A quantum simulation once presented as beyond the practical reach of classical computing has been reproduced for major two- and three-dimensional cases using tensor-network algorithms โ with some of the first calculations running on a personal laptop. The result shows how rapidly improved mathematics and software can shift the boundary between classical and quantum computing.
Quantum computers are often expected to outperform classical machines on problems involving large numbers of interacting quantum particles.
The reason is simple but formidable: quantum systems can occupy enormously complicated combinations of states, and the mathematical object used to describe them โ the wave function โ grows exponentially as more particles are added.
For hundreds of entangled quantum bits, or qubits, directly storing the complete wave function on an ordinary computer can become effectively impossible.
Yet researchers at the Simons Foundationโs Flatiron Institute and Boston University have demonstrated that the full wave function does not always have to be stored directly.
By using sophisticated mathematical structures called tensor networks, they compressed the relevant quantum information into a much smaller representation. Combined with an algorithm known as belief propagation, the method allowed them to simulate large quantum systems using surprisingly modest classical computing resources.
The work was published in Science on May 21, 2026.
The challenge involved hundreds of interacting qubits
The problem centered on quantum spin glasses โ disordered systems whose components interact in complicated and sometimes competing ways.
The simulated qubits were arranged in two- and three-dimensional geometries, including square-like and three-dimensional lattice structures.
A conventional computer bit has one of two values: 0 or 1.
A qubit behaves differently.
Quantum mechanics allows it to exist in a superposition of states, and multiple qubits can also become entangled. When particles are entangled, the description of one particle cannot generally be separated from the rest of the quantum system.
This creates a serious computational challenge.
For a system containing (N) qubits, the number of amplitudes required for a completely general wave function grows as (2^N).
At 10 qubits, that is manageable.
At 100 qubits, the number is already unimaginably large.
At several hundred qubits, explicitly storing the entire state is out of reach for conventional hardware.
That explosive growth is one of the fundamental reasons quantum many-body simulations are considered promising applications for quantum computers.
A 2025 experiment claimed a beyond-classical result
The new work was motivated by a Science paper published online in March 2025 and in print in April 2025.
In that study, Andrew D. King and colleagues used D-Waveโs Advantage2 quantum annealing system to simulate the dynamics of quantum spin-glass models.
After comparing their results with several leading classical approaches, the researchers argued that their quantum annealer could address some simulations that remained outside the practical reach of the classical techniques they tested.
That made the problem an attractive test for scientists at the Center for Computational Quantum Physics, or CCQ, at the Flatiron Institute.
Rather than assuming the classical limit had been reached, Joseph Tindall and his colleagues asked whether a different classical algorithm could do better.
The answer turned out to be yes โ for substantial classes of the two- and three-dimensional simulations.
Tensor networks became the key
The central tool was a tensor network.
Tensor networks provide a way of representing extremely complicated quantum states without explicitly storing every possible amplitude in the full wave function.
Instead, the quantum state is broken into smaller mathematical objects called tensors, which are connected in a network.
The connections encode correlations and entanglement between different parts of the system.
Tindall has compared the idea to compressing a large computer file.
The complete wave function may be enormous, but if much of its information has exploitable structure, a tensor network can represent it far more efficiently.
This does not mean every quantum state can be compressed easily.
Some highly entangled states remain extremely difficult for tensor-network methods.
But many physically important quantum systems contain enough structure that compression can work remarkably well.
The researchers designed their networks so that their geometry reflected the actual lattices being simulated.
That lattice-specific approach was crucial to their success.
A decades-old algorithm helped control the complexity
Tensor networks alone were not the entire solution.
The researchers also turned to belief propagation, an algorithm with roots going back decades and long used in areas including information theory, statistics, error correction, and artificial intelligence.
Belief propagation works by passing information between connected parts of a network so that local calculations can approximate properties of the full system.
The Flatiron team adapted the technique to quantum tensor networks.
During the simulated quantum evolution, belief propagation allowed the researchers to update the network without repeatedly performing prohibitively expensive calculations involving the entire structure.
More sophisticated versions of the method were then used to calculate physical observables from the final quantum states.
This combination proved far more computationally efficient than many older approaches to three-dimensional quantum dynamics.
Some of the first calculations ran on a laptop
Perhaps the most attention-grabbing part of the work is that Tindall performed many of the initial simulations on a personal laptop using ITensor, a tensor-network software library developed and maintained by researchers associated with the Flatiron Institute.
The larger study did not rely exclusively on a laptop, and describing the entire result as a laptop replacing a quantum computer would therefore be an exaggeration.
But the fact that meaningful calculations from a problem previously associated with specialized quantum hardware could begin on personal computing hardware illustrates just how efficient the new algorithm had become.
The published work reports state-of-the-art accuracy using modest classical computing resources and scalable simulations involving hundreds of qubits.
Entanglement was the main obstacle
Quantum entanglement is one of the primary reasons these simulations are difficult.
Suppose hundreds of quantum spins evolve together.
As time passes, interactions cause the state of one region to become increasingly correlated with the state of another.
Eventually, describing a single qubit independently no longer captures the physics.
The wave function must encode an enormous web of correlations.
Tensor networks work by retaining the parts of that entanglement structure that matter most while compressing information that contributes less to the desired observables.
The amount of entanglement determines how difficult that compression becomes.
If entanglement grows too rapidly, the tensor network must also grow more complex, increasing computational cost.
The achievement of the new work was showing that for the spin-glass dynamics investigated, the entanglement could be handled efficiently enough to obtain high-quality results in two and three dimensions.
The classical results matched important quantum-computer results
The researchers tested their methods in several ways.
For smaller systems where more direct calculations were possible, the tensor-network simulations produced accurate results.
They also checked whether the simulations reproduced theoretical expectations for the underlying physics.
Most importantly, their results agreed closely with important observables previously obtained from the D-Wave quantum annealer for the cases they simulated.
The team was also able to study systems containing hundreds of qubits and recover expected universal behavior associated with KibbleโZurek physics โ a framework describing how systems develop defects and correlations when they are driven through phase transitions.
The paper therefore did more than reproduce numerical values.
It showed that the classical simulations could capture meaningful large-scale physical behavior.
Does this mean the earlier quantum-advantage claim was completely overturned?
Not quite.
This is where an important scientific nuance is necessary.
The Simons Foundation described the result as overturning the earlier claim of quantum supremacy.
However, authors of the earlier D-Wave study argued in a formal response that Tindall and colleagues had not reproduced every part of their original work.
They noted that the classical study did not tackle the most complicated lattice geometry, the largest three-dimensional instances, the longest simulation times, or all of the observables included in their quantum experiment.
Their position is therefore that the new classical result substantially narrows the regime in which the beyond-classical claim can be made, rather than eliminating it altogether.
That distinction matters.
The new research clearly demonstrates that classical algorithms can reach much further into this problem than earlier comparisons suggested.
But it does not establish that every calculation performed by the quantum annealer can already be duplicated efficiently on a laptop or conventional computer.
The boundary between classical and quantum computing keeps moving
This episode illustrates a broader difficulty in demonstrating quantum advantage.
To show that a quantum computer has achieved something practically inaccessible to classical machines, researchers must compare it with the best classical algorithms available.
But classical algorithms do not stand still.
A quantum experiment can motivate classical researchers to develop new mathematical shortcuts, compression techniques, approximations, or software that dramatically improve conventional performance.
This has happened repeatedly in quantum-computing research.
A quantum processor reaches a new milestone.
Classical researchers study the problem.
A better classical algorithm is developed.
The threshold for quantum advantage moves again.
This competition can be scientifically productive because both sides improve.
Quantum and classical computing are not simply rivals
Tindall and his colleagues emphasize that classical and quantum approaches can complement one another.
Classical simulations are essential for benchmarking quantum machines.
If researchers cannot independently check at least some of a quantum computer's output, determining whether it is correct becomes difficult.
At the same time, quantum hardware gives classical researchers new problems and new regimes to investigate.
An experiment performed on a quantum processor may reveal exactly where existing classical techniques fail โ providing clues for the development of better algorithms.
In this case, the D-Wave experiments became a demanding benchmark for tensor-network researchers.
The result was a new classical technique capable of simulating quantum dynamics at scales that had previously looked far less accessible.
Why tensor networks are so powerful
Tensor networks succeed because physics often contains structure.
A generic state of hundreds of qubits may require an impossibly large number of values to describe exactly.
But physical systems are not always generic.
Interactions are frequently local.
Entanglement can follow recognizable patterns.
Symmetries can reduce complexity.
Only certain quantities may actually need to be calculated.
Tensor networks exploit these features.
Instead of asking a classical computer to memorize every possible configuration of a quantum system, researchers encode the most important correlations in a structured mathematical network.
The price is that the result is often approximate rather than an exact representation of the complete wave function.
But if the approximation converges and can be checked against theory, smaller exactly solvable systems, or independent calculations, it can still provide highly accurate physical predictions.
Three-dimensional quantum simulations are especially difficult
Tensor-network techniques are well established for one-dimensional quantum systems.
One-dimensional chains can often be represented efficiently using matrix product states, a particularly successful form of tensor network.
Two dimensions are harder.
Three dimensions are harder still.
The network contains many loops and interconnected paths, making the calculation required to extract information from it โ known as tensor-network contraction โ increasingly expensive.
That is why the new work is notable.
Rather than applying only established one-dimensional methods, the researchers constructed lattice-specific two- and three-dimensional tensor networks and combined them with efficient belief-propagation techniques.
The study describes this as a scalable approach for large-scale two- and three-dimensional quantum dynamics.
Better software can be as important as better hardware
The result also highlights an often-underappreciated part of computational science: software engineering.
A powerful mathematical idea is not automatically useful.
Researchers must translate it into software that handles memory efficiently, performs tensor operations rapidly, remains numerically stable, and can scale to large systems.
The Flatiron Institute has invested heavily in scientific software libraries such as ITensor precisely for this reason.
The CCQ's broader mission includes developing the algorithms, theories, and codes needed to tackle the quantum many-body problem and predict the properties of materials and molecules.
In this case, that combination of mathematics and software allowed relatively ordinary classical hardware to perform calculations previously considered exceptionally challenging.
The implications extend beyond quantum-computer benchmarking
The researchers are interested in the technique for more than settling arguments over quantum advantage.
Quantum dynamics sits at the heart of many problems in condensed-matter physics.
Scientists would like to predict how strongly interacting quantum materials evolve, how phase transitions occur, how disorder changes physical behavior, and how systems respond when driven far from equilibrium.
Better tensor-network algorithms could make some of these questions accessible without requiring quantum hardware.
They may also contribute to optimization research because spin-glass systems are closely related to mathematical problems in which researchers seek the best solution from a vast number of possibilities.
The next target is even harder: moving electrons
The simulations in the current study focus on qubit-like spin systems.
The researchers now want to move toward a significantly tougher problem: systems in which electrons can move from site to site.
That change introduces additional complexity.
Electrons are fermions, meaning their quantum states obey special exchange rules. Their movement, interactions, charge, and spin must all be treated consistently.
Yet these are precisely the kinds of models needed to understand many real materials.
Strongly correlated electron systems are connected to some of condensed-matter physics' biggest unsolved questions, including unconventional superconductivity and exotic magnetic phases.
If tensor-network methods can be extended efficiently into this regime, the scientific payoff could be substantial.
A laptop did not make quantum computers obsolete
The result should not be interpreted as evidence that quantum computers are unnecessary.
Nor does it show that every large quantum system can suddenly be simulated on consumer hardware.
The researchers solved a specific and carefully structured class of quantum-dynamics problems using methods designed to exploit the geometry and compressibility of those systems.
Many other quantum problems remain extraordinarily difficult for classical computers.
Some may ultimately provide robust quantum advantages.
What the work demonstrates is that researchers must be cautious about declaring where the classical boundary lies.
A calculation that appears impossible with today's classical algorithms may become manageable when someone discovers a better representation of the problem.
The real breakthrough is compression
At the heart of the achievement is a deceptively simple idea.
The researchers did not make an ordinary laptop capable of storing an exponentially huge quantum wave function.
They found a way not to store that wave function directly.
By representing its important structure through tensor networks and using belief propagation to update and interrogate that representation efficiently, they dramatically reduced the computational burden.
That is the deeper lesson.
Scientific computing advances not only when processors become faster, but also when researchers find better ways to describe the problem.
In this case, sophisticated mathematics allowed classical computers to enter territory that had recently been portrayed as belonging to quantum hardware.
The result does not end the race between classical and quantum computing.
It makes the race much more interesting.
Journal references
Joseph Tindall, Antonio Francesco Mello, Matthew Fishman, E. Miles Stoudenmire, and Dries Sels. โDynamics of disordered quantum systems with two- and three-dimensional tensor networks.โ Science, 392(6800), 868โ872 (2026), published May 21, 2026. The DOI was verified against the Science bibliographic record indexed by PubMed and corresponds to this exact article.
https://doi.org/10.1126/science.adx2728
Andrew D. King et al. โBeyond-classical computation in quantum simulation.โ Science, 388(6743), 199โ204 (2025), published online March 12, 2025. The DOI and publication details were independently verified through PubMed.