D-Wave and the Classical Counterattack

Most weeks I read a paper claiming something new was built. This week I read one claiming that something already built was not as far out of reach as advertised. In March 2025, D-Wave reported in Science that its Advantage2 annealing processor had simulated quantum spin-glass dynamics that the leading classical methods it tested could not match in reasonable time.¹ In May 2026, Science published a paper by Joseph Tindall and colleagues at the Flatiron Institute, SISSA, and Boston University that reproduced a real slice of that simulation classically, sometimes with lower error than the machine.² I spent the week on the Tindall paper, then read D-Wave's reply. What I found was not a winner, but a lesson in how the boundary between quantum and classical gets drawn, and who has to defend it. This is a first-pass notebook investigation centered on one new paper and the debate around it: what the simulation did, what it reached, how D-Wave responded, and what it might mean for an investor watching the annealing story. It is my own opinion, written to learn in public, and it is not a valuation and not financial advice. Do your own research.

What D-Wave claimed

Both papers study the same system: a lattice of tiny magnets, called spins, each able to point up or down. Every neighboring pair is joined by a coupling $J_{ij}$ saying whether the pair would rather agree or disagree, and how strongly. In a spin glass those couplings are random, and the result is frustration. No arrangement satisfies every link at once.

Add quantum mechanics, and the energy becomes

$$H(s) = \mathcal{J}(s) \sum_{\langle ij \rangle} J_{ij}\,\sigma^z_i \sigma^z_j \;+\; \Gamma(s) \sum_{i} \sigma^x_i .$$

The operator $\sigma^z_i$ reads whether spin $i$ is up or down along the axis that carries the answer. The operator $\sigma^x_i$ flips it, and that is the quantum ingredient: it lets a spin sit in both states at once. $\Gamma(s)$ is the strength of that flipping, called the transverse field, $\mathcal{J}(s)$ is the overall strength of the couplings, and $s = t/t_a$ is the fraction of the run completed, with $t_a$ the total annealing time.

The experiment is a quench. Start with $\Gamma$ large, every spin flipping freely and no order anywhere, then lower $\Gamma$ and raise $\mathcal{J}$ over a few nanoseconds. The system crosses a quantum phase transition into the frozen, glassy phase, and correlations spread as it goes. The question is not which arrangement has the lowest energy. It is what the whole state does on the way down.

Why the simulation is hard

Following that descent exactly is brutal. A quantum state of $N$ spins takes $2^N$ numbers to write down, hopeless past a few dozen spins. Classical methods survive because physically relevant states obey an area law: entanglement across a boundary grows with the size of the boundary, not the volume inside it. A tensor network exploits that. Instead of one impossibly long list, the state becomes a mesh of small arrays, one per qubit, joined by links, and each link's size, its bond dimension, sets how much correlation the description can hold.

D-Wave tested the leading families: matrix product states (MPS), which string the tensors into a one-dimensional chain and so sit awkwardly on a 3D lattice; projected entangled pair states (PEPS), the higher-dimensional version; and neural quantum states (NQS), which swap the mesh for a trained neural network. Their most dramatic resource estimates came from extrapolating the MPS bond dimension required to match the processor: on the order of a million years of Frontier supercomputer time per input, with memory beyond that machine's 700 petabytes and electricity beyond annual global consumption. The original paper also noted that this was not an intrinsic lower bound for every possible classical method.¹

Those are the numbers that made the claim land. They are also, on close reading, claims about the methods tested.

Hand-drawn blue-pen infographic titled Keep The Lattice, Compress The State. On the left a tangled cluster of up and down spin arrows is labeled full state, 2 to the N numbers. An arrow marked compress plus truncate points right to a clean regular mesh of small circles joined by short links, labeled tensor network, N site tensors. Three small lattice icons run along the bottom, labeled cylinder, diamond and dimerized cubic, and a red caption on the right reads same shape as the problem. — Figure 1: The classical shortcut. Instead of storing every number in the quantum state, a tensor network keeps a compressed description whose geometry matches the programmed problem, so the approximation no longer fights the shape of the problem.

How the Tindall method worked

The first move is easy to state and consequential: shape the network like the programmed lattice. Instead of folding a three-dimensional lattice into a one-dimensional chain, the authors build the tensor network with the geometry of the problem itself, cylindrical, diamond, or dimerized cubic. Each tensor sits where its qubit sits, each link runs where a coupling runs, and the compression stops fighting the geometry.²

Matching the shape is not enough. Every time step applies operations that enlarge the links, and left alone they would swell until the description became unaffordable. Something has to decide, over and over, what to throw away.

That something is belief propagation, or BP, an algorithm borrowed from statistical inference. Rather than re-examining the whole network to judge which parts of a link matter, BP asks each region's neighbors for a compressed summary of themselves, then ranks the new information against those summaries. It is like editing one chapter of a book from notes supplied by the surrounding chapters instead of rereading the whole book. Keep the strongest contributions, discard the rest, shrink each link back to size. It is dimensionality reduction, applied to quantum correlations.

Compression was never the hard part. Knowing what to throw away was.

The payoff is affordability. Bond dimensions of 32 for the cylinder, 16 for the diamond, and 8 for the dimerized cubic lattice were used in the main benchmarks, with costs scaling roughly linearly in system size and 3D diamond lattices reaching 900 qubits. One annotated example took about 53 minutes on a CPU to evolve and 7.3 seconds on a GPU per correlator.²

Reading answers out is a separate problem. Plain BP is not accurate enough at the end, so the authors switch to more careful schemes: boundary MPS message passing around the cylinder, loop-corrected BP on the 3D lattices. Which one works depends on geometry, because short loops make approximate contraction harder.

Hand-drawn blue-pen infographic titled The Traffic Cop Of Correlations. On the left a tangled mesh with many outward arrows is labeled growing correlations. In the center a cartoon traffic officer labeled BP receives three small note cards labeled summaries from neighbors, and an arrow to a trash can is labeled trim the rest. A red arrow labeled keep the strongest leads right to a clean orderly mesh labeled manageable network. — Figure 2: Belief propagation is the traffic cop. At each step it asks the surrounding regions for compressed summaries, keeps the correlations that matter most, and shrinks the rest back to size, which is what kept the simulation affordable as entanglement grew.

What it reproduced, and what it left open

The scoring is a normalized correlation error: the gap between the simulated two-spin correlations and a trusted ground truth, measured against the size of those correlations. On the cylindrical and diamond lattices the classical errors came in well below the annealer's and appeared controllable as system size grew. On the dimerized cubic lattice they were comparable.²

That is a genuine result and should not be shrunk. It shows the methods D-Wave benchmarked were not the whole classical frontier.

But the boundary matters as much as the number. What was reproduced is pair correlations, how two spins tend to align, and the spin-glass order parameter built from them, essentially the average of those squared correlations across the lattice. Not full output samples. Not the four-spin statistics D-Wave used, through the Binder cumulant, to verify critical scaling at large sizes. Nor the densest biclique geometry, nor the longest anneals, nor the low-precision couplings where every link has equal strength and only its sign varies, and where correlations therefore spread fastest.

This paper did not classically replace the machine. It reproduced selected observables from specific finite-time anneals.

Hand-drawn blue-pen infographic titled Where The Line Falls, split by a single dividing line. The side headed covered in Tindall shows small cylinder, diamond and dimerized cubic lattice icons with the labels 7 to 20 nanoseconds and pair correlations. The side headed left open in Tindall shows a dense biclique graph icon with the labels 40 nanosecond anneals, plus-or-minus J couplings, full samples and four-spin statistics. A red caption at the bottom reads frontier moved, not settled. — Figure 3: The boundary is the story. The classical method reached three lattice geometries and the pair correlations built on them, while the densest graph, the longest anneals, the fastest-spreading couplings, and the higher-order statistics stayed on the other side of the line.

D-Wave's response, and why the dispute continues

D-Wave's reply came in two parts, and the first surprised me.

Their arXiv comment opens by calling the classical simulations impressive, a significant improvement on the state of the art, in some cases lower in error than the quantum results.³ Only then does it draw the line, listing what was left untouched, and concluding not that nothing had changed but that these advances "narrow the parameter space" in which beyond-classical computation had been demonstrated.

Then came evidence, in a follow-up with an inverted premise.⁴ Unable to compute a classical ground truth for the larger systems, the authors used the quantum processor itself as the reference against which to measure the classical method's error. As annealing time grew from 7 ns to 40 ns, that error grew with it. Loop corrections, useful when correlations are short-ranged, became less effective and in some regimes counterproductive. And one concrete prediction failed: the classical authors had expected the cubic-dimer error to fall once the shortest loops disappeared at larger size. Within error bars it did not.

Classical computers usually supplied the reference. This time the quantum processor became the approximate reference.

That inversion deserves a flag, because it rests on assumptions. Using the processor as the reference requires that the quantum and classical errors be statistically independent, and that the machine's own error stay roughly flat as systems grow. The authors argue for both, but this is an approximate reference, not a certified one. It is still a striking reversal. Classical computers used to supply the ground truth that quantum machines were measured against.

The debate has not stopped. A June 2026 preprint by Dries Sels, one of the Tindall co-authors, goes at the biclique geometry directly with a semiclassical method.⁵ I have not read it closely enough to judge it. That it exists at all, aimed at one of the remaining gaps, is the point.

So here is my takeaway. The Tindall paper did not debunk D-Wave. It showed that the methods D-Wave originally tested did not define the complete classical frontier, which is a different and narrower thing. The hardest cases in D-Wave's original benchmark remained outside the scope of the Tindall paper, and the dispute is still live.

Where I land as a physicist-investor

D-Wave moved too fast in that abstract. "The classical methods we tested failed" and "quantum annealers may reach beyond classical computation" are different statements, and a benchmark tells you about the methods inside the benchmark. The Tindall paper moved the boundary. It did not settle it.

The company's own disclosures are more measured than the headline. Its annual report states that advances in classical computing may prove more robust for longer than currently anticipated, that no current quantum computer including its own has reached broad quantum advantage and may never do so, and that achieving such an advantage would not by itself mean outperforming classical computers on tasks other than the one used to demonstrate it.⁶ Those are company disclosures rather than independent findings, but they frame the problem honestly.

What the week really changed is the question I care about. I came in asking who wins this benchmark. That is the smaller question. If a machine's advantage is strongest where the problem already speaks its hardware's native language, then the advantage is local by construction. It is a corner, not a kingdom.

The market may have many workbenches rather than one throne.

Which is why I no longer assume this is a winner-takes-most market. Different architectures may find different workloads where their trade-offs become strengths. But a niche is worth something only if somebody pays for it. A technology does not need to dominate every workload. It needs a measurable advantage that a customer will buy. The market may have many workbenches rather than one throne, and the investor's question becomes which workbench generates real revenue, and from whom.

The battle is not over. What has changed is that the burden of proof is no longer one-sided.

Sources & notes

A. D. King, et al. (D-Wave Quantum Inc. and collaborators), "Beyond-classical computation in quantum simulation," Science 388, 199-204 (2025). DOI: 10.1126/science.ado6285. Published online 12 March 2025. The quoted conclusion is from the abstract; the Frontier time, memory, and electricity estimates are from Fig. 5D and the accompanying discussion. The earlier line of work is A. D. King, et al., "Quantum critical dynamics in a 5,000-qubit programmable spin glass," Nature 617, 61-66 (2023).
J. Tindall, A. F. Mello, M. Fishman, E. M. Stoudenmire, D. Sels, "Dynamics of disordered quantum systems with two- and three-dimensional tensor networks," Science 392, 868-872 (2026). DOI: 10.1126/science.adx2728. Preprint: arXiv:2503.05693. The paper read this week. Source for the lattice-matched tensor-network ansatz, the belief-propagation evolution scheme and the boundary-MPS and loop-corrected contraction methods, the bond dimensions of 32, 16 and 8, the roughly linear scaling in system size, the 3D diamond lattices of up to 900 qubits, the GPU timings, and the per-lattice correlation-error comparisons against the annealer.
A. D. King, et al., "Comment on: 'Dynamics of disordered quantum systems with two- and three-dimensional tensor networks' arXiv:2503.05693," preprint at arXiv:2504.06283 (2025). D-Wave's direct response. The itemized list of what was and was not reproduced, and the "narrow the parameter space" framing, are from the abstract and its numbered points. The same comment addresses a separate classical challenge, L. Mauron and G. Carleo, arXiv:2503.08247 (2025). A preprint, not yet peer-reviewed.
A. Nocera, J. Raymond, W. Bernoudy, M. H. Amin, A. D. King, "Evaluating classical simulations with a quantum processor," preprint at arXiv:2508.15759 (2025). D-Wave's follow-up study. Source for the growth of classical error as annealing time rises from 7 ns to 40 ns, the diminishing and sometimes counterproductive value of loop corrections, the absence of the predicted error decrease at larger system size, and the two stated assumptions behind using the processor as an approximate reference. A preprint, not yet peer-reviewed.
D. Sels, "Truncated Wigner dynamics of biclique quantum spin glasses," preprint at arXiv:2606.20187 (2026). The newest classical work, aimed at the biclique geometry. A preprint, not yet peer-reviewed, and noted here only to show that the exchange has continued.
D-Wave Quantum Inc., Annual Report on Form 10-K for the fiscal year ended December 31, 2025. The language on advances in classical computing appears in the market-opportunity risk discussion; the language on broad quantum advantage, and on advantage in one task not implying advantage in others, appears in the quantum-advantage risk factor. Company disclosure, cited to convey D-Wave's own stated framing, not independent analysis.