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

Conference: ICML 2026
arXiv: 2606.02785
Code: None (Repository not released)
Area: Physics / Hybrid Quantum-Classical Learning / HEP + Molecular Chemistry
Keywords: Variational Quantum Circuits, Quantum Fisher Information Matrix, Multi-modal Representations, Particle Transformer, DimeNet++

TL;DR

Quiver feeds categorical inputs into an additional Variational Quantum Circuit (VQC) to extract the Quantum Fisher Information Matrix (QFIM) as a "Quantum Geometric View." This view is then injected into classical backbones via cross-attention (for Transformers) or residual gating (for GNNs), achieving consistent improvements across two distinct physical tasks: JetClass top quark tagging and QM9 HOMO-LUMO gap regression.

Background & Motivation

Background: Jet tagging in high-energy physics and property prediction in molecular chemistry (QM9) both represent high-dimensional structured data problems. Mainstream approaches such as Particle Transformer (~2.14M parameters) and geometric/equivariant GNNs like DimeNet++ have approached SOTA on their respective benchmarks.

Limitations of Prior Work: These models are trained entirely in classical feature spaces. For samples requiring higher-order or non-local correlations—such as color-singlet \(W\) jets vs. color-connected QCD jets, or electronic structures in QM9 that depend on many-body correlations—models can only learn these associations implicitly through increased capacity rather than having them explicitly exposed.

Key Challenge: Classical feature engineering (kinematics, structural descriptors) is inherently poor at expressing many-body coherence correlations. Simply increasing model capacity or data volume does not efficiently bridge this structural blind spot. A fundamentally different geometric perspective is needed that is complementary to, rather than redundant with, classical features.

Goal: To decompose the problem into two sub-tasks: (1) How to use quantum circuits to extract "geometric correlation structures" from classical inputs into a compact, system-agnostic tensor; (2) How to integrate this tensor into existing SOTA classical backbones with minimal parameter cost and physical alignment.

Key Insight: A Variational Quantum Circuit \(|\psi(\boldsymbol{\Theta})\rangle=U(\boldsymbol{\Theta})|0\rangle^{\otimes N}\) encodes inputs into Hilbert space, where the parameter manifold naturally carries the Fubini-Study metric, which is equivalent (up to a factor of 4) to the Quantum Fisher Information Matrix (QFIM). The diagonal terms of the QFIM represent "single-parameter sensitivity," while off-diagonal terms represent "coherence coupling"—a geometric encoding of many-body correlations that can be computed using classical simulators (e.g., PennyLane).

Core Idea: Use the "Quantum Fisher View" as a second modality complementary to the classical view. Once fused, the classical backbone can directly consume quantum geometric information rather than learning it implicitly from scratch.

Method

Overall Architecture

Quiver = Classical Input → Task-specific VQC → QFIM Measurement → Modality Fusion Layer → Classical SOTA Backbone. Each application uses a dedicated quantum encoding: 1P1Q (one qubit per particle) for jets, and a new 2A2Q (two-qubit blocks per bonded atom pair) for molecules. Fusion mechanisms are differentiated by backbone: Transformers use cross-attention via sequence concatenation, while GNNs use residual gating on edge states modulated by the QFIM. The entire VQC is simulated classically on PennyLane, with QFIMs pre-computed and cached.

Key Designs

  1. Quantum Fisher View: Extracting Many-body Coherence with VQC:

    • Function: Maps classical input \(x\) to a parameterized quantum state \(|\psi(\boldsymbol{\Theta}(x),\boldsymbol{\theta})\rangle\), then calculates the QFIM \(F_{ij}(\boldsymbol{\theta};x)=4\,\text{Re}[\langle \partial_i\psi|\partial_j\psi\rangle-\langle\partial_i\psi|\psi\rangle\langle\psi|\partial_j\psi\rangle]\) at a fixed reference point \(\boldsymbol{\theta}_0\) to obtain an input-dependent relation tensor.
    • Mechanism: Diagonal elements \(F_{ii}\) represent the circuit's local sensitivity to \(\theta_i\), serving as "per-qubit dynamic importance"; off-diagonal elements \(F_{ij}\) are non-zero only when two directions act on overlapping qubit subsystems, thus directly encoding "coherently coupled input dimensions." For 1P1Q encoding (10 particles × 3 rotation parameters per qubit), the QFIM is a 30×30 real symmetric matrix stored as 90 channels × 10 particles. For 2A2Q, a 10-qubit × 2-layer × 3-rotation setup yields a 60×60 matrix organized into 10×10 blocks of 6×6, where sub-block \(Q_{ij}\) corresponds to the coupling of atom pair \((i,j)\).
    • Design Motivation: The QFIM is an intrinsic geometry of the parameter manifold, independent of the measurement basis. Off-diagonal elements naturally flag "joint behaviors," corresponding to physically meaningful high-order correlations that are difficult for classical features to capture.

  2. 2A2Q Molecular Encoding: Rotation-Invariant Pairwise Quantum Encoding:

    • Function: A novel molecular quantum encoding for the QM9 task that avoids coordinate dependency and integrates chemical bond information into entanglement operations.
    • Mechanism: Each heavy atom is assigned one qubit. First, single-atom embeddings \(R_Y(w_{\text{atom}}^j)|0\rangle\) are applied. Then, for each pair of bonded atoms where \(d_{ij}<d_{\text{CUTOFF}}=1.7\,\text{Å}\), three angles \(\omega_1^{(ij)}=e_{d_1}(1-d_{ij}/d_{\text{CUTOFF}})\cos\theta_{ij}\), \(\omega_2^{(ij)}=e_{\text{bond}}^{(ij)}\pi\), and \(\omega_3^{(ij)}=e_{d_2}(1-d_{ij}/d_{\text{CUTOFF}})\cos\phi_{ij}\) are used for joint encoding and entanglement: \(\mathcal{U}_{ij}=(I_{YY}(\omega_3)I_{ZZ}(\omega_2)I_{XX}(\omega_1))(R_Y\otimes R_Y)|00\rangle\). Finally, \(R_Z R_Y R_Z\) rotations are applied to each qubit. Predictions are derived from the expectation value of Hamiltonian \(\mathcal{H}=\sum_i c_i Z_i\), trained with Huber loss.
    • Design Motivation: Directly encoding Cartesian coordinates onto qubits introduces reference frame dependency. By merging "encoding + entanglement" into pairwise operations, the distance \(d_{ij}\) is naturally invariant. \(e_{\text{bond}}\) allows the entanglement strength to learn chemical bond types, making the QFIM sub-blocks reflect bonding correlations.

  3. Differentiated Architectural Injection: Cross-attention and Gated Residuals:

    • Function: Fuses the QFIM modality with classical modalities across different backbones without significant parameter increases.
    • Mechanism: For Particle Transformer, the 90 QFIM channels for each particle \(i\) are embedded into a 128-d token \(q_i=\text{MLP}_{\text{QFIM}}(\mathbf{Q}[:,i])\) and appended to the classical sequence, forming an input of length \(2P\). For DimeNet++, a residual gate \(\tilde{x}_{ij}^{(l)}=(1+\alpha\cdot\Theta(Q_{ij}))x_{ij}^{(l)}\) modulates edge states, where \(\alpha\) is a globally learnable scalar initialized to zero, and \(\Theta(Q_{ij})\in[-1,1]\) is processed from the 6×6 QFIM sub-block via a small CNN and \(\tanh\).
    • Design Motivation: Transformers possess a natural cross-attention mechanism, making sequence concatenation intuitive. GNNs lack this; therefore, "zero-initialized residual gates" ensure that when \(\alpha=0\), Quiver is equivalent to the baseline, guaranteeing that improvements stem strictly from QFIM information.

Loss & Training

Standard cross-entropy is used for JetClass binary classification. Huber loss is used for QM9 regression (robust to outliers, combining \(\ell_2\) and \(\ell_1\)). VQCs are simulated in PennyLane, and QFIMs are computed using standard implementations. Both tasks are evaluated over multiple seeds (5 for JetClass, 10 for QM9).

Key Experimental Results

Main Results 1: JetClass Top Quark vs. QCD Classification

Feature Set Model Params AUC ↑ 1/ε_B @ ε_S=0.5 ↑
Kin ParT 5M 0.97832 ± 0.00004 176 ± 1
Kin Quiver 5M 0.98070 ± 0.00003 240 ± 1
Full ParT 5M 0.99235 ± 0.00003 1306 ± 8
Full Quiver 5M 0.99244 ± 0.00003 1362 ± 28
Full ParT 0.1M 0.98875 ± 0.00008 570 ± 13
Full Quiver 0.1M 0.98893 ± 0.00005 590 ± 7

Using only kinematic features, Quiver with 5M parameters increases the QCD rejection rate from 176 to 240 (+36%). With full features, it increases from 1306 to 1362 (+4%), with a parameter cost of only +7% (2.14M → 2.29M).

Main Results 2: QM9 HOMO-LUMO Gap Regression

Model Params Test MAE (meV) ↓ Paired Δ MAE (meV) Rel. Decrease
DimeNet++ 1.886M 72.42 ± 1.52 — —
𝒬DimeNet++ (Quiver) 1.891M 67.92 ± 1.98 4.50 ± 2.46 6.21%

With a parameter increase of only 0.27%, the 10-seed paired \(t\)-test yielded \(t_9=5.78, p<10^{-3}\), indicating statistical significance.

Key Findings

  • Improvements across both tasks are "persistent": JetClass training curves show that the \(\Delta\) MAE between 𝒬DimeNet++ and DimeNet++ remains positive across all epochs.
  • Gains do not vanish as the classical model scales: Quiver performs better across 0.1M, 0.5M, and 5M parameter ranges, suggesting QFIM provides "information" rather than just "capacity."
  • Relative improvements of several percentage points are achieved with minimal parameter costs (+0.27% to +7%), providing evidence for "quantum advantage without quantum speedup."
  • Success across both architectures (Transformer and GNN) validates the claim of being "architecture-agnostic."

Highlights & Insights

  • QFIM as a Modality, Not Auxiliary Loss: Unlike most hybrid quantum-classical methods that treat VQCs as part of an end-to-end chain, Quiver extracts QFIM as "data" to be consumed by classical SOTA models. This decoupling allows the method to run on classical simulators today without relying on NISQ hardware.
  • Zero-Initialized Gating Design: Initializing \(\alpha\) to 0 ensures baseline equivalence, making the argument that "Quiver's gains originate from QFIM information" rigorous at the design level.
  • 2A2Q Encoding: By encoding chemical bonds into entanglement strength \(e_{\text{bond}}\), the quantum circuit acts as a "physics-aware feature extractor."
  • Cross-domain Stability: Stable improvements in High Energy Physics (Transformer) and Molecular Chemistry (GNN) suggest that Quantum Fisher geometry encodes a "domain-agnostic many-body correlation structure."
  • "Harvesting the Future": Demonstrates quantifiable performance gains for large models using classically simulated VQCs, offering an immediately applicable direction for quantum machine learning in the pre-fault-tolerant era.

Limitations & Future Work

  • Classical simulation costs limit the qubit count to \(\leq 10\), necessitating the exclusion of many particles in JetClass and hydrogen atoms in QM9.
  • QFIMs are calculated at a fixed reference \(\boldsymbol{\theta}_0\) rather than being jointly optimized with the downstream model. The challenge for joint optimization lies in backpropagating through QFIM measurements.
  • The paper does not extensively discuss the storage or time costs of QFIM pre-computation, leaving its scalability to industrial-sized datasets (e.g., full JetClass) insufficiently proven.
  • Comparison with classical baselines is relatively narrow, focusing only on ParT and DimeNet++, lacking horizontal comparisons with other "explicit higher-order" methods like EFN or PointNet++.
  • vs. Bal et al. 2025 (1P1Q): Adopts their 1P1Q encoding but innovates by using QFIM as a fused view rather than using VQC directly for prediction, bypassing the "VQC performs worse than SOTA" bottleneck.
  • vs. Classical Multi-modal Fusion: Unlike typical image/text fusion, Quiver's second modality is generated from the first via a physically interpretable transform, eliminating cross-modality alignment issues.
  • vs. Increasing Model Capacity: Comparisons with wider baselines and the marginal 0.27% parameter increase in 𝒬DimeNet++ confirm that gains come from informational content, not just parameter count.