annealing hardware represents a specialized approach to quantum computing — distinct from the gate-model systems built by IBM and Google — that had never before successfully argued for quantum advantage on a problem of genuine scientific relevance.

What Tensor Networks Actually Do

The Flatiron team's method exploits a fundamental fact about entanglement in physical systems: information does not distribute randomly. In a spin glass system evolving under a quantum quench — a sudden change in the applied magnetic field — the entanglement between distant qubits builds up gradually and tends to obey what physicists call an area law: entanglement entropy across any cut through the system scales with the boundary of the region, not its volume.

Tensor networks exploit this structure. Rather than storing the full quantum wave function — which grows exponentially with the number of qubits, making it unmanageable for hundreds of particles — they compress the wave function into a network of small, interconnected tables of numbers called tensors. The specific class of tensor networks used for three-dimensional systems is called projected entangled pair states, or PEPS. First developed by Frank Verstraete and Ignacio Cirac in 2004 for static ground-state problems, PEPS had not previously been applied at scale to three-dimensional quantum dynamics — the type of calculation D-Wave's 2025 paper targeted.

"It's a zip file for the wave function where you've taken all this information, and you've compressed it into this mathematical data structure full of these small tables of numbers that are interconnected to each other," said Joseph Tindall, associate research scientist at the CCQ and first author of the new study, speaking to ScienceDaily.

Extending this approach to three dimensions was not straightforward. Contracting a 3D tensor network is a computationally hard problem in the worst case. "Working with these objects — especially in three dimensions — is very untrodden," Tindall said. "You need sophisticated codes and algorithms to deal with them; it's a software engineering challenge in itself."

A 1982 Algorithm Finds New Purpose

For the most computationally accessible calculations — the regime where D-Wave's spin glass dynamics obeys the area law and entanglement has not yet propagated throughout the full system — the team did not use the most expensive tensor network contraction methods. Instead, they adapted belief propagation, a message-passing inference algorithm first formalized by Judea Pearl in 1982 for probabilistic graphical models, and later applied to error-correcting codes. Belief propagation is the same algorithmic family underlying many modern machine learning methods, including factor graphs and loopy message passing used in probabilistic neural networks.

The tradeoff is explicit: belief propagation is an approximation that is exact on tree-structured graphs but becomes inexact on graphs with cycles. For the spin glass problems D-Wave's 2025 paper targeted, the Flatiron team found that belief propagation was accurate enough to match D-Wave's results while running at dramatically lower computational cost. "It's a little more approximate than some of the other methods, but it's way cheaper, and we can run it much more directly on lots of harder problems," said Miles Stoudenmire, CCQ research scientist and co-author. Tindall ran many of the early calculations on a personal laptop using ITensor, a tensor network software library developed at the CCQ.

The simulations matched D-Wave's published outputs on spin glass dynamics across the square, cubic, and diamond lattice geometries the 2025 paper had tested — and agreed with theoretical predictions on smaller problem instances where the correct answer is known analytically.

Where the Classical Method Breaks Down

D-Wave's formal response, issued May 26, identified four specific limitations it says the Flatiron paper does not address.

First, observables discrepancy: the Flatiron team's belief propagation approach did not compute the full-state data profiles and higher-order fourth-order physical observables that D-Wave's quantum processor produced. Second, geometric omissions: the classical simulations did not attempt the most complex non-planar biclique lattice topologies included in D-Wave's original paper. Third, scale limitations: the classical method operated on localized sub-sectors of the problem rather than the maximum-size three-dimensional geometries D-Wave had evaluated. Fourth, coupling inefficiencies: belief propagation breaks down under strongly coupled spin glass configurations where quantum correlations propagate rapidly across the full lattice.

This last point is the most technically significant. The area-law assumption that makes PEPS efficient — and that makes belief propagation a reliable approximation — holds for weakly entangled, weakly coupled systems in early-time evolution. For strongly coupled spin glasses, where entanglement builds up faster and the dynamics become more complex, the area law breaks down, and both the PEPS approximation and the belief propagation contraction become unreliable. D-Wave argues these are precisely the regimes where its hardware demonstrated genuine quantum advantage.

D-Wave stated in its press release that prior analysis suggested matching its largest reported problem instances using matrix product state methods — a related class of tensor networks — would require "nearly a million years on the Frontier supercomputer." The Quantum Computing Report, an independent industry analyst, assessed the result as establishing "a more refined and rigorous baseline" rather than a definitive refutation.

This Has Happened Before

The dispute between Flatiron and D-Wave follows a pattern that has recurred throughout the short history of quantum supremacy claims. When Google announced in 2019 that its Sycamore processor had performed a random circuit sampling task in 200 seconds that would take a classical supercomputer 10,000 years, IBM disputed the estimate within days, arguing the same problem could be solved in 2.5 days on its Summit machine using a better classical algorithm. In June 2024, a team using a cluster of 1,432 GPUs performed the same sampling task faster than Google's original hardware, calling it "the first strong refutation" of the 2019 result.

Wikipedia's quantum supremacy article now explicitly notes: "Due to unpredictable possible improvements in classical computers and algorithms, quantum supremacy may be temporary or unstable." The Tindall 2026 paper is cited in that same article as having "questioned" D-Wave's 2025 advantage.

This pattern carries an important structural implication. The algorithms most likely to fall to classical simulation are those that target structured physical problems — the type that clever classical approaches can exploit. Random circuit sampling is hard to simulate classically precisely because it has no exploitable structure. D-Wave's transverse-field Ising dynamics has structure — and that structure is what PEPS and belief propagation were designed to exploit.

What Comes Next, and Why It Matters Beyond the Debate

The Flatiron team has already described its next target: systems where electrons move between sites, as they do in real superconductors and quantum materials. These fermionic systems introduce additional mathematical constraints — the Pauli exclusion principle, antisymmetric wave functions — that make classical simulation harder even with tensor networks.

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

The value of classical simulation advances extends beyond proving or disproving any single quantum advantage claim. Accurate classical simulations allow researchers to validate quantum hardware results by providing an independent reference; they help quantum hardware teams identify which problems genuinely exceed classical reach; and they can accelerate the understanding of quantum materials including high-temperature superconductors — without requiring access to a cryogenic quantum processor. "That can help guide us, and it can also help guide quantum computing researchers, because, obviously, the barrier for entry for us to simulate certain things is a lot easier than for them, because we don't have to build a quantum computer. I can just write some code and press 'run' on my personal computer," Tindall told ScienceDaily.

There is also a broader convergence underway that neither the pro-quantum nor the pro-classical framing captures well. The belief propagation algorithm family now doing quantum physics work is the same family underlying probabilistic graphical models in machine learning. The ITensor software library sits at an intersection of condensed matter physics and computational linear algebra that, five years ago, would not have been recognizable to researchers in either field alone. Whether or not quantum hardware eventually produces problems that no classical algorithm can touch, the methods being developed to test that claim are already generating tools useful for materials science, optimization, and computational physics independently of the outcome.

Frequently Asked Questions

Does this result mean quantum computers are not useful?

No, and the Flatiron team has been explicit on this point. The result challenges a specific claim about a specific class of problems — the non-equilibrium dynamics of transverse-field Ising spin glasses in two and three dimensions, as implemented on D-Wave's annealing hardware. It says nothing about gate-model quantum computers (IBM, Google, IonQ), about Shor's factoring algorithm, about quantum chemistry, or about any application where quantum hardware's advantage has not yet been tested against current classical algorithms. D-Wave itself operates commercially across optimization, drug discovery, and logistics tasks where annealing provides practical value regardless of the supremacy debate.

What exactly is quantum annealing, and is it the same as what IBM and Google build?

Quantum annealing is a specialized quantum computing paradigm optimized for a specific class of problems: finding the lowest-energy configuration of a system with many interacting variables, which maps to certain optimization and simulation tasks. D-Wave is the only major company building quantum annealers commercially. IBM, Google, IonQ, and most other quantum computing companies build gate-model quantum computers, which are more general-purpose and use a different physical architecture. The Flatiron result applies to quantum annealing dynamics specifically — it does not extend automatically to gate-model systems.

Why does the belief propagation algorithm matter beyond this specific result?

Belief propagation is the algorithmic foundation of a wide class of probabilistic inference methods, including many used in machine learning and network analysis. The fact that it can now be applied to quantum tensor networks — enabling classical computers to efficiently approximate 3D quantum dynamics — suggests that the conceptual tools of classical AI and quantum physics are converging. The implication is not just that this one result was possible; it is that the boundary between what classical computation can and cannot reach may be more permeable than the quantum hardware industry's marketing has suggested, and that advances in classical probabilistic methods will continue to redefine that boundary.

What should I know before drawing conclusions from quantum advantage claims?

Every major quantum supremacy claim since 2019 has been contested by improved classical algorithms, typically within months of the original publication. The field now commonly distinguishes between "quantum advantage" (a quantum device outperforms current classical methods on a given problem) and "quantum supremacy" (a quantum device solves a problem no classical computer could solve in any feasible time). The latter is a much harder claim to make, because it requires proving a lower bound on all possible classical algorithms — something no one has yet done for a physically relevant problem. When evaluating a quantum advantage announcement, ask: what specific problem, at what specific size, against what specific classical baseline, with what specific level of independent classical verification?