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Quantum Engineering Year 2: Advanced Quantum Science Month 25Day 695

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Year 2·Month 25·Week 4

Day 695: Quantum Hamming Bound & Perfect Codes

Day 695 of 2,016~16 min read

Learning Objectives

•**Derive** the quantum Hamming bound using sphere-packing arguments
•**Calculate** the bound for specific code parameters
•**Define** perfect quantum codes and identify examples
•**Explain** why the [[5,1,3]] code is both MDS and perfect
•**Compare** Singleton and Hamming bounds
•**Understand** the rarity of perfect quantum codes

Today's Schedule (7 hours)

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On this page

1 The Sphere-Packing Argument Classical Intuition Quantum Extension 2 The Quantum Hamming Bound Statement Derivation 3 Interpretation and Examples Single-Error Correction t 1 Double-Error Correction t 2 d 5 4 Perfect Quantum Codes Definition Characterization Known Perfect Quantum Codes Why Perfect Codes Are Rare 5 The 513 Code Doubly Optimal Stabilizer Structure Syndrome Table 6 Comparing Singleton and Hamming Bounds Which Is Tighter Example n 15 d 3 Example n 5 d 3 7 Beyond Non-Degenerate Codes Limitation of Hamming Bound Example Degenerate Violation Quantum Mechanics Connection Measurement and Distinguishability Holevo Bound Connection

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