Fifteen researchers, most at the Technical University of Denmark, reported in Science on September 25, 2025, that they had used entangled light to learn a physical process with 11.8 orders of magnitude fewer samples than classical methods without entanglement need. On December 5, three researchers published a benchmark in Scientific Reports in which classical models scored highest on four of five ordinary datasets.
In quantum machine learning, quantum computers learn patterns from data. In peer-reviewed journals, researchers have reported quantum learning advantages on data from quantum systems and on constructed problems. A learning advantage means a quantum learner needs fewer samples or other resources than classical methods. The bond-trading gains that HSBC and IBM reported are in a preprint.
Proofs on constructed problems
In Nature Physics in 2021, Yunchao Liu, Srinivasan Arunachalam, and Kristan Temme constructed a classification problem and proved that a quantum kernel method classifies its data with high accuracy. In a kernel method, a computer classifies data by measuring how similar data points are. In the quantum version, a quantum computer takes those measurements.
They showed that no classical learner can classify the data “inverse-polynomially better than random guessing,” assuming discrete logarithms are hard to compute classically. Some public-key cryptography depends on the same hardness assumption.
In a December 2025 paper in Nature Communications, Laura Lewis, Dar Gilboa, and Jarrod McClean proved an exponential quantum advantage for learning periodic neurons, a class of shallow neural networks, in a theoretical framework called the quantum statistical query model.
In a July 2025 paper in npj Quantum Information, Haimeng Zhao of Caltech and Dong-Ling Deng of Tsinghua University proved an unconditional, noise-resistant advantage on a classical task they designed. A quantum model with a constant number of parameters solves the task, while commonly used classical models need parameters that grow at least linearly with its size. They tested the quantum model in experiments on IonQ’s Aria trapped-ion computer.
Experiments on quantum data
In a 2022 Science paper, a team led by Hsin-Yuan Huang reported that, for some tasks, quantum machines can learn about physical systems from exponentially fewer experiments than conventional approaches. The team used 40 physical superconducting qubits and 1,300 quantum gates in its experiments.
In a February 2025 preprint of their Science paper, the photonic team described the learned process as a shift in the state of light across 100 optical modes. They reported about 5 dB of two-mode squeezing, a measure of how strongly two light beams are entangled.
Benchmarks on classical data
In the Scientific Reports paper, Savita Kumari Sheoran and two colleagues compared classical models with quantum classifiers on five public datasets, adding noise and class imbalance. They simulated the quantum models with the PennyLane software framework.
Logistic regression scored highest on three datasets, random forest on a wine dataset, and a quantum support vector machine on a breast cancer dataset, at 98.7% accuracy.
A variational quantum classifier is a quantum circuit whose settings are trained like a neural network’s weights. In the benchmark, one scored 14% to 23% on the wine data and below 50% on both two-class datasets. The authors wrote that it was “very prone to noise with lower stability.”
In a May 27, 2026, preprint, Sudip Vhaduri, Ryan Gammon, and Sayanton Dibbo compared models on the MNIST handwritten-digit dataset, simulating every quantum model in PennyLane. The authors wrote that their quantum network needed about 94% fewer parameters than the classical one at higher feature counts. Both exceeded 0.96 accuracy at 64 features and 60,000 samples. The authors also reported that their simulated quantum support vector machine consistently beat the classical version in accuracy, reaching about 0.90 against about 0.85 at 1,000 samples.
On a GPU, the simulated quantum network took about 121,169 seconds, or nearly 34 hours, at the top feature count. The classical network took 95 to 98 seconds across the feature counts tested. The 34-hour figure is the cost of simulating the quantum network on classical hardware and does not measure a run on a quantum processor.
HSBC and IBM researchers described a bond-trading trial in a September 2025 preprint. They transformed production-scale intraday trade data on IBM Heron processors and fed the output to classical models that estimate fill probabilities for corporate bond trade inquiries. A fill probability is the chance that a customer trades at the price the bank quoted.
The authors reported a relative gain of up to about 34% in out-of-sample test scores, compared with models given the original data or features from a noiseless quantum simulation. They suggested that “inherent noise in current quantum hardware contributes to this effect.” In a release, HSBC reported “up to 34% improvement in predicting the probability of winning customer inquiries” from the trial.
In an April 2026 preprint, Haimeng Zhao, Hsin-Yuan Huang, and five co-authors at Caltech, Google Quantum AI, MIT, and Oratomic described quantum oracle sketching, a way to load classical data into a quantum computer from random samples. They claimed an exponential advantage in machine size for some classification and dimension-reduction tasks.
In numerical simulations on gene-expression and movie-review data, they reported size reductions of four to six orders of magnitude with fewer than 60 logical qubits. A logical qubit is an error-protected unit of quantum information encoded across many physical qubits.
Hardware requirements
Liu and colleagues wrote that their circuits can be efficiently implemented on a fault-tolerant quantum computer, one that corrects its own errors as it runs.
In a March 2026 preprint, Vasilis Belis and Maria Schuld of Xanadu, with colleagues at CERN and Italian institutions, proposed learning over permutations with a quantum Fourier transform. They wrote that further work is needed to estimate the circuit’s resources under realistic hardware constraints. They wrote that they had “not yet provided empirical evidence” comparing the model with classical methods.
Update, November 12, 2026: In a July 19 revision of their preprint, Vhaduri and colleagues replaced the 94% parameter figure in the abstract with “superior parameter and memory efficiency at higher feature counts, while incurring higher runtime.”
