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Quantum Engineering Year 2: Advanced Quantum Science Month 31Day 858

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Year 2·Month 31·Week 3

Day 858: Eastin-Knill Proof

Day 858 of 2,016~18 min read

Learning Objectives

•**Follow** the complete proof of the Eastin-Knill theorem
•**Explain** the cleaning lemma and its role in the proof
•**Derive** the discreteness condition from error detection
•**Apply** the proof technique to specific codes
•**Understand** the Bravyi-Koenig extension for topological codes
•**Connect** the proof to the Clifford hierarchy

Today's Schedule (7 hours)

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On this page

Overview Schedule Learning Objectives Proof Setup The Framework Key Definitions The Cleaning Lemma Statement Proof of Cleaning Lemma The Main Proof Theorem Statement Proof The Discreteness Gap Explicit Bound Proof Sketch Example Steane Code Gap Extension The Bravyi-Koenig Theorem Topological Codes The Clifford Hierarchy Implications Alternative Proof Approach Cartan Decomposition The Lie Algebra Approach Transversal Generators Error Detection Kills Off-Diagonal Terms Conclusion Proof for Specific Codes Case Study The Steane Code Case Study Surface Code Worked Examples Example 1 Verify the Infinitesimal Constraint Example 2 Compute the Discreteness Gap Example 3 The 15-Qubit Reed-Muller Code Anomaly Practice Problems Level 1 Direct Application Level 2 Intermediate Level 3 Challenging Computational Lab Summary Key Formulas Proof Summary Main Takeaways Daily Checklist Preview Day 859

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