Quantum Neural Networks (QNN)
A QNN is a feature map plus a trainable observable, optimized against a supervised loss the same way VQE optimizes against energy. Key insight: it’s the same five-step variational loop as VQE — prepare a parameterized state, measure an observable, classically optimize — except the loss is a supervised prediction error instead of , and (distinctively) the observable itself can be trainable, not just the circuit. sQUlearn’s construction uses (a SummedPaulis observable): the readout weighting on each qubit is a learned parameter, so the model isn’t just choosing how to prepare a state, it’s also choosing how to weigh what it measures — a second trainable stage VQE doesn’t have (VQE’s observable, the Hamiltonian, is fixed by the physics problem, not learned).
The loop
1. Encode input x through a feature map circuit (see [[Data Encoding Circuits (Feature Maps)]])
2. Prepare |ψ(x, θ)⟩ — encoding + trainable ansatz layers
3. Measure the trainable observable O(θ_readout) = θ_0·I + Σ θ_i·Z_i
4. Loss = SquaredLoss(prediction, y_true); classical optimizer (e.g. Adam) updates θ
5. Repeat over the training set
Training under the noise you’ll deploy on
A concrete result worth internalizing: a QNN trained directly on a noisy backend (e.g. FakeTorino) measurably outperforms a QNN trained on an ideal noiseless simulator and only evaluated on the noisy backend afterward. The optimizer, given noisy feedback throughout training, finds parameters that are robust to that specific noise profile — noise-aware training is a real, usable technique, not just theoretical hardening. Same broader lesson as elsewhere in this vault (see the Error Mitigation section generally, e.g. NoiseLearnerV3 and Pauli-Lindblad Models): don’t treat “ideal circuit, apply mitigation after” and “expose the noise during optimization” as equivalent — they aren’t.
Related
- Data Encoding Circuits (Feature Maps)
- Variational Quantum Eigensolver (VQE) — same loop shape, energy loss instead of supervised loss, fixed observable instead of trainable
- Parameterized Circuits
- Quantum Kernel Methods — the other QML approach: no circuit parameters trained at all
Self-Check
- How does a QNN’s training loop resemble VQE’s, and where does it diverge?
- Why does making the readout observable trainable give the model something VQE’s fixed Hamiltonian observable doesn’t have?
- Why would a QNN trained directly on a noisy backend outperform one trained ideally and only evaluated on noise afterward?