Exercises: Quantum Classifiers

These need pennylane and scikit-learn. Solutions to starred exercises are in Answers to Selected Exercises.

A standing rule for this chapter: no accuracy comparison counts until it has been run on at least five independent datasets and splits, with an error bar. Case Study 1 is what happens otherwise.

Choosing a dataset

33.1 ★ Fit four classical baselines to iris-binary and confirm all reach 1.0000. Explain why a quantum result on that dataset carries no information.

33.2 ★ Sweep make_moons noise from 0.05 to 0.45 and report the best classical accuracy at each. At what noise level does the problem stop being trivially solved?

33.3 ★ Do the same for make_circles. Which dataset family leaves more room, and why?

33.4 ★★ LogisticRegression underperforms kNN by 0.18 on moons at noise 0.30. Explain in terms of the decision boundary, and say what that gap represents for a quantum model.

33.5 ★★ Construct a dataset where kNN performs badly and a linear model does well. What structure did you need?

33.6 ★★★ Design a dataset that would genuinely favour a quantum model. What property would it need, and is that property natural or constructed? (Compare Chapter 32 §32.6's separations.)

Data re-uploading

33.7 ★ Implement the single-qubit re-uploading circuit. Confirm it has 3 parameters per layer.

33.8 ★ Train it on moons at noise 0.30 with 2, 4, and 8 layers. Report accuracy for each.

33.9 ★★ Show that a single-qubit circuit without re-uploading — encode once, then rotate — cannot represent a non-linear boundary. Measure its accuracy.

33.10 ★★ Trace the composition: write out what a 2-layer re-uploading circuit computes as a function of $x$. Where does the non-linearity come from?

33.11 ★★ Does accuracy improve monotonically with layers? Plot it, and explain any plateau.

33.12 ★★★ Implement the original paper's version, which uses a richer per-layer encoding $R(\theta + w \odot x)$. Does it outperform the simple version measured here?

33.13 ★★★ How many layers does the universality theorem require for a given approximation error? Look it up, and compare with the number that actually helps here.

Replication

33.14 ★ Train the 1-qubit and 4-qubit models on random_state=0 and reproduce the 0.13 gap.

33.15 ★ Now run ten seeds. Report mean, std, min, and max for each model.

33.16 ★ Confirm that split 0 is the 1-qubit model's maximum and the 4-qubit model's minimum. What does that tell you?

33.17 ★★ Compute the paired difference across splits with a standard error. Is it significant at 2σ? At 1σ?

33.18 ★★ How many splits would you need to resolve a true difference of 0.02, given the observed spread? Compute it.

33.19 ★★ Compare re-splitting one dataset ten ways against generating ten datasets. Do the error bars differ? Which is the right protocol and why?

33.20 ★★★ Write compare_models(scores_a, scores_b, ...) that refuses fewer than five replicates, and make the refusal message quote the measurement that motivates it.

The classical bar

33.21 ★ Compare the 1-qubit model against kNN across ten splits. Report the difference and its significance.

33.22 ★ Compare it against LogisticRegression. What does the result say about the universality theorem?

33.23 ★★ Tune kNN's k and SVC's hyperparameters by grid search. Does the gap widen?

33.24 ★★★ Give the quantum model more layers, more steps, and a better optimizer. How much of the 0.0626 gap to kNN can you close, and what does it cost?

The decision rule and the bills

33.25 ★ Compute $|\langle Z\rangle|$ for every test prediction and report the distribution. How many points sit within 0.05 of the boundary?

33.26 ★ Sweep the shot count from 100 to 100,000 and report mean accuracy, standard deviation, and label flips against the exact prediction.

33.27 ★★ Write shots_for_decision(margin_profile, tolerated_flips) deriving the shot count from the margins. Compare its answer with the measured sweep.

33.28 ★★ Construct a dataset where many points sit near the boundary. How does the required shot count change?

33.29 ★★ Compute the inference cost for 1 million predictions at 100, 1,000, and 10,000 shots. Then measure kNN.predict on 1 million points and state the ratio.

33.30 ★★★ For a model serving $P$ predictions per month over $M$ months, at what $P \cdot M$ does inference cost exceed training cost? Solve it symbolically and evaluate for this chapter's classifier.

Project

33.31 ★★ (Project Checkpoint) Build vqelab/classifiers.py with compare_models, SplitComparison, margin_profile, shots_for_decision, training_cost, inference_cost, and ClassifierReport. Write tests asserting:

  1. compare_models returns INSUFFICIENT_REPLICATES for a single split, and the reason quotes what replication showed.
  2. The refusal holds for every count below the minimum.
  3. The single-split gap is 0.1313 and the ten-split difference is $+0.0202 \pm 0.0170$, not significant.
  4. Split 0 is the 1-qubit maximum AND the 4-qubit minimum.
  5. kNN beats the quantum model significantly, at more than five standard errors.
  6. The quantum model ties with LogisticRegression.
  7. Comparison requires paired lengths; the σ threshold is adjustable.
  8. margin_profile reports the boundary distribution and rejects empty input.
  9. shots_for_decision demands more shots when points sit nearer the boundary.
  10. Tolerating one flip substantially lowers the count.
  11. This chapter's data needs under 2,000 shots.
  12. Inference costs 27.8 QPU hours per million predictions at 1,000 shots, and 277.8 at 10,000.
  13. training_cost reproduces Chapter 32's 2,058,000,000 shots and 2.38 QPU-days.
  14. ClassifierReport is not deployable-claim-supported without an inference cost, and says so.
  15. Nor without enough replicates.
  16. The cheap model really does use a quarter of the parameters and qubits.

Tests 1, 3, 4, 12 and 14 are the ones this chapter exists to encode.

33.32 ★★★ Extend the module with lifetime_cost(report, predictions_per_month, months) reporting the training and inference totals side by side, and the month at which inference overtakes training.

Going further

33.33 ★★ Multi-class classification needs a different readout. Implement one-vs-rest and a multi-qubit measurement scheme, and compare on three-class iris.

33.34 ★★★ Class imbalance interacts with shot noise: the minority class has fewer samples and its decision margins matter more. Construct an imbalanced dataset and measure how the shot requirement changes.

33.35 ★★★ Apply Chapter 32 §32.8's seven-question checklist to this chapter's own results. Which questions does this chapter answer, and where is it weakest?