The clearest short introduction to the stabiliser formalism, written by the person who invented it. Read this before attempting the surface code literature.
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Fault tolerance & the threshold theorem
The result that if physical error rates are below a threshold, arbitrarily long quantum computations become possible with only polylogarithmic overhead — plus the machinery (transversal gates, magic-state distillation) that gets you there.
Learn first:Surface codes
Why it matters
The threshold theorem is why large-scale quantum computing is believed possible at all. Magic-state distillation is why T gates dominate fault-tolerant resource estimates.
After this you will be able to
- State and interpret the threshold theorem
- Explain transversal gates and the Eastin–Knill theorem
- Estimate resources for a fault-tolerant algorithm
Start here
3 best places to start
Hand-picked and ordered. If you only have time for one, take the first.
Surface codes: Towards practical large-scale quantum computation
Fowler, Mariantoni, Martinis & Cleland (2012)
The standard reference on surface codes, written as a tutorial rather than a terse paper. Where every industrial fault-tolerance roadmap ultimately points.
How to factor 2048 bit RSA integers in 8 hours using 20 million noisy qubits
Craig Gidney & Martin Ekerå (2019)
The resource-estimate paper everyone cites when asked "when will RSA break?". A masterclass in turning an asymptotic algorithm into concrete hardware requirements.
Also covering this
2 more resources
Ph219 Quantum Computation lecture notes
John Preskill (Caltech)
Preskill's notes have taught much of the field. Chapter 10 on quantum error correction is, for many researchers, the definitive introduction to the subject.
Quantum error correction below the surface code threshold
Google Quantum AI (2024)
The first convincing demonstration that adding more physical qubits made the logical qubit better rather than worse. Arguably the most important experimental result of the decade so far.