VQC Embedding vs Classical Embedding on QM9 — a Confound-Controlled Negative

Summary

Controlled-substitution test of whether a shallow variational quantum circuit (VQC) used as the embedding layer inside a fixed transformer beats a classical linear embedding of equal output dimension on QM9 HOMO-LUMO gap prediction. Only the embedding changes. Result: the VQC (8 qubits, depth 2) is ~13% worse than the linear baseline (0.717 vs 0.634 eV test MAE, >3 sigma); a matched classical bottleneck (Linear 14->8->64) reaches 0.699 eV, showing ~78% of the VQC's deficit is the dimensional bottleneck, not quantum structure. Across a 30-config sweep (qubits 4/6/8/10 x depth 1/2/3 x entanglement linear/circular/all), 0/30 beat the linear baseline and 0/30 beat even the matched classical bottleneck. A clean, reviewer-proof negative at this scale and circuit family.

Claims

• —On QM9 HOMO-LUMO gap with a fixed 4-layer/4-head/d=64 transformer (only the embedding swapped; subset 5000, 40 epochs, 3 seeds), a shallow VQC embedding (8 qubits, depth 2, linear entanglement) yields test MAE 0.717 +/- 0.024 eV versus the classical linear embedding's 0.634 +/- 0.017 eV -- ~13% worse, a >3 sigma separation.
• —A classical control matched to the VQC's dimensional bottleneck (Linear 14->8->64, the VQC architecture minus the quantum circuit) yields 0.699 +/- 0.013 eV. The 8-dimensional bottleneck therefore accounts for ~78% of the VQC's deficit relative to the full linear map; the quantum circuit itself adds only ~0.018 eV over its matched classical twin (within ~1 sigma -- roughly neutral).
• —Across a 30-configuration sweep (qubits {4,6,8,10} x depth {1,2,3} x entanglement {linear, circular, all-to-all}), 0/30 configurations beat the linear baseline (0.634 eV) and 0/30 beat even the matched classical bottleneck (0.699 eV); the best VQC configuration was 0.721 eV (6 qubits, depth 2, all-to-all).
• —Conclusion at this scale and circuit family: the VQC feature map provides no representational advantage over a classical map of equal output dimension. Its disadvantage is overwhelmingly the dimensional bottleneck, not the quantum structure -- it is on par with (marginally below) a classical bottleneck of the same width.

Assumptions

• —The 30 sweep configurations are single-seed, so per-configuration MAE carries ~0.03 eV noise; fine-grained ranking within the sweep is coarse. The conclusion does not depend on exact ordering -- even the best VQC (0.721) is well above the linear baseline (0.634) and the matched classical bottleneck (0.699), each established with 3-seed error bars.
• —This is a deliberately small-scale probe (5000 molecules, 40 epochs, n <= 10 qubits, depth <= 3, classical simulation via PennyLane lightning.qubit with adjoint differentiation). It does not rule out a VQC-embedding advantage at larger model/data scale, deeper circuits, different encodings, or other circuit families; it characterizes this controlled substitution at this scale.
• —All quantum circuits are classically simulated; no quantum hardware is used. The VQC routes information through an n_qubit-dimensional bottleneck (pre-linear -> angle encoding -> variational layers -> Pauli-Z expectations -> post-linear), which is the structural reason the bottleneck control is the decisive comparison.

Context

Reproducibility

Structural Metrics