Organizations: IBM Research · Applied Quantum Algorithms ⟨aQaL⟩, Leiden University, The Netherlands · Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, United Kingdom · The University of Chicago
Machine learning is widely regarded as a promising application in which quantum computing could offer a computational advantage. However, general quantum machine learning models with provable guarantees on performance, trainability, and efficient inference remain limited. In this work, we introduce quantum Hamiltonian embeddings, a supervised learning framework in which a classical training dataset is encoded into a single effective, parameterized Hamiltonian. Training optimizes the parameters of this data-dependent Hamiltonian, whose approximate ground state defines the classification rule. Once the ground state has been approximated using a quantum algorithm, inference can be performed either classically using a classical representation of the learned state, or with a quantum representation of the state. We establish efficient methods for gradient estimation and prove favorable generalization guarantees. Furthermore, under standard cryptographic assumptions, we show that Hamiltonian embeddings can realize hypothesis functions beyond the reach of efficient classical algorithms. We then demonstrate the framework at scale on IBM Heron processors using a standard benchmark dataset. Together, these results establish Hamiltonian embeddings as a new framework for quantum machine learning, combining theoretical guarantees with a path for experimental investigations at scale on quantum hardware.
Figures & tables
Figure 1: a) Test accuracy versus system size for various methods on a binary classification problem from the Fashion-MNIST dataset. The methods are: best-performing classical baseline (gray), classical iterative solver (orange) used in HE framework, quantum kernel based on a product-state embedding (blue), 1 -reduced density matrix ( 1 -RDM) projected quantum kernel with the same embedding [ 15 ] (green), and the 1 -RDM projected quantum kernel based on the E2 embedding of Ref. [ 15 ] (pink). b) Held-out test accuracy of the five runs with the highest validation accuracy at n=50 . In this plot, we additionally show the accuracy obtained by the HE model trained on hardware (yellow).
Appendix figures & tables8 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 2: Left: fidelity between the final exact learned ground state and its MPS approximation obtained by Schmidt truncation at maximum bond dimension χ , using the fixed qubit ordering of the simulation. Right: test accuracy obtained by replacing the exact learned state with the corresponding truncated MPS state.
Figure 3: Test accuracy Aη as a function of bond dimension at system sizes n∈{12,24,36,48,60} , averaged over 10 trials, with (a) η=0.1 , (b) η=0.01 . (c) Minimum energy seen during training as a function of bond dimension. (d) The signed margin y(x)\expectationvalueU(x)†Z1U(x) .
Figure 4: Accuracy of surrogate models. The green curve shows the surrogate that optimizes the mS,z from Eq. 239 , keeping cS,z(α) fixed by taking α=α∗ obtained from the Hamiltonian embedding model. The orange curve optimizes over both mS,z and α (which determines cS,z(α) ). Also shown in orange dots for comparison are the best (over bond dimensions χ ) accuracies for the Hamiltonian embedding model for each trial.
Figure 5: A comparison of the test accuracy from the Hamiltonian embedding model, when used in conjunction with a classical iterative solver (blue circles) and the SKQD method on hardware (orange squares) for the dataset of Fig. 1 (b).
Figure 6: Cross-seed transfer at n=50 . Each cell is the test accuracy of a model trained on one seed (row) and evaluated on another seed’s held-out data split (column); the red boxes mark the diagonal, where a model is tested on its own split.
Figure 7: Training examples needed to reach 90% test accuracy for Hamiltonian embedding and classical surrogate with λ=0.1 . The sweep uses 20 independent seeds. Central values are the first crossings of the mean test-accuracy curve across the 20 seeds, while the error bars are the standard deviations of the seed-wise first-crossing values. The bars therefore show seed-to-seed variability rather than a conventional confidence interval.
Model / λ
n=4
6
8
10
12
14
16
18
20
HE
56
72
96
140
144
168
192
216
240
10−8
160
216
320
400
432
560
640
648
800
10−7
160
216
320
400
432
560
640
648
800
10−6
160
216
320
400
432
560
640
648
800
10−5
160
216
320
400
432
560
640
648
800
10−4
160
216
320
400
432
560
640
648
800
Appendix
Table 1: First tested training size L for which the mean test accuracy over all seeds reaches 90% . The λ sweep uses 20 independent seeds and 5000 test points per run.
4
6
8
10
12
14
16
18
20
mean ratio
L90HE
56
72
96
140
144
168
192
216
240
–
L90ridge,best
128
216
256
320
384
448
512
576
640
2.62×
Appendix
Table 2: Hamiltonian embedding compared with the best ridge threshold observed over the tested λ grid. The final column gives the ridge-to-HE ratio averaged over system sizes.
Many practically relevant applications of quantum machine learning involve classical data, for which performance depends critically on how inputs are embedded into quantum states. Yet the use of a fixed embedding circuit ansatz remains standard practice. We propose an energy-based generative learning framework that synthesizes gate sequences to optimize embedding structures and refine data-tailored parameters, using a fidelity-based surrogate objective to guide the search toward improved class distinguishability. Empirically, the method improves classification performance across diverse settings, while also revealing datasets where architecture search within the present embedding family yields only limited additional gains. We explain this saturation by deriving bounds on the achievable empirical risk in terms of the Wasserstein distance in the input space, showing that classical data geometry provides an a priori diagnostic for regimes in which substantial gains from embedding optimization are unlikely. The results establish a practically useful and theoretically motivated framework for searching effective quantum data embeddings through generative optimization, with the attainable gains diagnosed through the geometry of the underlying classical data.
Jaewoong Heo, Daniel K. Park
Department of Statistics and Data Science, Yonsei University, Seoul 03722, Republic of Korea · Department of Applied Statistics, Yonsei University, Seoul 03722, Republic of Korea · Department of Quantum Information, Yonsei University, Seoul 03722, Republic of Korea
Autoencoders transformed classical machine learning by solving the curse of dimensionality, enabling principled weight initialization and learning compact, structured representations. In this work, we extend this paradigm to quantum machine learning by introducing a variational autoencoder framework that learns task-specific quantum embeddings of classical data. We demonstrate that high-dimensional datasets, including ImageNet, can be compressed into a 13-qubit quantum representation while remaining reconstructable through a learned decoder. On MNIST (3 vs 5), our approach achieves 98.5% validation accuracy using a circuit-centric quantum classifier, within 1.2 percentage points of a classical neural network baseline (99.7%) and more than 30 percentage points above a naive amplitude-embedding approach. Unlike amplitude embeddings, which require full quantum state tomography for recovery, or angle embeddings, which generally rely on circuit inversion under restrictive assumptions, the proposed framework reconstructs the original data from only a polynomial number of measurements. The framework was further validated on IBM quantum hardware, confirming that the learned embeddings remain stable and reconstructable under real device noise.
Aldo Lamarre, Dominik Šafránek
Department of Electrical Engineering, KAIST, Daejeon, South Korea · Center for Theoretical Physics of Complex Systems, Institute for Basic Science (IBS), Daejeon - 34126, Korea · Faculty of Mathematics and Physics, Charles University, Ke Karlovu 3, 121 16 Praha 2, Czech Republic
In supervised quantum machine learning with classical data, classification performance critically depends on the geometry induced by quantum data embedding. Existing data-adaptive approaches rely on pairwise fidelity evaluations between samples, which scale as O(B2) for B samples and create a computational bottleneck in iterative embedding optimization. To address this issue, we propose a prototype-guided framework for efficient data-adaptive quantum embedding. Using class-prototype separation and sample-to-prototype compactness, we derive a measurable lower certificate for class-ensemble distinguishability and construct a prototype-based objective requiring only O(B) fidelity evaluations. We integrate this objective into an end-to-end pipeline that combines task-specific quantum circuit architecture optimization with continuous refinement of the data-embedding parameters. Across five benchmark datasets with up to 20 qubits, our method retains competitive quantum kernel support vector machine (SVM) performance while reducing fidelity-evaluation cost, with particularly strong gains over budget-matched pairwise optimization on T-MNIST. Geometric analyses examine how the learned embeddings relate to kernel alignment and classification margins, revealing dataset-dependent benefits and limitations. We further derive a prototype-based classifier in the reproducing kernel Hilbert space that provides an explicit witness for the SVM objective and a low-cost proxy for model selection, and extend the framework to multiple prototypes for multimodal class structures.
Seokhoon Jeong, Daniel K. Park
Department of Statistics and Data Science Yonsei University Republic of Korea · Department of Statistics and Data Science Department of Applied Statistics Department of Quantum Information Yonsei University Republic of Korea