1. The input: qubits on a lattice with X and Z stabilizers
The state to classify is a topologically ordered state, such as the toric code, defined by stabilizer operators. X-type stabilizers act on vertices, Z-type stabilizers on plaquettes. Click any stabilizer below to highlight the qubits it acts on.
Click a purple vertex or green plaquette to see which four qubits its stabilizer measures. A state in the topological phase satisfies all stabilizers on average; perturbations degrade them.
2. The hybrid pipeline
The key idea: a shallow quantum circuit compresses the many-body state into a few measured bits, and a classical neural network learns to map those measurement statistics to a phase label. Click each stage to expand it.
3. Live simulator: can the hybrid network tell the phase?
Drag the perturbation strength. The demo emulates measurement statistics of stabilizer expectation values (with shot noise) and feeds them through a tiny trained classifier running right here in your browser. Near the phase transition the decision becomes uncertain, exactly as in the paper's phase-boundary experiments.
Classifier output
Phase diagram: classifier confidence vs perturbation
Each run adds a point. The crossing at P = 0.5 locates the phase transition (near g ≈ 0.5 in this toy model, mirroring how the paper extracts phase boundaries from network confidence).
4. Why hybrid?
Quantum part
A purely classical network would need exponentially many measurement outcomes to reconstruct a many-body state. A shallow quantum circuit can instead process the state coherently, extracting nonlocal order-parameter information, such as stabilizer or string-operator correlations, with only a few final measurements.
Classical part
Real hardware is noisy and the transition region is subtle. A conventional neural network is cheap, robust, and easy to train on the noisy measurement statistics, learning a decision boundary that generalizes across perturbation strengths, exactly what phase recognition requires in an experiment.
Together they form an end-to-end phase recognizer that is trainable, noise-tolerant, and implementable on today's devices, and that is the central result demonstrated in the paper.