Further Reading: Building Complex Circuits

Tagged Tier 1 (confident it exists and recommended) and Tier 2 (real and worth seeking, but verify the current version or URL).

The primary sources

• The qiskit.circuit documentation on Parameter, ParameterVector, and ParameterExpression. The authoritative account of binding semantics — including the sort order that §8.2's hazard depends on. Worth reading the parameters property's documentation carefully; the ordering is stated there and almost nobody reads it until after they have been bitten. Tier 1.
• The qiskit.circuit.library documentation. The full catalogue: ansätze, feature maps, arithmetic circuits, state preparation, and standard gates. Check that you are reading the function-based pages, not the deprecated class ones — the version selector matters here more than almost anywhere else in the library. Tier 1.
• The Qiskit 2.1 release notes, on the circuit-library deprecations. The rationale for replacing blueprint circuits with functions, in the maintainers' own words. Useful context for why the change was worth the churn, and a good example of a library correcting an abstraction that had outlived its usefulness. Tier 2 — find the version-specific page.

On ansatz design

This is an active research area with no settled answers, which §8.8's 🔬 Honest Assessment says plainly. The papers below are the ones worth knowing.

• Kandala et al., "Hardware-efficient Variational Quantum Eigensolver for Small Molecules and Quantum Magnets" (2017), Nature 549, 242. The paper that introduced the hardware-efficient ansatz and the term. Read it for the design rationale — the ansatz is shaped by the device, not by the chemistry, and the paper is explicit that this is a trade. Tier 1.
• Sim, Johnson, and Aspuru-Guzik, "Expressibility and Entangling Capability of Parameterized Quantum Circuits" (2019), Advanced Quantum Technologies 2, 1900070. A systematic comparison of ansatz families on two measurable axes. This is the paper to reach for when you want to justify an ansatz choice quantitatively rather than by convention, and it is the source of the expressiveness-versus-trainability framing that Chapter 35 develops. Tier 1.
• Cerezo et al., "Variational Quantum Algorithms" (2021), Nature Reviews Physics 3, 625. The standard review. Its ansatz section surveys the families — hardware-efficient, problem-inspired, UCCSD, HVA — with their costs and known pathologies. The best single starting point. Tier 1.
• McClean et al., "Barren Plateaus in Quantum Neural Network Training Landscapes" (2018), Nature Communications 9, 4812. Why more expressive is not better. Directly relevant to §8.7's entanglement-pattern advice and to §8.8's ry-versus-ryrz conclusion. Measured in Chapter 16 §16.6 and derived in Chapter 32 §32.5. Tier 1.

On the single-qubit cost result (§8.8)

The finding that ryrz is free deserves its source.

• McKay et al., "Efficient Z-Gates for Quantum Computing" (2017), Physical Review A 96, 022330. Virtual $Z$-gates — why rz costs nothing. Already recommended in Chapter 3 and worth re-reading here, because §8.8's surprise is a direct consequence. Tier 1.
• Nielsen and Chuang §4.2, on the $Z$–$Y$ decomposition. The reason an arbitrary single-qubit gate costs a fixed number of pulses regardless of complexity. Chapter 3's Case Study 2 is this result applied. Tier 1.

On the ordering hazard (Case Study 1)

Not a quantum problem, and the best writing about it is general.

• Anything on "primitive obsession" and on encoding invariants in types. The fix — replace a family of stringly-named parameters with a structured ParameterVector — is a textbook instance. Chapter 7's Case Study 1 made the same argument about the layout trap, and the pattern recurring twice in two chapters is itself the lesson. Tier 2.
• Any discussion of "natural sort" versus lexicographic sort. The specific mechanism (theta10 < theta2) is a well-known nuisance in file naming, version strings, and test ordering. Recognizing it as a known class of bug makes it much easier to spot. Tier 2.
• Chapter 27 §27.6 of this book, on tests that can never fail and why they are worth keeping. The probe test from Case Study 1 is the motivating example. Tier 1.

Forward references

• Chapter 24 puts this ansatz inside a working optimization loop.
• Chapter 28 reduces its depth, with measurements.
• Chapter 33 §33.2 covers feature maps properly — why zz_feature_map's parameters are data rather than free variables.
• Chapter 36 §36.4 compares this hardware-efficient ansatz against the chemistry-inspired UCCSD, which is the comparison Case Study 2's hardest question asks you to design.
• Appendix A condenses the circuit-library API to a reference card.

Where to go next. If one thing: Sim, Johnson, and Aspuru-Guzik on expressibility. It turns "which ansatz?" from a matter of convention into a measurement, which is the same move Case Study 2 makes for cost.

Then Chapter 9 — mid-circuit measurement, classical feedforward, and teleportation.