Further Reading: Quantum Error Correction in Code

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

The founding papers

Shor, "Scheme for reducing decoherence in quantum computer memory" (1995), Physical Review A 52, R2493. The nine-qubit code, and the paper that made quantum computing look possible rather than merely interesting. Short. Read it for the concatenation idea — §25.5's structure is right there. Tier 1.
Steane, "Error correcting codes in quantum theory" (1996), Physical Review Letters 77, 793. The $[[7,1,3]]$ code — same distance, two fewer qubits. Tier 1.
Calderbank and Shor (1996) and Steane (1996) on CSS codes. The construction that turns pairs of classical codes into quantum ones, and the reason the field could move quickly. Tier 1.
Gottesman, "Stabilizer Codes and Quantum Error Correction" (1997), PhD thesis, arXiv:quant-ph/9705052. The stabilizer formalism, from its author. Still the clearest single source on §25.6, and it contains the Gottesman–Knill theorem that makes Chapter 11's stabilizer simulator work. Tier 1.

On the threshold

The section to read if you read only one. §25.8's fan is the whole of fault tolerance in one table, and these are the sources that establish it properly.

Aharonov and Ben-Or, "Fault-tolerant quantum computation with constant error" (1997). One of the independent proofs of the threshold theorem. The formal version of "below threshold, any logical error rate is reachable." Tier 1.
Knill, Laflamme, and Zurek on the accuracy threshold (1998), and Kitaev's contemporaneous work. The other roots of the same result. Tier 1.
Preskill, "Reliable Quantum Computers" (1998), arXiv:quant-ph/9705031. The best-written introduction to what the threshold means, as opposed to how it is proved. Read this before the proofs. Tier 1.
Any treatment of circuit-level noise versus code-capacity noise. This is exactly §25.9's distinction — the difference between assuming perfect syndrome extraction and not — and it is the single most important thing to understand about quoted threshold numbers. Tier 2.

On the surface code

Fowler, Mariantoni, Martinis, and Cleland, "Surface codes: Towards practical large-scale quantum computation" (2012), Physical Review A 86, 032324. The standard reference, and unusually practical: stabilizers, decoding, logical operations, and resource estimates. If you read one thing about the surface code, read this. Tier 1.
Kitaev's original toric code paper (1997/2003). Where the idea comes from, framed topologically. Harder, and worth it for the intuition about why locality buys you a threshold. Tier 1.
Google Quantum AI, "Quantum error correction below the surface code threshold" (2024), Nature. The distance-7 below-threshold demonstration, with $\Lambda \approx 2$. Read the actual $\Lambda$ and compare it against §25.8's table — that comparison is the whole story about how far there is still to go. Tier 1, but check for newer results; this number moves.
Literature on minimum-weight perfect matching and neural-network decoders. The decoder is a real engineering problem: it must run in real time, inside the coherence window, forever. §25.7's feedforward requirement made concrete. Tier 2 — active.

On magic states and the T-gate problem

Bravyi and Kitaev, "Universal quantum computation with ideal Clifford gates and noisy ancillas" (2005), Physical Review A 71, 022316. Magic state distillation. This is why Chapter 15's resource estimates were $T$-count-dominated and why Chapter 22's rotationCount and rotationDepth mattered. Tier 1.
Current work on magic state cultivation and lower-overhead distillation. The overhead has come down substantially and is still moving. Tier 2 — active.
Chapter 15's resource estimator, re-run on Chapter 23's Shor circuit. The fault-tolerant cost of everything Part IV built. Tier 1.

On qLDPC codes — the live alternative

Recent work on quantum low-density parity-check codes, including the IBM bivariate-bicycle codes and related constructions. The claim is a far better encoding rate than the surface code — many logical qubits per physical block instead of one. The catch is connectivity: they need longer-range couplings than a 2D grid provides, which is exactly the trade §25.10 describes. Tier 2 — the fastest-moving area in the field, and the reason to check dates on anything you read about QEC overhead.

Simulation and tooling

Stim (Gidney), a fast stabilizer circuit simulator. Purpose-built for QEC: millions of shots of circuit-level noise on large codes, far beyond what Aer's stabilizer method is meant for. If you do more than this chapter's exercises, you want Stim. Tier 1.
PyMatching, the standard minimum-weight perfect matching decoder, designed to pair with Stim. Tier 1.
Qiskit's noise-model documentation, specifically on which instructions errors can attach to. Case Study 1 in one page. Tier 1.

Backward references

Chapter 11 — Gottesman–Knill and the stabilizer method, without which none of this chapter's simulations run.
Chapter 12 — the measured median two-qubit gate error of 0.0078 that decides §25.9.
Chapter 15 — the resource estimator, and where the millions of physical qubits come from.

Forward references

Chapter 26 — Part V opens on debugging, and §25.3's blind spot is the best worked example in the book of a bug that hides inside a correct-looking number.
Chapter 27 — testing, where this chapter's "a test that cannot fail is not evidence" becomes a discipline.

Where to go next. If one thing: Preskill's "Reliable Quantum Computers." It explains what a threshold is better than anything written since, and §25.8's table will make far more sense afterwards.

If two: add Fowler et al. on surface codes for the practical version, then read the Google 2024 result with §25.8's $\Lambda$ column open beside it. $\Lambda \approx 2$ against the table's 28 at $p = 0.01$ tells you exactly what "below threshold" has and has not bought.

Then Chapter 26 — Part V, and the discipline that catches mistakes like this chapter's before they reach a slide deck.