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.
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
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
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