Quiz: Hybrid Architectures

Answers with explanations at the end.

1. Every algorithm in Parts IV and VI is hybrid. What is the interesting question instead?

2. Give the three placements of the quantum/classical boundary this book has measured, and the cost each incurs.

3. What is the encoding cost for quantum data, and why?

4. Chapter 32's $\mathcal{O}(N)$ bound is information-theoretic. Why does it not apply here?

5. Why is the transverse-field Ising model a good test rather than a demonstration?

6. The classifier reached train 1.0000 and test 0.6429. What should be concluded, and what should be reported?

7. How many real numbers is a 6-qubit state?

8. What did the classical baselines score, and which won?

9. Which representation beat which, and why is that direction damaging to the quantum claim?

10. State the general lesson about savings.

11. Why is this obstacle different from every other one in Part VI?

12. Where is the boundary, and which other chapters does it govern?

13. Comparing shadows and direct estimation at the same shots per observable gives what result?

14. Why is that comparison unfair?

15. Give the result at equal total budget.

16. Explain the reversal arithmetically.

17. How does the shadow advantage scale with the number of observables, and where is the breakeven?

18. Why are shadows the only technique in this book that helps with the shot budget?

19. In a naive hybrid loop, what fraction of wall-clock is computation, and what is the fix?

20. Summarize Part VI.

Answers

1. Where to put the boundary. You have no choice about being hybrid; you do have a choice about how much work sits on each side, and each placement trades one cost for another.

2. Variational (Ch. 32/33): the quantum part evolves and is optimized → barren plateaus, and shots × samples × steps. Kernel (Ch. 34): the quantum part only evaluates → convex training, but $n^2$ evaluations. Quantum data (Ch. 35): the quantum part supplies the input too → no encoding cost.

3. Zero. Chapter 32's amplitude encoding charges $N - \log_2 N - 1$ two-qubit gates per sample per epoch — 57 for 64 amplitudes, 63,840 over a full training run. Quantum data pays none of it, because the state is already in the register.

4. Because the bound is about loading: an arbitrary $N$-dimensional state has $N$ independent real parameters, so getting them into a register costs $\mathcal{O}(N)$ gates. A state produced by a physical process was never loaded — it was prepared by the dynamics. There is nothing to encode.

5. Because it is a real physics question with a known answer: the ground state is ordered for $h<1$, disordered for $h>1$, with a genuine quantum phase transition at $h=1$. The labels are not invented for the demonstration.

6. Overfitting — 36 parameters on 28 training samples. Both numbers should be reported; quoting only the 1.0000 would be the failure Chapter 33 §33.3 and Chapter 34 §34.6 both catalogued.

7. $2 \times 2^6 = $ 128 — a 64-dimensional complex vector. An ordinary feature vector.

8. LogisticRegression 0.6429 on both representations; SVC and kNN 0.5714 on full amplitudes and 0.7857 on probabilities. The best classical result, 0.7857, beat the quantum classifier's 0.6429.

9. Probabilities beat full amplitudes (0.7857 against 0.5714). It is damaging because it closes off the natural rescue — "the classical model does not see the full quantum state." It does, and it does better when it sees less: the phase information adds 64 dimensions of noise.

10. A saving is only a saving relative to what the alternative pays. The encoding cost is what it takes to get $N$ numbers into a quantum register; a classical model has no register and simply reads them. Eliminating a cost your competitor never paid is not an advantage.

11. Because it gets better with scale rather than worse. Chapter 32's encoding cost grows with $N$, Chapter 33's inference bill with usage, Chapter 34's Gram matrix as $n^2$, and plateaus and concentration exponentially in qubits. This one disappears: at 50 qubits the state is $2.3\times10^{15}$ real numbers and there is no classical feature vector to hand a baseline.

12. Around 30–35 qubits, where statevector simulation dies (Chapter 26 §26.2 measured 16 GB at 30 qubits). The same boundary governs Chapter 21's Grover crossover, Chapter 23's Shor, Chapter 25's threshold, and Chapter 30's XEB — the quantum case begins where classical simulation ends.

13. Shadows look about 2.5× worse — 0.0542 against 0.0202 at 1,000 shots, stable across three shot counts and twelve repetitions each.

14. Because direct estimation runs one circuit per observable. "1,000 shots each" gives direct estimation $1{,}000 \times 17 = 17{,}000$ total shots against the shadow method's 1,000. It silently gives one side seventeen times the budget.

15. Shadows win at every budget, by 1.5–1.8×: 0.0431 against 0.0645 at 1,700 total shots, 0.0172 against 0.0284 at 8,500, and 0.0087 against 0.0157 at 34,000.

16. Error scales as $1/\sqrt{N}$, so splitting a budget 17 ways costs $\sqrt{17} = 4.1\times$. Shadows pay only about 2.5× for reusing one record. The difference — roughly 1.6× — is the win.

17. The $\sqrt{n}$ splitting penalty grows (2.0× at 4 observables, 4.1× at 17, 31.6× at 1,000) while the shadow penalty stays at ~2.5×. The breakeven is around four to eight observables, low enough that most realistic workloads are past it.

18. Because every other technique tried to reduce shots per estimate and ran into $1/\epsilon^2$ — Chapter 24's mitigation, Chapter 27's tolerance, Chapter 32's plateau product, Chapter 34's concentration product. Shadows increase the estimates per shot instead. That does not repeal $1/\epsilon^2$; it changes what a fixed budget buys.

19. Under 0.1%. Chapter 31 measured the circuit at 7.10 μs while queue and network latency on a shared device is seconds, so a 24-parameter, 60-step optimization spends ~2,940 seconds waiting and 0.02 seconds computing. The fix is co-locating the classical update with the quantum device, which is what hybrid runtimes do.

20. No quantum model in Part VI beat a tuned classical baseline on any dataset, and three of four tied with logistic regression. Alongside that: the techniques are real and some are elegant (single-qubit universality, kernel convexity, classical shadows), the negative results are the field's most valuable output (barren plateaus, the input problem, concentration, dequantization — all found by people trying to make QML work), and the quantum-data case survives the scaling. If a practical advantage is demonstrated, current evidence points there.