README.md

---
license: cc-by-4.0
language: en
pretty_name: "QM7b — Quantum-Augmented (QParquet v1.0)"
tags:
  - chemistry
  - molecular-property
  - quantum-chemistry
  - quantum-machine-learning
  - quantum-kernels
  - qm7b
  - benchmark
size_categories:
  - 1K<n<10K
task_categories:
  - tabular-regression
  - tabular-classification
---

# QM7b — Quantum-Augmented Dataset

A quantum-augmented edition of QM7b (Montavon et al. 2013, [arXiv:1305.7074](https://arxiv.org/abs/1305.7074)) — 7,211 small organic molecules with 14 computed quantum-chemistry properties (atomization energies, polarizabilities, HOMO/LUMO eigenvalues, ionization potentials, electron affinities, excitation energies) at PBE0 / ZINDO / GW / SCS levels of theory.

Each molecule carries its packed heavy-atom Coulomb-matrix features, the 14 original DFT-computed property targets, a derived HOMO–LUMO gap at GW level, and a **quantum-derived label `y_q`** produced by a Heisenberg-model quantum kernel on a 7-qubit simulator. The kernel matrix `K_q` and the per-sample 1-RDM observables are precomputed and shipped alongside, so downstream consumers can train against quantum-geometric structure without running any quantum circuit themselves.

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

---

## Headline result

On the 6,041-molecule subset of QM7b with exactly 7 heavy atoms (C, N, O, S, Cl), N_train=300 / N_test=100 stratified by HOMO–LUMO-gap quartile, the quantum kernel exposes a label channel that classical kernels on the same Coulomb features cannot represent.

**Quantum-kernel separation on a structural label channel**

| measurement | value | interpretation |
|---|---|---|
| accuracy of quantum-kernel SVC on a quantum-derived label channel | **0.81** | quantum kernel fits its own geometric label direction |
| accuracy of classical-RBF SVC on the same labels | **0.49** (chance) | classical RBF cannot represent that direction |
| prediction-accuracy advantage of quantum kernel over classical RBF on the label channel | **+0.32** | head-to-head moat metric ([Huang et al. 2021 §IV](https://arxiv.org/abs/2011.01938)) |
| kernel-space geometric difference `g(K_Q, K_RBF)` | **19.71** vs threshold **√N = 17.32** | quantum and classical kernels are structurally distinct ([Schuld 2024](https://arxiv.org/abs/2403.07059); [Huang 2021](https://arxiv.org/abs/2011.01938) Fig. 1) |
| sample-complexity ratio `s_classical / s_quantum` derived from kernel-target alignment | **~1,500×** | quantum kernel needs ~1,500× less data to reach the same alignment on `y_q` |

**Shuffled-label null** (n = 30 permutations of training labels, fixed test labels):

| measurement | real | null (mean ± std) | z-score |
|---|---|---|---|
| quantum SVC accuracy | 0.81 | 0.506 ± 0.067 | **+4.54** |
| classical SVC accuracy | 0.49 | 0.495 ± 0.058 | −0.09 |
| accuracy advantage | +0.32 | +0.011 ± 0.094 | **+3.27** |

The quantum SVC accuracy is 4.5 σ above shuffled chance; the head-to-head advantage is 3.3 σ above the shuffled null. The classical SVC is at chance whether the training labels are shuffled or not, confirming `y_q` is genuinely quantum-geometric and not a memorisation artifact.

---

## Compressed-representation regression — head-to-head on the 14 original properties

Per-property kernel ridge regression on each original DFT property, plus the derived HOMO–LUMO gap, head-to-head against classical RBF on the same 28-dimensional packed Coulomb features. The quantum kernel uses **7 qubits** — a **4× compression** of the feature representation.

| target index | observed range | classical MAE | quantum MAE | Δ (q − c) | 2 σ_classical_CV | within tolerance |
|---|---|---|---|---|---|---|
| T00 (atomization energy, PBE0) | −2213 to −410 | 119.25 | 332.32 | +213.07 | 28.35 | — |
| T01 (polarizability, PBE0) | 3.4 to 39.7 | 3.156 | 3.188 | +0.032 | 0.478 | ✓ |
| T02 (polarizability, SCS) | 0.05 to 3.35 | 0.205 | 0.202 | −0.003 | 0.021 | ✓ |
| T03 (HOMO, GW) | −16.0 to −7.5 | 0.985 | 1.657 | +0.672 | 0.167 | — |
| T04 (LUMO, GW) | −2.3 to +4.2 | 1.187 | 1.159 | −0.028 | 0.156 | ✓ |
| T05 (IP, ZINDO) | 1.5 to 36.8 | 1.632 | 1.701 | +0.069 | 0.328 | ✓ |
| T06 (EA, ZINDO) | 6.9 to 15.7 | 0.964 | 1.609 | +0.644 | 0.169 | — |
| T07 (E1, ZINDO) | −4.0 to +2.9 | 1.322 | 1.236 | −0.086 | 0.219 | ✓ |
| T08 (Emax, ZINDO) | −10.95 to −5.12 | 0.613 | 1.153 | +0.541 | 0.088 | — |
| T09 (Imax, ZINDO) | −3.81 to +0.41 | 0.621 | 0.632 | +0.010 | 0.037 | ✓ |
| T10 (HOMO, ZINDO) | −14.1 to −7.0 | 0.659 | 1.442 | +0.783 | 0.100 | — |
| T11 (LUMO, ZINDO) | −1.84 to +1.96 | 0.405 | 0.425 | +0.020 | 0.080 | ✓ |
| T12 (uncertain — see column-mapping note) | 2.53 to 17.17 | 0.741 | 2.295 | +1.554 | 0.332 | — |
| T13 (uncertain — see column-mapping note) | 2.43 to 16.46 | 0.599 | 2.490 | +1.891 | 0.298 | — |
| **gap_GW** (LUMO_GW − HOMO_GW) | 7.34 to 15.24 | 1.844 | 2.240 | +0.396 | 0.344 | — |

The quantum kernel matches classical RBF (Δ within ±2 σ of classical 5-fold CV variance) on **7 of the 15 targets**: both polarizabilities, LUMO_GW, IP_ZINDO, the first ZINDO excitation, the absorption intensity, and LUMO_ZINDO. It underperforms on the absolute atomization energy, both HOMO eigenvalues, electron affinity, the maximal absorption energy, and the HOMO–LUMO gap (which inherits the HOMO error). Original-task regression is reportable for the targets where it matches classical RBF; the dataset's value as a quantum-augmented release rests on the structural label channel, not on universal regression dominance.

---

## Column-mapping caveat — names array in source `.mat` is scrambled

The `names` field in the upstream `qm7b.mat` file is character-array corrupted by `scipy.io.loadmat` and is not human-readable as shipped. The 14 property indices in this datacard follow the canonical ordering from **Montavon et al. 2013, Table 1** ([arXiv:1305.7074](https://arxiv.org/abs/1305.7074)).

Indices with unambiguous numeric ranges — T03 / T04 (HOMO / LUMO GW eigenvalues), T10 / T11 (HOMO / LUMO ZINDO) — are confidently labelled. **T12 and T13** have positive ranges that do not match HOMO / LUMO PBE0 (eigenvalues should be negative for HOMO), so we mark them *uncertain*. Consumers needing exact column semantics should cross-reference Montavon 2013.

---

## What this dataset adds over a classical QM7b

| field | classical QM7b | this dataset |
|---|---|---|
| Coulomb matrices + 14 property targets | ✓ | ✓ |
| packed heavy-atom features (28-D) | — | ✓ |
| `K_q` — precomputed quantum kernel matrix (N × N float32) | — | ✓ |
| `y_q` — quantum-derived labels in {−1, +1} | — | ✓ |
| `observables_1rdm` — per-sample 1-RDM Pauli expectations | — | ✓ |
| derived `gap_GW` target | — | ✓ |
| validated schema + provenance metadata | — | ✓ |

The added columns express geometric structure in a 7-qubit Hilbert space (a Heisenberg model on the molecular bond graph) that classical kernels on the same Coulomb features do not capture — the shuffled-null and head-to-head numbers above quantify how much.

---

## Schema (QParquet v1.0)

QParquet v1.0 ships a kernel-centric schema; classical features and targets remain joinable from the upstream QM7b source by `input_id` (SHA-1 of the packed Coulomb-matrix 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<float32> | (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-derived labels |
| `observables_1rdm` | list<float32> | (N, 21) | per-sample 1-RDM Pauli ⟨X_j⟩, ⟨Y_j⟩, ⟨Z_j⟩ for j ∈ [0, 7) |
| file-level `qparquet_metadata` | JSON (parquet key-value metadata) | — | encoding, n_qubits, backend, full evaluation report, per-property MAE table, shuffled-null z-scores, citations |

Validation enforced 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)`.

To recover the classical features and DFT property targets, join by `input_id` against the upstream QM7b `.mat` file (`scipy.io.loadmat("qm7b.mat")`); the hashing is deterministic on `features_packed`. The full classical view is not duplicated in this artifact — its value is the quantum-augmented columns.

---

## Loading

```python
import numpy as np
import pandas as pd
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("qm7b_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 (encoding, evaluation, citations, …)
import json
meta = json.loads(table.schema.metadata[b"qparquet_metadata"].decode())

# Train on the quantum-derived 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:]))

# Original-task regression: join with upstream QM7b for classical targets
# (download qm7b.mat from quantum-machine.org/data/qm7b.mat — link in metadata)
```

The dataset is a drop-in for scikit-learn precomputed-kernel pipelines: load `K_q`, train. No quantum hardware or simulator required at inference time.

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

```python
import relab

# Quantum kernel matrix
K_q = relab.kernel(features_scaled, domain="molecular", n_qubits=7)

# Quantum-derived labels
y_q = relab.fit(features_scaled, domain="molecular", n_qubits=7)
```

---

## Methodology

- **Encoding**: a Heisenberg model on the molecular bond graph (`XX + YY + ZZ` couplings, one qubit per heavy atom). Coulomb off-diagonals `J_ij` map to bond couplings; diagonals `h_i = ½ Z_i^2.4` map to local fields. Reference: [arXiv:2407.14055](https://arxiv.org/abs/2407.14055) (Heisenberg encoding for graph-structured data).
- **Why this encoding for molecular data**: Coulomb sub-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](https://arxiv.org/abs/1109.2618)) destroy. Validated on QM7 atomization-energy regression prior to this dataset.
- **Feature scaling**: `MinMaxScaler` to `[−π, π]` per [Schuld, Sweke, Meyer 2021](https://arxiv.org/abs/2008.08605) Fourier-bandwidth constraint.
- **Quantum-label construction**: generalised Rayleigh quotient on `K_q` against the classical-RBF kernel `K_c`, threshold at the median back-projection ([Huang et al. 2021 §IV](https://arxiv.org/abs/2011.01938)). Test-set extension via quantum-kernel interpolation — `K_q` is the only kernel that can faithfully generalise the quantum label direction.
- **Backend**: Apple Silicon Metal GPU via the Zilver MLX simulator (open-source v0.3.2). Statevector-exact at 7 qubits. Cross-verified against a pure-NumPy reference at atol = 1e-4.

### What this kernel is, in plain language

The kernel compresses 28-dimensional Coulomb features into a 7-qubit Hilbert space and measures molecular similarity as the fidelity of two Heisenberg-evolved states. At seven qubits, the kernel is *classically tractable in practice* — the full N × N matrix is computable in tens of seconds on a laptop. 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; that distinctness is what the head-to-head and shuffled-null numbers above quantify.

For the asymptotic-hardness question see Tang's body of work on dequantisation ([arXiv:1807.04271](https://arxiv.org/abs/1807.04271); [arXiv:1910.06151](https://arxiv.org/abs/1910.06151)) and the QSVT framework ([Gilyén, Su, Low, Wiebe 2019](https://arxiv.org/abs/1806.01838)). The plain Heisenberg fidelity kernel is BQP-complete worst-case ([Janzing & Wocjan 2007](https://arxiv.org/abs/quant-ph/0610203)) but admits no published Tang-style classical sampling algorithm; we do not make an asymptotic-hardness claim at seven qubits. The QSVT spectral-filter upgrade — block-encoding the bond Hamiltonian and applying a HOMO–LUMO gap-midpoint projector polynomial — *is* provably not dequantisable ([Lin & Tong 2020](https://arxiv.org/abs/2002.12508); [Martyn et al. 2021](https://arxiv.org/abs/2105.02859)) and is on the ReLab roadmap; it is not the kernel shipped in this dataset.

---

## Reproduction

Headline run:
- N_train = 300, N_test = 100, stratified across HOMO–LUMO-gap quartiles
- RNG seed = 42
- Backend: Zilver MLX simulator on Apple Silicon Metal GPU, max_qubits = 25
- `K_q` computed in 17.5 s; per-property KRR sweep in sub-second

---

## Citation

```bibtex
@dataset{relab_qm7b_quantum_2026,
  title  = {QM7b — Quantum-Augmented (QParquet v1.0)},
  author = {ReLab (Sirius Quantum)},
  year   = {2026},
  source = {derived from Montavon et al. 2013, arXiv:1305.7074},
  note   = {Quantum kernel matrix and quantum-derived labels via a Heisenberg model on the molecular bond graph (7 qubits, one per heavy atom).}
}
```

If you build on this dataset, please also cite the upstream QM7b source (Montavon 2013) and the ReLab engine that generated the quantum-augmented columns.

## References

- Montavon, Rupp, Gobre, Vazquez-Mayagoitia, Hansen, Tkatchenko, Müller, von Lilienfeld 2013 — [arXiv:1305.7074](https://arxiv.org/abs/1305.7074) — QM7b dataset
- Rupp, Tkatchenko, Müller, von Lilienfeld 2012 — [arXiv:1109.2618](https://arxiv.org/abs/1109.2618) — Coulomb-matrix representation
- Huang et al. 2021 — [arXiv:2011.01938](https://arxiv.org/abs/2011.01938) §IV — head-to-head benchmark, geometric difference threshold, sample-complexity bound
- Schuld, Sweke, Meyer 2021 — [arXiv:2008.08605](https://arxiv.org/abs/2008.08605) — Fourier-bandwidth scaling
- Schuld 2024 — [arXiv:2403.07059](https://arxiv.org/abs/2403.07059) — geometric advantage `g(K_Q, K_C)`
- Zhao et al. 2026 — [arXiv:2604.07639](https://arxiv.org/abs/2604.07639) — compression-match framework
- Gilyén, Su, Low, Wiebe 2019 — [arXiv:1806.01838](https://arxiv.org/abs/1806.01838) — QSVT
- Janzing & Wocjan 2007 — [arXiv:quant-ph/0610203](https://arxiv.org/abs/quant-ph/0610203) — BQP-completeness of Hamiltonian overlap
- arXiv:2407.14055 — graph-Hamiltonian encoding for structured data