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SSL4Q: Semi-Supervised Learning of Quantum Data with Application to Quantum State Classification
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Proposes SSL4Q, a mean-teacher style semi-supervised transformer with shadow embeddings and permutation-equivalence attention for classifying quantum states from unordered random-Pauli measurement records.
Problem
Classifying quantum states from measurement data normally requires many labeled examples, whose labeling cost grows exponentially with qubit count; semi-supervised learning for quantum data is largely unexplored.
Approach
Combines shadow-style embedding of Pauli bases and outcomes with a permutation-equivalence attention layer (no positional encoding) and a teacher-student EMA consistency framework with Gaussian-noise augmentation, so that unlabeled measurement records contribute regularization.
Key results
- On Heisenberg model with L=20-50, mask rate 0.9 and M=16, SSL4Q reaches roughly 72-84% accuracy versus about 55-68% for kernel baselines and 61-74% for RNN/LLM4QPE in Table 1.
- On VQC-MNIST with mask rate 0.95, SSL4Q achieves 99.24% accuracy versus 95.74% for QCNN and 93.24% for CNN.
- Under depolarizing noise p=0.1, SSL4Q keeps 97.68% accuracy while QCNN drops to 72.34%.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use SSL4Q with semi-supervised consistency training instead of the strongest supervised baseline | Heisenberg model classification · Heisenberg model, M=16, L=20 | accuracy: 63.90 -> 71.97 | Helped |
| use SSL4Q with semi-supervised consistency training instead of the strongest supervised baseline | Heisenberg model classification · Heisenberg model, M=16, L=30 | accuracy: 64.47 -> 68.63 | Helped |
| use SSL4Q with semi-supervised consistency training instead of the strongest supervised baseline | Heisenberg model classification · Heisenberg model, M=16, L=40 | accuracy: 64.53 -> 72.87 | Helped |
| use SSL4Q with semi-supervised consistency training instead of the strongest supervised baseline | Heisenberg model classification · Heisenberg model, M=16, L=50 | accuracy: 67.60 -> 71.40 | Helped |
| use SSL4Q instead of QCNN | MNIST VQC classification · MNIST VQC, mask rate 0.6 | accuracy: 97.40 -> 99.20 | Helped |
| use SSL4Q instead of QCNN | MNIST VQC classification · MNIST VQC, mask rate 0.7 | accuracy: 96.80 -> 99.15 | Helped |
| use SSL4Q instead of QCNN | MNIST VQC classification · MNIST VQC, mask rate 0.8 | accuracy: 96.95 -> 99.20 | Helped |
| use SSL4Q instead of QCNN | MNIST VQC classification · MNIST VQC, mask rate 0.9 | accuracy: 96.38 -> 99.10 | Helped |
| use SSL4Q instead of QCNN | MNIST VQC classification · MNIST VQC, mask rate 0.95 | accuracy: 95.74 -> 99.24 | Helped |
| use SSL4Q instead of QCNN under depolarizing noise | VQC classification · VQC noisy MNIST, p=0.1 | accuracy: 72.34 -> 97.68 | Helped |
| add consistency-loss training to the supervised-only SSL4Q variant (ablation) | SSL4Q · Heisenberg model, M=16, L=20 | accuracy: 64.07 -> 71.97 | Helped |
| add consistency-loss training to the supervised-only SSL4Q variant (ablation) | SSL4Q · MNIST VQC, mask rate 0.95 | accuracy: 98.96 -> 99.24 | Helped |
Surprising findings
- The fully supervised SSL4Q variant alone already outperforms LLM4QPE and RNN, suggesting the PEA architecture contributes beyond semi-supervision.
Limitations
- Evaluated only on simulated data (Heisenberg/transverse-field Ising and VQC-MNIST), with binary MNIST classification for the VQC case.
- Hyperparameters such as Gaussian noise std and consistency weight were tuned on one configuration, and the authors note further tuning may help.
- Baselines are fully supervised only, as no prior semi-supervised method for quantum many-body problems was available.
Experimental setup
Datasets: MNIST simulated transverse-field Ising (Heisenberg) quantum system measurement records
Benchmarks: Heisenberg model quantum state classification (simulated) MNIST 0-vs-1 VQC-encoded quantum states
Metrics: classification accuracy
Baselines: Gaussian Kernel NTK RNN LLM4QPE CNN QCNN
Builds on: mean teacher classical shadow LLM4QPE Transformer
Closest papers by meaning
- Purest Quantum State Identification (NeurIPS 2025 · similarity 0.86)
- Online Learning of Pure States is as Hard as Mixed States (NeurIPS 2025 · similarity 0.85)
- Rethink the Role of Deep Learning towards Large-scale Quantum Systems (ICML 2025 · similarity 0.84)
- Limitations of measure-first protocols in quantum machine learning (ICML 2025 · similarity 0.84)
- QCircuitBench: A Large-Scale Dataset for Benchmarking Quantum Algorithm Design (NeurIPS 2025 · similarity 0.84)
- Reinforced Learning Explicit Circuit Representations for Quantum State Characterization from Local Measurements (ICML 2025 · similarity 0.84)
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