Quantum ML research, Neural collapse in VQCs
QNC
A research paper showing that training a variational quantum classifier rotates its states rather than contracting them, and where collapse can still appear.
- ~10⁻¹⁵Relative change when checking the exact-rotation result at 4, 6 and 8 qubits: machine precision
- 2–3Orders of magnitude the measurement-visible projection decays during training
- 52Preregistered predictions across 10 stages, each threshold committed before its metric
- 2IBM quantum hardware backends used for verification: ibm_marrakesh and ibm_kingston
How it works
- Preregister52 predictions over 10 stages, thresholds committed before metrics are computed
- Train VQCsPennyLane classifiers on MNIST and Fashion-MNIST, two encodings
- Measure geometryFull-Hilbert-space collapse witnesses and the measurement-visible projection
- ProveProposition 3: training conjugates every state by one global unitary
- Hardware checkVerification on ibm_marrakesh and ibm_kingston
- Classical controlMatched classical MLP trained for comparison
Problem
Neural Collapse is a geometry that classical networks reach late in training: samples of the same class squeeze together. Prior work (Du et al., PRL 2023) proved that variational quantum circuits reach it, but only at the static global optimum. I wanted to know whether it actually emerges while a VQC is trained, and by what mechanism.
Approach
I ran it as preregistered science. Before computing any metric I wrote down 52 predictions across 10 stages, each with a committed threshold, then scored every one CONFIRMED, REFUTED or PARTIAL against a frozen spec. The code carries 173+ tests and runtime physics assertions on every state: Hermitian, trace 1, positive semidefinite.
Experiments use PennyLane’s default.qubit at 4, 6 and 8 qubits, on MNIST and Fashion-MNIST under two encodings, repeated with a second optimizer and a second ansatz family. A matched classical MLP is the control. The result was also verified on two IBM quantum hardware backends, ibm_marrakesh and ibm_kingston.
Results
The main result is Proposition 3: what looks like collapse in a trained VQC is an exact rotation, not a contraction. Training conjugates every sample’s state by one global unitary, so the geometry between states is conserved. The relative change is ~10⁻¹⁵, machine precision, at 4, 6 and 8 qubits.
That is why every full-Hilbert-space collapse witness is a null result. What does change is the projection a measurement can see, which decays by 2–3 orders of magnitude. The classical MLP does the opposite: its within-class scatter grows 27–163× in absolute terms.
The paper is “Geometry Conservation in Variational Quantum Classifiers and Where Neural Collapse Can Appear” (earlier title “Collapse by Rotation”). I am first author (GCET), with Puja S Prasad (corresponding author) and Partha Pratim Roy of IIT (ISM) Dhanbad. It was submitted to Springer’s Quantum Machine Intelligence in August 2026 and is in major revision after 19 referee comments. The repo is public on GitHub.
What I learned
The null result is the interesting part. I set out to measure collapse in the state space and found conservation instead. The collapse lives only in what measurement can see, and the proof explains why.
Preregistration changed how I read my own numbers. With thresholds fixed in advance, a REFUTED score is information, not a prompt to adjust the metric until it passes.
The work continues: follow-up research on confidence-controlled Pauli-path simulation for VQC training is in progress.
Limits, stated not hidden
- The paper is not accepted yet: it is in major revision after 19 referee comments.
- The simulation evidence covers 4, 6 and 8 qubits; I make no claim about larger circuits.
- Every full-Hilbert-space collapse witness came back null; collapse appears only in the measurement-visible projection.
- Predictions were scored CONFIRMED, REFUTED or PARTIAL, and refuted ones stay in the record.