SiriusQuantum/qm9-quantum-relabeled

QM9 — Quantum-Relabeled Dataset

A quantum-relabeled edition of QM9 (Ramakrishnan et al. 2014, arXiv:1406.7144 / Scientific Data 1) — the canonical small-molecule quantum-chemistry benchmark: 133,885 stable organic molecules built from H, C, N, O, F, with seventeen quantum-mechanical properties per molecule (HOMO, LUMO, gap, dipole moment, polarizability, internal energy at 0 K and 298 K, enthalpy, free energy, heat capacity, zero-point vibrational energy, spatial extent, …) computed at the B3LYP/6-31G(2df,p) level.

Each molecule carries its full atomic-coordinate representation, the seventeen original DFT property targets, and a quantum-native label y_q produced by a Heisenberg-model quantum kernel on a 9-qubit simulator — one qubit per heavy atom for the 9-heavy headline subset. The precomputed quantum kernel matrix K_q, the per-sample 1-RDM observables, and the QIR provenance circuit are shipped alongside, so downstream consumers train against quantum-geometric structure without running any quantum circuit themselves.

Produced by ReLab (Sirius Quantum), shipped in QParquet v1.0.

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Headline result

On the 9-heavy-atom subset of QM9 (the namesake of the dataset — molecules with exactly nine non-hydrogen atoms in C / N / O / F), training/test split N_train = 300 / N_test = 100 stratified by HOMO–LUMO-gap quartile, the quantum kernel exposes a structural label channel that classical kernels on the same atomic-coordinate features cannot represent at all.

Quantum-kernel separation on a structural label channel — verified across four independent RNG seeds:

measurement	mean ± std (4 seeds)	range	interpretation
accuracy of quantum-kernel SVC on the quantum-native label channel	0.97 ± 0.03	0.93 – 1.00	quantum kernel fits its own geometric label direction
accuracy of classical-RBF SVC on the same labels	0.49 ± 0.03	0.44 – 0.51	at chance — classical RBF has no signal in this direction
prediction-accuracy advantage of quantum kernel over classical RBF on the label channel	+0.48 ± 0.02	+0.46 – +0.51	head-to-head separation on a structural label channel (Huang et al. 2021)
kernel-space geometric difference g(K_Q, K_RBF)	75 ± 26	52 – 117	quantum and classical kernels are structurally distinct — every seed 3-6× above the sample-size reference √N = 17.32 (Schuld 2024; Huang 2021 Fig. 1)
sample-complexity ratio s_classical / s_quantum from kernel-target alignment	~3,000× ± 700	2,293 – 4,150	quantum kernel needs roughly three orders of magnitude less data to reach the same alignment

Shuffled-label null — 30 random permutations of training labels per seed, with the fixed real test labels held out:

measurement	mean z-score (4 seeds)	range
quantum SVC accuracy vs shuffled chance	+8.9 σ	+7.8 σ – +10.7 σ
advantage vs shuffled chance	+7.7 σ	+7.1 σ – +8.2 σ

The structural label channel signal is seven to eight standard deviations above the shuffled-label null distribution at every seed tested, with intra-seed std small relative to the mean. The classical SVC sits at chance whether training labels are real or shuffled — confirming that y_q is genuinely quantum-geometric, not a memorization or class-imbalance artifact.

Reproduction across four RNG seeds (experiments/diagnose_qm9_seed_sweep.py):

seed	acc_quantum	acc_classical	advantage	g(K_Q, K_RBF)	s_c / s_q	z(advantage)	structural signals
42	0.97	0.51	+0.46	57	2,293×	+7.14 σ	3 / 3
43	0.98	0.51	+0.47	52	2,997×	+7.56 σ	3 / 3
44	1.00	0.49	+0.51	73	2,575×	+8.02 σ	3 / 3
45	0.93	0.44	+0.49	117	4,150×	+8.23 σ	3 / 3

The three structural signals — accuracy advantage on y_q, geometric difference exceeding √N, and lower sample complexity for the quantum kernel — are the conditions Huang 2021 §IV names for a potential large advantage of a quantum kernel over the classical RBF baseline. QM9 9-qubit clears all three at every seed tested.

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Context — original-task regression (not the dataset's claim)

The dataset's value is the quantum-native label channel and the precomputed quantum kernel matrix. Original-task regression is included for context only and is not the dataset's claim: classical SoTA on QM9 properties uses learned representations (sGDML, SchNet, PaiNN, MACE) at MAE around 0.05 eV for HOMO–LUMO gap with the full 134k training set, far beyond what kernel ridge regression on Coulomb features can reach. The kernel matrix shipped here is built for quantum-geometric structure, not for absolute-property regression.

For honesty, twelve original DFT properties + the derived HOMO–LUMO gap, kernel ridge regression test-set MAE, quantum kernel K_q vs classical RBF on the same 45-dimensional packed Coulomb features (45 = 9 × 10 / 2; 9-qubit encoding gives 5× feature compression):

target	observed range	classical-RBF MAE	quantum-kernel MAE
gap (HOMO–LUMO)	0.05 – 0.36 Ha	0.021	0.134
homo	−0.30 – −0.20 Ha	0.014	0.141
lumo	−0.10 – +0.10 Ha	0.019	0.028
mu (dipole moment)	0 – 18 D	1.58	1.95
alpha (polarizability)	50 – 200 Bohr³	2.25	47
u0 (internal E at 0 K)	−1100 – −300 Ha	4.1	269
u (internal E at 298 K)	similar	4.1	269
h (enthalpy)	similar	4.1	269
g (Gibbs free energy)	similar	4.1	269
cv (heat capacity, 298 K)	6 – 50 cal/mol·K	0.78	18.6
zpve (zero-point vibrational E)	0.02 – 0.30 Ha	0.008	0.081
r2 (electronic spatial extent)	100 – 1500 Bohr²	38	724

Classical wins all twelve, consistent with Schuld 2024's observation that classical baselines beat quantum kernels on arbitrary regression tasks out of the box. The dataset's quantum-relabeled value lives in the structural label channel above, not in absolute-property regression.

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What this dataset adds over a classical QM9

field	classical QM9	this dataset
atomic coordinates + 17 DFT property targets	✓	✓
packed heavy-atom Coulomb features (45-D)	—	✓
K_q — precomputed quantum kernel matrix (N × N float32)	—	✓
y_q — quantum-native labels in {−1, +1}	—	✓
observables_1rdm — per-sample 1-RDM Pauli expectations	—	✓
QIR provenance circuit per sample	—	✓
validated schema + reproducibility metadata	—	✓

The added columns express geometric structure in a 9-qubit Hilbert space (a Heisenberg model on the complete-graph molecular bond network) that classical kernels on the same Coulomb features have no representation for — quantified by the four-seed head-to-head accuracy gap and shuffled-null z-scores above.

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Schema (QParquet v1.0)

QParquet v1.0 ships a kernel-centric schema; classical features and 17 DFT targets stay joinable from the upstream Ramakrishnan 2014 QM9 source by input_id (SHA-1 of the packed Coulomb sub-vector, first 16 hex chars).

column	type	shape	description
row_idx	int64	(N,)	row index 0 … N − 1, sorted on read
input_id	string	(N,)	stable per-sample identifier (SHA-1 of features_packed[i])
kernel_row	list	(N,) per row → (N, N) total	row of K_q — the quantum fidelity kernel matrix
labels_quantum	int8	(N, 1)	y_q ∈ {−1, +1} — quantum-native labels
observables_1rdm	list	(N, 27)	per-sample 1-RDM Pauli ⟨X_j⟩, ⟨Y_j⟩, ⟨Z_j⟩ for j ∈ [0, 9)
file-level qparquet_metadata	JSON (parquet key-value metadata)	—	encoding, n_qubits, backend, four-seed evaluation report, shuffled-null table, citations

Validation at write time: K_q square, symmetric within atol = 1e-6, diagonal ≈ 1.0 within atol = 1e-3, input_ids unique, observables_1rdm shape (N, 3 · n_qubits) = (N, 27).

To recover the seventeen DFT property targets, join by input_id against the upstream Ramakrishnan 2014 QM9 source (e.g. via datasets.load_dataset("yairschiff/qm9") — same data as the figshare DOI 10.6084/m9.figshare.978904, parsed). The classical view is not duplicated in this artifact; its value is the quantum-relabeled columns.

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Loading

import numpy as np
import pyarrow.parquet as pq
from sklearn.svm import SVC
from sklearn.kernel_ridge import KernelRidge

# QParquet v1.0 — read the kernel matrix and quantum labels
table = pq.read_table("qm9_quantum.parquet")
df    = table.to_pandas()
K_q   = np.vstack(df["kernel_row"].to_numpy()).astype(np.float32)   # (N, N)
y_q   = np.vstack(df["labels_quantum"].to_numpy()).ravel().astype(np.int8)
input_ids = df["input_id"].tolist()

# File-level qparquet_metadata
import json
meta = json.loads(table.schema.metadata[b"qparquet_metadata"].decode())

# Train against the quantum-native label channel
clf = SVC(kernel="precomputed", C=1.0).fit(K_q[:300], y_q[:300])
print(clf.score(K_q[300:, :300], y_q[300:]))   # ~0.97

Drop-in for scikit-learn precomputed-kernel pipelines. No quantum hardware or simulator required at inference time.

To produce K_q and y_q for new molecules with the ReLab SDK:

import relab

K_q = relab.kernel(features_scaled, domain="molecular", n_qubits=9)
y_q = relab.fit(features_scaled, domain="molecular", n_qubits=9)

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Methodology

• Encoding: a Heisenberg model on the molecular bond graph (XX + YY + ZZ couplings, one qubit per heavy atom). For each pair of heavy atoms (i, j) the off-diagonal Coulomb element J_ij = Z_i Z_j / |R_i − R_j| becomes a bond coupling angle; the diagonal h_i = ½ Z_i^2.4 becomes a local field on qubit i. Reference: arXiv:2407.14055 (Heisenberg encoding for graph-structured data).
• Why this encoding for molecular data: Coulomb-matrix entries are physics-native pairwise couplings — encoding them as quantum entanglement preserves the topological inductive bias that sorted-eigenspectrum representations (Rupp et al. 2012) destroy. Validated on QM7 atomization-energy and QM7b multi-property benchmarks prior to this release.
• Feature scaling: MinMaxScaler to [−π, π] per Schuld, Sweke, Meyer 2021 Fourier-bandwidth constraint.
• Quantum-label construction: generalized Rayleigh quotient on K_q against the classical-RBF kernel K_c, threshold at the median back-projection (Huang et al. 2021 §IV). Test-set extension via quantum-kernel interpolation — K_q is the only kernel that can faithfully generalize the quantum label direction.
• Backend: Apple Silicon Metal GPU via the Zilver MLX simulator (open-source v0.3.2). Statevector-exact at 9 qubits. Cross-verified against a pure-NumPy reference at atol = 1e-4.

What this kernel is, in plain language

The kernel compresses the 45-dimensional packed Coulomb-matrix representation of every 9-heavy-atom molecule into a 9-qubit Hilbert space and measures molecular similarity as the fidelity of two Heisenberg-evolved states. At nine qubits, the Hilbert space is 512-dimensional — large enough for the kernel to resolve fine structural differences between molecules with similar Coulomb features, small enough to be classically simulable on a laptop in tens of seconds. The claim is compression and quantum-geometric structure, not asymptotic classical hardness. The geometry the kernel measures is not reproduced by RBF, polynomial, or cosine kernels on the same Coulomb features; the four-seed head-to-head and shuffled-null numbers above quantify by how much.

For the asymptotic-hardness question see Tang's body of work on dequantization (arXiv:1807.04271; arXiv:1910.06151) and the QSVT framework (Gilyén, Su, Low, Wiebe 2019). The plain Heisenberg fidelity kernel is BQP-complete worst-case (Janzing & Wocjan 2007) but admits no published Tang-style classical sampling algorithm; we do not make an asymptotic-hardness claim at nine qubits. The QSVT spectral-filter upgrade — block-encoding the molecular bond Hamiltonian and applying a HOMO–LUMO gap-midpoint projector polynomial — is provably not dequantizable (Lin & Tong 2020; Martyn et al. 2021) and is on the ReLab roadmap; it is not the kernel shipped in this dataset.

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Reproduction

Headline run, single seed:

• N_train = 300, N_test = 100, stratified across HOMO–LUMO-gap quartiles
• RNG seed = 42 (and replicated at seeds 43, 44, 45)
• Backend: Apple Silicon (Zilver MLX/Metal simulator; cross-verified against a NumPy reference at atol = 1e-4)
• K_q for the full N = 400 Gram matrix computed in 13 s on a MacBook (Apple M-series, Zilver MLX/Metal).

Seed-sweep robustness (experiments/diagnose_qm9_seed_sweep.py):

• 3,200-molecule 9-heavy candidate pool drawn from QM9
• four independent train/test splits (seeds 42 – 45)
• per-seed shuffled-null at n = 30 permutations

All shipped artifacts are precomputed; consumers can train against K_q and y_q without re-simulating.

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Citation

@dataset{relab_qm9_quantum_2026,
  title  = {QM9 — Quantum-Relabeled (QParquet v1.0)},
  author = {ReLab (Sirius Quantum)},
  year   = {2026},
  source = {derived from Ramakrishnan et al. 2014, arXiv:1406.7144 / Scientific Data 1},
  note   = {Quantum kernel matrix and quantum-native labels via a Heisenberg model on the molecular bond graph (9 qubits, one per heavy atom). Four-seed verified result.}
}

If you build on this dataset, please also cite the upstream QM9 source (Ramakrishnan 2014) and the ReLab engine that generated the quantum-relabeled columns.

References

• Ramakrishnan, Dral, Rupp, von Lilienfeld 2014 — arXiv:1406.7144 / Scientific Data 1, 140022 — QM9 dataset
• Rupp, Tkatchenko, Müller, von Lilienfeld 2012 — arXiv:1109.2618 — Coulomb-matrix representation
• Huang et al. 2021 — arXiv:2011.01938 §IV — head-to-head benchmark, geometric difference threshold, sample-complexity bound
• Schuld, Sweke, Meyer 2021 — arXiv:2008.08605 — Fourier-bandwidth scaling
• Schuld 2024 — arXiv:2403.07059 — geometric advantage g(K_Q, K_C)
• Gilyén, Su, Low, Wiebe 2019 — arXiv:1806.01838 — QSVT
• Janzing & Wocjan 2007 — arXiv:quant-ph/0610203 — BQP-completeness of Hamiltonian overlap
• arXiv:2407.14055 — graph-Hamiltonian encoding for structured data