QUIVER: Quantum-Informed Views for Enhanced Representations in Large ML Models

Abstract: Large machine learning models benefit substantially from multimodal inputs that provide a complementary view of the same example. We introduce QUIVER (QUantum-Informed Views for Enhanced Representations, a paradigm that enriches classical data-driven features with a quantum Fisher view: a geometrically motivated, basis-independent summary of higher-order correlations captured by a variational quantum circuit (VQC) trained to perform the same task. Unlike classical feature augmentation, the quantum Fisher information matrix encodes the intrinsic geometry of the learned quantum state manifold. While this feature map, motivated by quantum information theory, is ordinarily non-trivial to model classically, it can surface statistical structure that additional classical data or model capacity finds difficult to learn. This makes the quantum Fisher view a genuinely complementary modality rather than a redundant one. We demonstrate that QUIVER improves standard performance metrics on two benchmark datasets from very different fields: QM9 for predicting molecule properties, and JetClass for predicting jet flavor at the Large Hadron Collider (LHC). The core contribution, however, is domain-agnostic: the quantum Fisher view can be fused into a broad class of model architectures via targeted modifications to the base architecture, to incorporate information about the quantum geometry of the problem. These results demonstrate that quantum-geometric features, extracted from simulated variational circuits, can deliver measurable value for standard machine learning tasks, well before the advent of fault-tolerant quantum hardware.

• The paper introduces QUIVER, a hybrid quantum-classical method that fuses input-conditioned quantum Fisher information features from simulated variational circuits with transformer and graph neural network models.
• The paper reports improved performance on both benchmarks, including a jet-tagging AUC increase from 0.97832 to 0.98070 and a QM9 HOMO–LUMO MAE reduction from 72.42 to 67.92 meV.
• The paper shows that QFIM features can provide complementary information with minimal overhead, adding about 7% parameters to Particle Transformer and only 0.27% to DimeNet++, while remaining limited by classical simulation and 10-qubit encodings.

Overview

Quiver (QUantum-Informed Views for Enhanced Representations) proposes a hybrid quantum-classical feature augmentation scheme in which a classically simulated variational quantum circuit (VQC) supplies a "quantum Fisher view" of each input example, which is then fused into a large classical model (2606.02785). The central object is the quantum Fisher information matrix (QFIM) of the VQC's parameter manifold, which, up to a factor of four, coincides with the Fubini–Study metric on pure states. Because this matrix is input-conditioned—data enters through the state-preparation stage preceding trainable rotations—it encodes a basis-independent, geometric summary of higher-order correlations among input features. The authors' claim is that this view is genuinely complementary to classical features rather than redundant, and they support it empirically on two benchmarks from unrelated domains: top-quark jet tagging on JetClass and HOMO–LUMO gap regression on QM9. Notably, all quantum computations are simulated classically (PennyLane), so the method requires no quantum hardware.

Mathematical construction

A VQC prepares $$|\psi(\bm{\Theta})\rangle = U(\bm{\Theta})|0\rangle^{\otimes N}$$, where the angles $\bm{\Theta}(x)$ are functions of the classical input $x$. The QFIM,

$$F_{ij}(\bm{\theta};x) = 4\,\mathrm{Re}\left[\langle \partial_i \psi | \partial_j \psi\rangle - \langle \partial_i \psi|\psi\rangle\langle\psi|\partial_j\psi\rangle\right],$$

measures statistical distinguishability of infinitesimally separated parameter points. Its diagonal entries act as per-parameter sensitivity scores, while off-diagonal entries are non-zero precisely when two parameters act coherently on overlapping qubit subsystems—giving a compact relational tensor whose entries flag collective behavior among encoded input elements. This relational reading is what makes the QFIM directly consumable by attention layers or message-passing networks.

Two encodings are used. For jets, the paper adopts the one-particle–one-qubit ($1$P$1$Q) encoding, mapping each of the ten highest-$p_\mathrm{T}$ constituents (features $(p_\mathrm{T}, \eta, \phi)$) to a dedicated qubit. For molecules, the authors introduce a novel two-atom–two-qubit ($2$A$2$Q) embedding: ten qubits, one per heavy atom, with learnable per-species $\bm{\Theta}(x)$0 rotations and pairwise Ising-type entanglement blocks $\bm{\Theta}(x)$1 applied only to bonded pairs within a $\bm{\Theta}(x)$2 Å cutoff. The pairwise angles combine frame-invariant distances with residual frame-dependent angular terms—a trade-off the authors accept explicitly, since the VQC is not intended to be state-of-the-art in isolation. Predictions are read out via a weighted sum of Pauli-$\bm{\Theta}(x)$3 observables, trained with Huber loss.

Fusion mechanisms

The fusion is deliberately architecture-agnostic but implemented differently per backbone:

• Transformer (Particle Transformer): each particle slot receives all 90 channels of its reshaped $\bm{\Theta}(x)$4 QFIM as a feature vector, embedded by an MLP into a 128-dimensional token appended to the classical token sequence, yielding 20 tokens total. The Lorentz-vector pair bias is zero-padded to the doubled sequence length. Parameter count rises from 2.14M to 2.29M (+7%).
• GNN (DimeNet++): rather than adding a parallel QFIM branch—which the authors note could improve performance merely through generic capacity—they apply a residual multiplicative gate on directed edge states after the initial embedding and after each interaction block: $\bm{\Theta}(x)$5, where $\bm{\Theta}(x)$6 is produced by a small CNN over the $\bm{\Theta}(x)$7 QFIM sub-block followed by an MLP with $\bm{\Theta}(x)$8 output, and $\bm{\Theta}(x)$9 is a global learnable scalar initialized to zero. The resulting $x$0DimeNet++ adds only 0.27% parameters (1.891M vs. 1.886M).

The zero-initialized gate ensures the augmented network starts identical to the baseline, strengthening the attribution of any gain to the QFIM signal itself.

Results

Jet tagging. On binary classification of hadronic top-quark jets versus QCD multijet background, Quiver-augmented ParT outperforms ParT across all training sizes (0.1M, 0.5M, 5M jets) and both feature sets (kinematic-only and full). With kinematic-only features at 5M training examples, AUC improves from 0.97832 ± 0.00004 to 0.98070 ± 0.00003, and the QCD background rejection $x$1 at $x$2 improves from 176 ± 1 to 240 ± 1—a substantial gain given that the largest classical baseline already has more parameters than the Quiver variant. Gains persist, though attenuate, with full features (e.g., $x$3: 1306 ± 8 → 1362 ± 28 at 5M). The improvement is largest in the low-data regime, consistent with the quantum view supplying structure that would otherwise require additional capacity or data to learn implicitly.

Molecular regression. On QM9 HOMO–LUMO gap regression, $x$4DimeNet++ achieves a test MAE of 67.92 ± 1.98 meV versus 72.42 ± 1.52 meV for DimeNet++, a 6.21% relative reduction at 0.27% parameter overhead. Over ten paired seeds, the mean paired difference is $x$5 meV, with a paired $x$6-test giving $x$7 ($x$8). The paired-seed design (identical data splits and RNG state across variants) and the near-zero parameter overhead jointly support the claim that the QFIM contributes genuine information rather than generic capacity.

Limitations

The paper is candid about several constraints. Classical simulation cost restricts both encodings to 10 qubits, so only the ten highest-$x$9 jet constituents are used out of up to 150 available in JetClass, and molecules are truncated to ten heavy atoms with hydrogen information largely discarded. Consequently, absolute benchmark numbers fall below published state-of-the-art for both tasks—the authors position the results as a controlled methodological comparison under identical conditions rather than new records. The molecular encoding retains residual reference-frame dependence through pairwise angular features. Finally, the pipeline trains the VQC separately from the downstream model; a fully joint hybrid optimization, with the circuit updated using QFIM-based measurements rather than observable expectations, is identified as an open technical challenge that could yield further gains.

Conclusion

Quiver demonstrates that QFIM-derived features from small, classically simulated VQCs can be fused into large classical models—at modest parameter cost—and yield statistically significant improvements on both transformer- and message-passing-based architectures across HEP and chemistry benchmarks. The evidence for complementarity rests on paired-seed significance testing and negligible-overhead gating designs; whether these gains survive scaling beyond 10 qubits, and whether joint quantum-classical optimization amplifies them, remain open questions raised by the work itself.